Updated on 12 October 2008
Revision Points
Charge:
Due to gravitational force an electron is expected to attract another electron with a force of 5.5*10^-67 N.
But, an electron is found to repel another electron at 1 cm with a force of 2.3*10^-24 N. This extra force is called the electric force.
The electric force is very large compared to the gravitational force. The electrons must have some additional property apart from their mass, which is responsible for this electric force. This property is termed as charge.
Electric Force
Due to gravitational force an electron is expected to attract another electron with a force of 5.5*10^-67 N.
But, an electron is found to repel another electron at 1 cm with a force of 2.3*10^-24 N. This extra force is called the electric force.
Two kinds of charges
As a convention, the charge on proton is termed positive and the charge on electron is termed negative. The charges on proton and electron have the same strength but are of different nature. Two protons repel each other, two electrons repel each other, but a proton and an electron attract each other.
Units of charge: The SI unit of charge is coulomb abbreviated as C. 1 coulomb is defined as the charge flowing through a wire in 1 second if the electric current in it is 1 A.
The charge on a proton is
e = 1.60218*10^-19 C
The charge on an electron is the negative of this value.
Charge is quantized:
If protons and electrons are the only charge carriers in the universe, the charge on any object has to be in multiples of e only. So charge in quantized.
Charge is conserved:
It is not possible ot create or destroy net charge, even though we can destroy or create charged particle. For example in a beta decay process, a neutron converts itself into a proton and a fresh electron is created. Thus the net charge however remains zero before and after the event.
Frictional electricity:
When glass rod is rubbed with a silk cloth, electrons are transferred from the glass rod to the silk cloth. Due to excess electrons, silk cloth becomes negatively charged and due to excess protons, glass rod becomes positively charged. Thus due to friction, both bodies get charged.
Induction:
A redistribution of charge in a material due to the presence of a nearby charged body, is called induction.
Electric force and the Coulomb's law:
Coulomb established the mathematical formula for the electric force between two charges.
F = k*q1*q2/r²
q1,q2 charges
r = separation between charges
k = constant
In SI units k is measured to be 8.98755*10^9 N-m²/C²
The constant k is often written as 1/4πε0.
The constant ε0 is called the permittivity of the space and its value is
ε0 = 8.85419*10^-12 C²/N-m²
Electric Field:
A charge produces an electric field in the space around it and this electric field exerts a force on any charge placed on it.
The intensity of field is defined as Vector E = Vector F/q
We can interpret that intensity of field is force exerted by a charge (q1) on another charge (q2) per unit charge of the charge on which field is exerting the force (q2). Hence force exerted on a charge in an electric field is equal to intensity of electric field multiplied by the charge on which the force is exerted.
Electric field is a vector quantity. It has magnitude and direction.
Lines of electrical force:
The electric field in a region can be graphically represented by drawing lines. These lines are called lines of electric force or electric field lines. Lines of force are drawn in such a way that the tangent to a line of force gives the direction of the resultant electric field there.
For a point charge, electric field lines are straight lines originating from the charge iin all directions.
Electric potential energy:
Electric potential energy comes into picture as the configuration of the system of charges changes. As, in the system, charges exert electric forces on each other, if the position of one or more charges is changed, work needs to be done by the system or by the environment.
If work is done by the system, the change in potential energy is -W. This is so because if system does the work its potential energy decreases.
If we take the potential energy of the system to be zero when one of the charges is at a infinite distance or separation, the potential energy when the charge is brought up to a separation of r will be
U(r) = U(r) - U(∞) = q1q2/4πε0r.
The potential energy will have units or work or force*distance. That is why in the denominator only r is there. In force formula r² term is there.
If two positive charges are close together there is repulsion among them. Hence there is a potential energy in them. When they are very far apart the potential energy between the two positive charges is zero. Hence the assumption that potential energy of a system is zero when on the charges is at an infinite distance or separation is an appropriate assumption.
Electric potential:
Electric field (which creates electric force) can also be described by assigning a scalar quantity V at each point. This scalar quantity is termed as electric potential.
If a test charge is moved in an electric field from a point A to a point B while all the other charges in question remain fixed, if the electric potential energy of the system changes by Ub-Ua, we define the potential difference between the point A and point B as
Vb - Va = (Ub - Ua)/q
Potential difference is equal to change in potential energy for unit charge between two points A and B.
You can calculate potential energy difference between points A and B by multiplying potential difference by the charged moved from A to B.
Relation between electric field and potential
Scalar product of Field vector and displacement vectors gives potential.
Electric field (E) is force being applied by the charge divided by the test charge.
Potential is work done per unit charge.
So we can interpret it as Vector Force * vector distance/q = qE.r/q = E.r which is scalar product of field vector and displacement vectors.
Hence we can calculate potential V if we know E and r.
If we know V we can find E through the relation
Ex = - ∂V/∂x
Ey = -∂V/∂y
Ez = -∂V/∂z
We can find the x,y and z components of E.
E can be written as Ex i +Ey j + Ez k
dV can also be written as –Edr cos θ where θ is the angle between the field E and the small displacement dr.
If θ is equal to zero, dV = -Edr or –dV/dr is maximum. Thus the electric field is along the direction in which the potential decreases at the maximum rate.
The potential does not vary in a direction perpendicular to the electric field as cos θ = 0.
If we draw equipotential surface around a charge, component of the electric field parallel to an equipotential surface is zero as potential does not change along the surface. Electric field is perpendicular to the surface at the any point on the surface.
For a point charge, the electric field is radial and the equipotential surfaces are concentric spheres with centres at the charge.
Electric Dipole:
A combination of two charges +q and -q separated by a small distance of d constitutes an electric dipole.
Electric dipole moment:
It is defined as a vector p = q*distance vector d
where distance vector is the vector joining the negative charge to the positive charge.
The line along the direction of the dipole moment is called the axis of the dipole.
Electric potential due to a dipole at a point P
Point is at distance r from the centre of the diploe (d/2) and theline joining the point P to the centre of the dipole make an angle θ with the direction of dipole movement (from –q to +q)
Potential at P due to charge –q = - [1/4πε0][q/(r + (dcos θ)/2)]
Potential at P due to charge q = [1/4πε0][q/(r - (dcos θ)/2)]
Net potential due to q and –q = [1/4πε0](qd cos θ)/r²
qd can be replaced by p.
The general definition of electric dipole
V = pcosθ/4πε0r²
where p is magnitude of electric dipole moment defined above
Any charge distribution that produces electric potential given by above formula is called an electric dipole. The two charge system can be expressed as p = qd.
The potential at a distance r from a point charge q is given by V = [1/4πε0](q/r)
The potential at a distance r from an electric dipole with electric dipole moment of p is given by V = [1/4πε0](p cos θ)/r²)
Electric field due to a dipole
Er = [1/4πε0](2p cos θ)/r³
Eθ = [1/4πε0](p sin θ)/ r³
Resultant electric field at P = E= √ (Er²+ Eθ²)
= [1/4πε0](p/r³)√(3 cos²θ + 1)
The angle the resultant field makes with radial direction OP (O is the centre point of the dipole axis and P is that at which electric field is being calculated) is α.
tan α = Eθ/ Er = ½ tan θ or
α = tan-1 (½ tan θ)
Special cases
a. θ = 0. In this case P is on the axis of the dipole. This position is called an end-on position.
V = [1/4πε0](p cos θ)/r²) as θ = 0
V = [1/4πε0](p/r²)
General formula for E = [1/4πε0](p/r³)√(3 cos²θ + 1) as θ = 0
E = [1/4πε0](2p/r³)
b. θ = 90°. In this case P is on the perpendicular bisector of the dipole axis.
General formula for V = [1/4πε0](p cos θ)/r²) as θ = 90°.
V = 0
General formula for E = [1/4πε0](p/r³)√(3 cos²θ + 1) as θ = 90°,
E = [1/4πε0](p/r³)
Angle α is given by tan α = tan 90°/2 = ∞
Therefore α = 90°.
Torque on an electric dipole placed in an electric field.
If the dipole axis makes an angle θ with the electric field magnitude of the torque = | Γ| = pE sin θ
In vector notation Γ = p × E
Potential energy of a dipole placed in a uniform electric field
dipole axis makes an angle θ with the electric field magnitude of the torque
Change in potential energy = U(θ) – U(90°) = -pE cos θ = -p.E
If we choose the potential energy of the dipole to be zero when θ = 90° , above equation becomes
U(θ) = -pE cos θ = -p.E
Electric field inside a conductor
There can be no electric field inside a conductor in electrostatics. When electric field is applied from left to right some free electrons move toward the left creating a negative charge on the left surface. Due to which there will be positive charge on the right surface. Due to this charge buildup, coulomb attraction sets between these two charges and an electric field opposite in direction to the applied electric field is set up. The movement of free electrons continues till the applied electric field and the electric field due to redistribution of electrons are equal. Hence inside the conductor these two electric fields balance each other and there is no electric field inside a conductor in electrostatics.
Conductors, insulators, semiconductors
Conductors have free electrons that move throughout the body. When such a material is placed in an electric field, the free electrons move in a direction opposite to the field. The free electrons are called conduction electrons in this context.
In insulators, electrons are tightly bound to their respective atoms or molecules. So in an electric field, they can't leave their parent atoms. They are insulators or dielectrics.
In semiconductors, at 0 K there are no free electrons but as temperature raises, small number of free electrons appear (they are able to free themselves from atoms and molecules) and they respond to the applied electric field.
Sunday, March 23, 2008
Concept Review Ch.30 Gauss's Law
JEE Topics
Flux of electric field
Gauss’s law and
Gauss's law's application in simple cases, such as, to find field due to infinitely long straight wire, uniformly charged infinite plane sheet and uniformly charged thin spherical shell
=============
Flux of an electric field
If in a plane surface area of ∆s, a uniform electric field E exists, and makes an angle θ with the normal to the surface area (positive normal - you can arbitrarily decide which direction is positive), the quantity
∆Φ = E ∆s cos θ
is called the flux of the electric field through the chosen surface.
∆s is represented as a vector. Also it is area.
∆Φ = E.∆s
Where E and ∆s are vectors and ∆Φ is a scalar quantity.
As flux is a scalar quantity, it can be added arithmetically. Hence surfaces which are not on single plane, can be divided into small parts which are plane, the flux through each part can be found out and the sum is flux through the complete surface.
Non-uniform electric field can also be tackled that way. Divide the surface into parts over which the electric field is uniform and then find the flux in each part and sum them.
Using the techniques of integration flux over a surface is:
Φ = ∫ E.∆s
When integration over a closed surface is done a small circle is placed on the integral sign.( ∮).
Flux over a closed surface
Φ = ∮ E.∆s (* Intregration over a closed surface)
Solid angle:
Typical example is the angle in the paper containers used by moongfaliwalas.
Ω = S/r²
S = the area of the part of sphere intercepted by the cone
r = radius of the sphere assumed on which we are assuming the cone
A complete circle subtends an angle 2 π
Any closed surface subtends a solid angle 4 π at the centre.
How much is the angle subtended by a closed plane curve at an external point? Zero.
Gauss's law
The flux of the net electric field through a closed surface equals the net charge enclosed by the surface divided by ε0.
In symbols
∮ E.∆s (* Intregration over a closed surface) =
qin/ε0.
Where
qin = charge enclosed by the closed surface
ε0 = emittivity of the free space
It needs to be stressed that flux is the resultant of all charges existing in the space. But, its quantity is given the right hand side.
Applications of Gauss’s law
a. Charged conductor
The free electrons redistribute themselves to make the field zero at all the points inside the conductor. Any charge injected anywhere in the conductor must come over to the surface of the conductor so that the interior is always charge free.
If there is cavity inside the conductor, for example a hollow cylinder and a charge +q is placed in this cavity, as the inside of the conductor has to be charge free, negative charge appears on the inside of the cavity. If the conductor is neutral, a charge +q will appear on the surface.
b. Electric field due to a uniformly charged sphere
A total charge Q is uniformly distributed in a spherical volume of radius R. what is the electric field at a distance r from the centre of the charge distribution outside the sphere?
Through Gauss’s law we get E 4 πr² = Q/ ε0.
So E = Q/4π ε0 r²
The electric field due to a uniformly charged sphere at a point outside it, is identical with the field due to an equal point charge placed at the centre.
Field at an internal point
At centre E = O
At any other point r less than R radius of the sphere
E = Qr/4π ε0 R³
c. Electric field due to a linear charge distribution
The linear charge density (charge per unit length) is λ.
Electric field at a distance r from the linear charge distribution
E = λ/2π ε0r
d. Electric field due to a plane sheet of charge
Plane sheet with charge density (charge per unit area) σ.
Field at distance d from the sheet = E = σ/2ε0
We see that the field is uniform and does not depend on the distance from the charge sheet. This is true as long as the sheet is large as compared to its distance from P.
e. Electric field due to a charged conducting surface
To find the field at a point near this surface but outside the surface having charge density σ.
E = σ/ ε0
Notice in case of plane sheet of charge it σ/2ε0. But in the case of conductor is σ/ ε0
Spherical Charge Distributions
Useful results for a spherical charge distribution of radius R
a) The electric field due to a uniformly charged, thin spherical shell at an external point is the same as that due to an equal point charge placed at the centre of the shell.
b) The electric field due to a uniformly charged, thin spherical shell at an internal point is zero.
c) The electric field due to a uniformly charged sphere at an external point is the same as that due to an equal point charge placed at the centre of the sphere.
d) The electric field due to a uniformly charged sphere at an at an internal point is proportional to the distance of the point from the centre of the sphere. Thus it is zero at the centre and increases linearly as one moves out towards the surface.
e) The electric potential due to a uniformly charged, thin spherical shell at an external point is the same as that due to an equal point charge placed at the centre of the shell. V = Q/4πε0r
f) The electric potential due to a uniformly charged, thin spherical shell at an internal point is the same everywhere and is equal to the that at the surface. V = Q/4πε0R
g)The electric potential due to a uniformly charged sphere at an external point is the same as that due to an equal point charge placed at the centre of the sphere. V = Q/4πε0r
Electric potential energy of a uniformly charged sphere
Charge density ρ = 3Q/4π R³ (charge density is per unit volume)
Electric potential energy of the charged sphere = 3Q²/20πε0R
Electric potential energy of a uniformly charged, thin spherical shell
Electric potential energy of the thin spherical shell = Q²/8πε0R
Earthing a conductor
The earth is good conductor of electricity.
Earth’s surface has a negative charge of 1nC/m². All conductors which are not given any external charge are also very nearly at the same potential.
But the potential of earth is often taken as zero.
When a conductor is connected to earth, the conductor is said to be earthed or grounded and its potential will become zero.
In appliances, the earth wire is connected to the metallic bodies. If by any fault, the live wire touches the metallic body, charge flows to the earth and the potential of the metallic body remains zero. If it not connected to the earth, the user may get an electric shock.
Updated 12 October 2008
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Flux of electric field
Gauss’s law and
Gauss's law's application in simple cases, such as, to find field due to infinitely long straight wire, uniformly charged infinite plane sheet and uniformly charged thin spherical shell
=============
Flux of an electric field
If in a plane surface area of ∆s, a uniform electric field E exists, and makes an angle θ with the normal to the surface area (positive normal - you can arbitrarily decide which direction is positive), the quantity
∆Φ = E ∆s cos θ
is called the flux of the electric field through the chosen surface.
∆s is represented as a vector. Also it is area.
∆Φ = E.∆s
Where E and ∆s are vectors and ∆Φ is a scalar quantity.
As flux is a scalar quantity, it can be added arithmetically. Hence surfaces which are not on single plane, can be divided into small parts which are plane, the flux through each part can be found out and the sum is flux through the complete surface.
Non-uniform electric field can also be tackled that way. Divide the surface into parts over which the electric field is uniform and then find the flux in each part and sum them.
Using the techniques of integration flux over a surface is:
Φ = ∫ E.∆s
When integration over a closed surface is done a small circle is placed on the integral sign.( ∮).
Flux over a closed surface
Φ = ∮ E.∆s (* Intregration over a closed surface)
Solid angle:
Typical example is the angle in the paper containers used by moongfaliwalas.
Ω = S/r²
S = the area of the part of sphere intercepted by the cone
r = radius of the sphere assumed on which we are assuming the cone
A complete circle subtends an angle 2 π
Any closed surface subtends a solid angle 4 π at the centre.
How much is the angle subtended by a closed plane curve at an external point? Zero.
Gauss's law
The flux of the net electric field through a closed surface equals the net charge enclosed by the surface divided by ε0.
In symbols
∮ E.∆s (* Intregration over a closed surface) =
qin/ε0.
Where
qin = charge enclosed by the closed surface
ε0 = emittivity of the free space
It needs to be stressed that flux is the resultant of all charges existing in the space. But, its quantity is given the right hand side.
Applications of Gauss’s law
a. Charged conductor
The free electrons redistribute themselves to make the field zero at all the points inside the conductor. Any charge injected anywhere in the conductor must come over to the surface of the conductor so that the interior is always charge free.
If there is cavity inside the conductor, for example a hollow cylinder and a charge +q is placed in this cavity, as the inside of the conductor has to be charge free, negative charge appears on the inside of the cavity. If the conductor is neutral, a charge +q will appear on the surface.
b. Electric field due to a uniformly charged sphere
A total charge Q is uniformly distributed in a spherical volume of radius R. what is the electric field at a distance r from the centre of the charge distribution outside the sphere?
Through Gauss’s law we get E 4 πr² = Q/ ε0.
So E = Q/4π ε0 r²
The electric field due to a uniformly charged sphere at a point outside it, is identical with the field due to an equal point charge placed at the centre.
Field at an internal point
At centre E = O
At any other point r less than R radius of the sphere
E = Qr/4π ε0 R³
c. Electric field due to a linear charge distribution
The linear charge density (charge per unit length) is λ.
Electric field at a distance r from the linear charge distribution
E = λ/2π ε0r
d. Electric field due to a plane sheet of charge
Plane sheet with charge density (charge per unit area) σ.
Field at distance d from the sheet = E = σ/2ε0
We see that the field is uniform and does not depend on the distance from the charge sheet. This is true as long as the sheet is large as compared to its distance from P.
e. Electric field due to a charged conducting surface
To find the field at a point near this surface but outside the surface having charge density σ.
E = σ/ ε0
Notice in case of plane sheet of charge it σ/2ε0. But in the case of conductor is σ/ ε0
Spherical Charge Distributions
Useful results for a spherical charge distribution of radius R
a) The electric field due to a uniformly charged, thin spherical shell at an external point is the same as that due to an equal point charge placed at the centre of the shell.
b) The electric field due to a uniformly charged, thin spherical shell at an internal point is zero.
c) The electric field due to a uniformly charged sphere at an external point is the same as that due to an equal point charge placed at the centre of the sphere.
d) The electric field due to a uniformly charged sphere at an at an internal point is proportional to the distance of the point from the centre of the sphere. Thus it is zero at the centre and increases linearly as one moves out towards the surface.
e) The electric potential due to a uniformly charged, thin spherical shell at an external point is the same as that due to an equal point charge placed at the centre of the shell. V = Q/4πε0r
f) The electric potential due to a uniformly charged, thin spherical shell at an internal point is the same everywhere and is equal to the that at the surface. V = Q/4πε0R
g)The electric potential due to a uniformly charged sphere at an external point is the same as that due to an equal point charge placed at the centre of the sphere. V = Q/4πε0r
Electric potential energy of a uniformly charged sphere
Charge density ρ = 3Q/4π R³ (charge density is per unit volume)
Electric potential energy of the charged sphere = 3Q²/20πε0R
Electric potential energy of a uniformly charged, thin spherical shell
Electric potential energy of the thin spherical shell = Q²/8πε0R
Earthing a conductor
The earth is good conductor of electricity.
Earth’s surface has a negative charge of 1nC/m². All conductors which are not given any external charge are also very nearly at the same potential.
But the potential of earth is often taken as zero.
When a conductor is connected to earth, the conductor is said to be earthed or grounded and its potential will become zero.
In appliances, the earth wire is connected to the metallic bodies. If by any fault, the live wire touches the metallic body, charge flows to the earth and the potential of the metallic body remains zero. If it not connected to the earth, the user may get an electric shock.
Updated 12 October 2008
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Concept Review Ch.31 Capacitors
Revision points
Capacitor
A combination of two conductors placed clsoe toeach other is called a capacitor. Oneof the conductors is given a positive charge and the other a negative charge.
Capacitance
For a given capacitor, the charge Q on the capacitor is proportional to the potential difference V between the two plates
So Q α V
or Q = CV
C is called the capacitance of the capacitor.
SI unit of capacitance is coulomb/volt which is written as farad. The symbol F is used for it.
To put equal and opposite charges on the two conductors they may be connected to the terminals of a battery.
Calculation of Capacitance
For parallel plate capacitor
C = ε0A/d
A = area of the flat plates (each used in the capacitor)
d = distance between the plate
Spherical capacitor
It consists of a solid or hollow spherical conductor surrounded by another concentric hollow spherical conductor.
If inner sphere radius is R1 and Outer sphere radius is R2
Inner sphere is given positive charge and outer sphere negative charge.
C = 4πε0R1R2/[R2-R1]
If the capacitor is an isolated sphere (outer sphere is assumed to be at infinity, hence R2 is infinity and
C = 4πε0R1
V becomes Q/C = Q/4πε0R1
V = potential
Parallel limit: if both R1 and R2 are made large but R2-R1 = d is kept fixed
we can write
4πR1R2 = 4πR² = A; where R is approximately the radius of each sphere, and A is the surface area of the sphere.
C = ε0A/d; where A = 4πR1R2 = 4πR²
Cylindrical Capacitor
It consists of a solid or hollow cylindrical conductor surrounded by another concentric hollow cylindrical conductor.
If inner cylinder radius is R1 and Outer cylinder radius is R2 and length is l,
Inner cylinder is given positive charge and outer cylinder negative charge
C = 2πε0l/ln(R2/R1)
Combination of capacitors
Series combination
1/C = 1/C1 + 1/C2 + 1/C3 ...
Parallel combination
C = C1 + C2 + C3
Force between plates of a capacitor
Plates on a parallel capacitor attract each other with a force
F = Q²/2Aε0
Energy stored in a capacitor
Capacitor of capacitance C has a stored energy
U = Q²/2C = CV²/2 = QV/2
Where Q is the charge given to it.
Dielectric material
In dielectric materials, there are no free electrons. Electrons are bound to the nucleus in atoms. Basically they are insulators. But when a charge is applied, in these materials also atoms or molecules are oriented in a such way that there is an induced. For example, in the case of rectangular slab of a dielectric, if an electric field is applied from left to right, the left surface of the slab gets a negative charge, and the right surface gets positive charge.
Change in capacitance of a capacitor with dielectric in it.
The surface charge density of the induced charge can be related to a measure called Polarization P (which is dipole moment induced per unit volume - where is the dipole? in the diectric slab as the two sides have opposite charges)
If σp is the magnitude of the induced charge per unit area on the faces.
The dipole moment (q*Vr(d)) of the slab is then charge*l (distance between faces)
= σpAl.
where
A is area of cross section of the dielectric slab
As polarization is defined as dipole moment induced per unit volume,
P = σpAl/Al (Al = volume of slab)
= σp
The induced surface charge density is equal in magnitude to the polarization P.
Dielectric constant
Because of induced charge, electric field is produced in the slab which is against the field applied on the slab.
Resultant field = Applied field - induced field
Resultant field = Applied field/K
K is greater than 1 and is a constant for give materials. K is called the dielectric constant or relative permittivity of the dielectric.
Dielectric strength
If a very high electric field is created in a dielectric, electrons in valence shell may get detached from their parent atoms and move freely like in a conductor. This phenomenon is called is dielectric breakdown. The electric field at which breakdown occurs is called the dielectric strength of the material.
Capacitance of a parallel plate capacitor with dielectric
C = KC0
where C0 is capacitance of a similar capacitor without dielectric.
Because K>1, the capacitance of a capacitor is increased by a factor of K when the space between the parallel plates is filled with a dielectric.
Magnitude of induced charge in term of K
QP = Q[1 - (1/K)]
QP = induced charge in the dielectric
Q = Applied charge
K = dielectric constant
Gauss's law when dielectric materials are involved
∮KE.dS = Qfree/ε0 .....(31.14)
Where integration is over the surface, E and dS are vectors, Qfree is the free charge given (charge due to polarisation is not considered) and K is dielectric constant.
The law can also be written as
∮D.ds = Q(free) ...... (31.15)
where D = Eε0 + P; E and P are vectors
E = electric field and P is polarisation
Electric field due to a point charge placed inside a dielectric
E = q/4πε0Kr²
Energy in the electric field in a dielectric
u = ½Kε0E²
Corona discharge
If a conductor has a pointed shape like a needle and a charge given to it, the charge density at the pointed end will be very high. Correspondingly, the electric field near these pointed ends will be very high which may cause dielectric breakdown in air. The charge may jump from the conductor to the air. Often this discharge of charge inot air is accompanied by a visible glow surrounding the pointed end and this phenomenon is called corona discharge.
High voltage generator – Van de Graaff Generator
The apparatus transfers positive charge to a sphere continuously till the potential reaches to around 3*10^ 6 V at which point corona discharge takes place and hence no further charge can be transferred. The charge of course can be increased by enclosing the sphere in a highly evacuated chamber.
Capacitor
A combination of two conductors placed clsoe toeach other is called a capacitor. Oneof the conductors is given a positive charge and the other a negative charge.
Capacitance
For a given capacitor, the charge Q on the capacitor is proportional to the potential difference V between the two plates
So Q α V
or Q = CV
C is called the capacitance of the capacitor.
SI unit of capacitance is coulomb/volt which is written as farad. The symbol F is used for it.
To put equal and opposite charges on the two conductors they may be connected to the terminals of a battery.
Calculation of Capacitance
For parallel plate capacitor
C = ε0A/d
A = area of the flat plates (each used in the capacitor)
d = distance between the plate
Spherical capacitor
It consists of a solid or hollow spherical conductor surrounded by another concentric hollow spherical conductor.
If inner sphere radius is R1 and Outer sphere radius is R2
Inner sphere is given positive charge and outer sphere negative charge.
C = 4πε0R1R2/[R2-R1]
If the capacitor is an isolated sphere (outer sphere is assumed to be at infinity, hence R2 is infinity and
C = 4πε0R1
V becomes Q/C = Q/4πε0R1
V = potential
Parallel limit: if both R1 and R2 are made large but R2-R1 = d is kept fixed
we can write
4πR1R2 = 4πR² = A; where R is approximately the radius of each sphere, and A is the surface area of the sphere.
C = ε0A/d; where A = 4πR1R2 = 4πR²
Cylindrical Capacitor
It consists of a solid or hollow cylindrical conductor surrounded by another concentric hollow cylindrical conductor.
If inner cylinder radius is R1 and Outer cylinder radius is R2 and length is l,
Inner cylinder is given positive charge and outer cylinder negative charge
C = 2πε0l/ln(R2/R1)
Combination of capacitors
Series combination
1/C = 1/C1 + 1/C2 + 1/C3 ...
Parallel combination
C = C1 + C2 + C3
Force between plates of a capacitor
Plates on a parallel capacitor attract each other with a force
F = Q²/2Aε0
Energy stored in a capacitor
Capacitor of capacitance C has a stored energy
U = Q²/2C = CV²/2 = QV/2
Where Q is the charge given to it.
Dielectric material
In dielectric materials, there are no free electrons. Electrons are bound to the nucleus in atoms. Basically they are insulators. But when a charge is applied, in these materials also atoms or molecules are oriented in a such way that there is an induced. For example, in the case of rectangular slab of a dielectric, if an electric field is applied from left to right, the left surface of the slab gets a negative charge, and the right surface gets positive charge.
Change in capacitance of a capacitor with dielectric in it.
The surface charge density of the induced charge can be related to a measure called Polarization P (which is dipole moment induced per unit volume - where is the dipole? in the diectric slab as the two sides have opposite charges)
If σp is the magnitude of the induced charge per unit area on the faces.
The dipole moment (q*Vr(d)) of the slab is then charge*l (distance between faces)
= σpAl.
where
A is area of cross section of the dielectric slab
As polarization is defined as dipole moment induced per unit volume,
P = σpAl/Al (Al = volume of slab)
= σp
The induced surface charge density is equal in magnitude to the polarization P.
Dielectric constant
Because of induced charge, electric field is produced in the slab which is against the field applied on the slab.
Resultant field = Applied field - induced field
Resultant field = Applied field/K
K is greater than 1 and is a constant for give materials. K is called the dielectric constant or relative permittivity of the dielectric.
Dielectric strength
If a very high electric field is created in a dielectric, electrons in valence shell may get detached from their parent atoms and move freely like in a conductor. This phenomenon is called is dielectric breakdown. The electric field at which breakdown occurs is called the dielectric strength of the material.
Capacitance of a parallel plate capacitor with dielectric
C = KC0
where C0 is capacitance of a similar capacitor without dielectric.
Because K>1, the capacitance of a capacitor is increased by a factor of K when the space between the parallel plates is filled with a dielectric.
Magnitude of induced charge in term of K
QP = Q[1 - (1/K)]
QP = induced charge in the dielectric
Q = Applied charge
K = dielectric constant
Gauss's law when dielectric materials are involved
∮KE.dS = Qfree/ε0 .....(31.14)
Where integration is over the surface, E and dS are vectors, Qfree is the free charge given (charge due to polarisation is not considered) and K is dielectric constant.
The law can also be written as
∮D.ds = Q(free) ...... (31.15)
where D = Eε0 + P; E and P are vectors
E = electric field and P is polarisation
Electric field due to a point charge placed inside a dielectric
E = q/4πε0Kr²
Energy in the electric field in a dielectric
u = ½Kε0E²
Corona discharge
If a conductor has a pointed shape like a needle and a charge given to it, the charge density at the pointed end will be very high. Correspondingly, the electric field near these pointed ends will be very high which may cause dielectric breakdown in air. The charge may jump from the conductor to the air. Often this discharge of charge inot air is accompanied by a visible glow surrounding the pointed end and this phenomenon is called corona discharge.
High voltage generator – Van de Graaff Generator
The apparatus transfers positive charge to a sphere continuously till the potential reaches to around 3*10^ 6 V at which point corona discharge takes place and hence no further charge can be transferred. The charge of course can be increased by enclosing the sphere in a highly evacuated chamber.
Concept Review Ch.32 Electric Current in Conductors
Electric current
i = ∆Q/∆t
Q is charge, t is time
Current density
j = i/A
i is current and A is area of the conductor
Drift speed
A conductor contains free electrons moving randomly in a lattice of positive ions. Electrons collide with positive ions and their direction changes randomly. In such a random movement, from any area equal numbers of electrons go in opposite directions and due to that no net charge moves and there is no current. But when there is an electric field inside the conductor a force acts on each electron in the direction opposite to the field. The electrons get biased in their random motion in favour of the force. As a result electrons drift slowly in the direction opposite to the field.
If τ be the average time between successive collisions, the distance drifted during this period is
l = ½ a(τ) = ½ (eE/m)( τ) ²
The drift speed is
vd = l/ τ = ½ (eE/m)τ = kE
τ the average time between successive collisions, is constant for a given material at a given temperature.
Relation between current density and drift speed
j = i/A = ne vd
Ohm's law
j = σE
E is field and σ is electrical conductivity of the material.
Resistivity of a material ρ = 1/σ
Another form of Ohm's law
V = voltage difference between the ends of a conductor = El (l = length of the conductor)
V = Ri
R = resistance of the conductor = ρ*l/A
1/R is called conductance
Colour codes for resistors
An object of conducting material having a resistance of known value is called a resistor.
Resistors are given colour coding with colour bands that indicate its resistance and tolerance.
Band 1 and band 2 represent digit 1 and digit 2 of resistance.
Band 3 represent the multiplier like 10^n.
Fourth band represents tolerance. No band: 20%, silver 10% and gold 5%.
Temperature dependence of resistivity
As temperature of a resistor increases its resistance increases. The relation can be expressed as
R(T) = R(T0)[1 + α(T - T0)]
α is called temperature coefficient of resistivity.
Thermistors: Measure small changes in temperatures
Superconductors:
For these materials resistivity suddenly drops to zero below a certain temperature. For Mercury it is 4.2 K. For the super conducting material if an emf is applied the current will exist for long periods of time even for years without any further application of emf.
Scientists have achieved superconductivity at 125 K so far.
Battery
Battery is a device which maintains a potential difference between its two terminals A and B.
In the battery some internal mechanism exerts forces on the charges and drives the positive charges of the battery towards one side and negative charges of towards anther side.
Force on a positive charge q is Fb (a vector quantity). If a charge q is moved from one terminal (say B) to the other terminal say A through an external circuit, the work done by the battery force is Fb*d where d is distance between A and B.
The work done by the battery force per unit charge is
ε = W/q = Fb*d/q
This ε is called the emf of the battery. Please note that emf is work done/charge.
If nothing is externally connected
Fb = qE or
Fb*d = qEd = qV (because V =Ed)
V = potential difference between the terminals
As ε = W/q = Fb*d/q = qV/q = V
Therefore ε = V
Energy transfer in an electric circuit
Thermal energy produced in a resistor
U = i²Rt
Power developed = P = U/t = i²R = Vi
Effect of internal resistance of a battery
Potential difference applied across an external resistor
= emf of the battery - ir (r = internal resistance)
Kirchoff's laws
The junction law
The sum of all currents directed towards a point is a circuit is equal to the sum of all the currents directed away from the point.
The loop law
The algebraic sum of all the potential differences along a closed loop in a circuit is zero.
Combination of resistors in series
Equivalent resistance = R1 + R2 +R3+...
Combination of resistors in parallel
1/Equivalent resistance = 1/R1 + 1/R2 +1/R3+...
Division of current in resistors joined in parallel
i1/i2 = R2/R1
i1 = iR2/(R1 + R2)
Batteries connected in series
i = (ε1 + ε2)/(R + r0)
Where R = external resistance
r0 = r1 + r2 r1, r2 are internal resistances of two batteries
Batteries connected in parallel
Equivalent emf =ε0 = ε1r2 + ε2r1/(r1+r2)
where ε1, ε2 are emfs of of batteries , and r1, r2 are internal resistances.
equivalent internal restance = r0 = r1r2/(r1 + r2)
So i in the circurit = ε0/(R + r0)
Wheatstone bridge
It is an arrangement of four resistances, and one of them can be measured if the other are known resistances.
R1 and R2 are two resistances connected in series. R3 and R4 are the other two resistance conncted series. If the there is no deflection in the galvanometer
R4 = R3R2/R1
as R1/R2 = R3/R4
Ammeter
Used to measure current in a circuit. A small resistance is connected in parallel to the coil measuring current in an ammeter to reduce the overall resistance of ammeter.
Voltmeter
A resistor with a large resistance is connected in series with the coil.
When a volt meter is connected in parallel to the point between which the potential is to be measured, if a large resistance is connected, the equivalent resistance is less than the small resistance.
Charging of the capacitor
q = Є C(1 - e^-t/CR)
q is charge on the capacitor, t is time, Є = emf of the battery, C = capacitance, R is resistance of battery and connecting wires,
CR has units of time and is termed time constant. In one time constant τ (=CR) the charge accumulated becomes 0.63 Є C.
Discharging
q = Qe^-t/CR
where q is charge remaining on the capacitor
Q is the initial charge
In one time constant 0.63% is discharged.
Atmospheric electricity
At about 50 km above the earth’s surface, the air becomes highly conducting and thus there is perfectly conducting surface having potential of 400 kV with respect to earth and current (positive charge) comes down from this surface to earth.
Thunderstorms and lightning bring negative charge to earth.
Water vapour condenses to form small water droplets and tiny ice particles. A parcel of air (cloud) with these droplets and ice particles forms a thunderstorm. A matured thunderstorm is formed with its lower end at a height of 3-4 km above the earth’s surface and the upper end at about 6-7 km above the earth’s surface. Negative charge is at the lower end and positive charge is at the upper end of this thunderstorm. This negative charge creates a potential difference of 20 to 100 MV between these clouds and the earth. This cause dielectric breakdown of air and air becomes conducting.
There are number of thunderstorms every day throughout the earth. They charge the atmospheric batter by supplying negative charge to the earth and positive charge to the upper atmosphere.
It is discharged by this battery.
i = ∆Q/∆t
Q is charge, t is time
Current density
j = i/A
i is current and A is area of the conductor
Drift speed
A conductor contains free electrons moving randomly in a lattice of positive ions. Electrons collide with positive ions and their direction changes randomly. In such a random movement, from any area equal numbers of electrons go in opposite directions and due to that no net charge moves and there is no current. But when there is an electric field inside the conductor a force acts on each electron in the direction opposite to the field. The electrons get biased in their random motion in favour of the force. As a result electrons drift slowly in the direction opposite to the field.
If τ be the average time between successive collisions, the distance drifted during this period is
l = ½ a(τ) = ½ (eE/m)( τ) ²
The drift speed is
vd = l/ τ = ½ (eE/m)τ = kE
τ the average time between successive collisions, is constant for a given material at a given temperature.
Relation between current density and drift speed
j = i/A = ne vd
Ohm's law
j = σE
E is field and σ is electrical conductivity of the material.
Resistivity of a material ρ = 1/σ
Another form of Ohm's law
V = voltage difference between the ends of a conductor = El (l = length of the conductor)
V = Ri
R = resistance of the conductor = ρ*l/A
1/R is called conductance
Colour codes for resistors
An object of conducting material having a resistance of known value is called a resistor.
Resistors are given colour coding with colour bands that indicate its resistance and tolerance.
Band 1 and band 2 represent digit 1 and digit 2 of resistance.
Band 3 represent the multiplier like 10^n.
Fourth band represents tolerance. No band: 20%, silver 10% and gold 5%.
Temperature dependence of resistivity
As temperature of a resistor increases its resistance increases. The relation can be expressed as
R(T) = R(T0)[1 + α(T - T0)]
α is called temperature coefficient of resistivity.
Thermistors: Measure small changes in temperatures
Superconductors:
For these materials resistivity suddenly drops to zero below a certain temperature. For Mercury it is 4.2 K. For the super conducting material if an emf is applied the current will exist for long periods of time even for years without any further application of emf.
Scientists have achieved superconductivity at 125 K so far.
Battery
Battery is a device which maintains a potential difference between its two terminals A and B.
In the battery some internal mechanism exerts forces on the charges and drives the positive charges of the battery towards one side and negative charges of towards anther side.
Force on a positive charge q is Fb (a vector quantity). If a charge q is moved from one terminal (say B) to the other terminal say A through an external circuit, the work done by the battery force is Fb*d where d is distance between A and B.
The work done by the battery force per unit charge is
ε = W/q = Fb*d/q
This ε is called the emf of the battery. Please note that emf is work done/charge.
If nothing is externally connected
Fb = qE or
Fb*d = qEd = qV (because V =Ed)
V = potential difference between the terminals
As ε = W/q = Fb*d/q = qV/q = V
Therefore ε = V
Energy transfer in an electric circuit
Thermal energy produced in a resistor
U = i²Rt
Power developed = P = U/t = i²R = Vi
Effect of internal resistance of a battery
Potential difference applied across an external resistor
= emf of the battery - ir (r = internal resistance)
Kirchoff's laws
The junction law
The sum of all currents directed towards a point is a circuit is equal to the sum of all the currents directed away from the point.
The loop law
The algebraic sum of all the potential differences along a closed loop in a circuit is zero.
Combination of resistors in series
Equivalent resistance = R1 + R2 +R3+...
Combination of resistors in parallel
1/Equivalent resistance = 1/R1 + 1/R2 +1/R3+...
Division of current in resistors joined in parallel
i1/i2 = R2/R1
i1 = iR2/(R1 + R2)
Batteries connected in series
i = (ε1 + ε2)/(R + r0)
Where R = external resistance
r0 = r1 + r2 r1, r2 are internal resistances of two batteries
Batteries connected in parallel
Equivalent emf =ε0 = ε1r2 + ε2r1/(r1+r2)
where ε1, ε2 are emfs of of batteries , and r1, r2 are internal resistances.
equivalent internal restance = r0 = r1r2/(r1 + r2)
So i in the circurit = ε0/(R + r0)
Wheatstone bridge
It is an arrangement of four resistances, and one of them can be measured if the other are known resistances.
R1 and R2 are two resistances connected in series. R3 and R4 are the other two resistance conncted series. If the there is no deflection in the galvanometer
R4 = R3R2/R1
as R1/R2 = R3/R4
Ammeter
Used to measure current in a circuit. A small resistance is connected in parallel to the coil measuring current in an ammeter to reduce the overall resistance of ammeter.
Voltmeter
A resistor with a large resistance is connected in series with the coil.
When a volt meter is connected in parallel to the point between which the potential is to be measured, if a large resistance is connected, the equivalent resistance is less than the small resistance.
Charging of the capacitor
q = Є C(1 - e^-t/CR)
q is charge on the capacitor, t is time, Є = emf of the battery, C = capacitance, R is resistance of battery and connecting wires,
CR has units of time and is termed time constant. In one time constant τ (=CR) the charge accumulated becomes 0.63 Є C.
Discharging
q = Qe^-t/CR
where q is charge remaining on the capacitor
Q is the initial charge
In one time constant 0.63% is discharged.
Atmospheric electricity
At about 50 km above the earth’s surface, the air becomes highly conducting and thus there is perfectly conducting surface having potential of 400 kV with respect to earth and current (positive charge) comes down from this surface to earth.
Thunderstorms and lightning bring negative charge to earth.
Water vapour condenses to form small water droplets and tiny ice particles. A parcel of air (cloud) with these droplets and ice particles forms a thunderstorm. A matured thunderstorm is formed with its lower end at a height of 3-4 km above the earth’s surface and the upper end at about 6-7 km above the earth’s surface. Negative charge is at the lower end and positive charge is at the upper end of this thunderstorm. This negative charge creates a potential difference of 20 to 100 MV between these clouds and the earth. This cause dielectric breakdown of air and air becomes conducting.
There are number of thunderstorms every day throughout the earth. They charge the atmospheric batter by supplying negative charge to the earth and positive charge to the upper atmosphere.
It is discharged by this battery.
Concept Review Ch.33 Thermal and Chemical Effects of Electrical Curremt
Joule's laws of heating
Work done by the electric field on the free electrons in time t is
W = potential difference * charge
= V(it)
= (iR)it = i²Rt
Joules laws
1. The heat produced in a given resistor in a given time is proportional to the square of the current in it, i.e.,
H α i²
2. The heat produced in a given resistor by a given current is proportional to the time for which the current exists in it, i.e.,
H α t
3. The heat produced in a given resistor by a given current in a given time is proportional to its resistance.
H α R
These three are known as Joule’s laws.
Verification of Joule’s laws
Done by Joules calorimeter.
K-oil is taken in a copper calorimeter, current i1 is sent for a time t and the rise in temperature (Δ θ1) is noted. Later on the oil is allowed to cool to the room temperature and current i2 is sent for a time of ‘t’ same as last time. The rise in temperature (θ2) is noted. (Δ θ2)
It can be found that
Δθ1/ Δθ2 = i1²/i2²
This shows that Δθ α i²
Verification of the law H α t is made by taking reading of rise in temperature at regular time intervals. It can be found that temperature rise in uniform in every time period. It increases by equal amount in equal time periods.
Verification of H α R is done by taking two joule calorimeters. Different resistances R1 and R2 are dipped in the K-oil of different calorimeters and the system is connected to a battery. The initial temperatures are noted and after some time the final temperatures arenoted. It can be seen that
Δθ1/ Δθ2 = R1/R2.
Seebeck effect
If the junctions of two metallic strips (joined at the ends to form a loop) are kept at different temeratures , there is an electric current in the loop.
This effect is called Seebeck effect and the emf developed is called the Seebeck emf and thermo emf.
Peltier effect
This effect is reverse of Seebeck effect.
When the two junctions of a thermocouple (two different metallic strips joined at the ends to form a loop) are kept at the same temperature and electric current passed through them, it was observed that heat was produced at one junction and it was absorbed at the other junction. So one junction became hot and the other junction became cooler.
Thermoelectric series
Metals are arranged in a series called thermoelectric series, which may be used to predict the direction of current in a thermocouple in the temperature range 0°C to 100°C.
Antimony, nichrome, iron, zinc, copper, gold, silver, lead, aluminium, mercury, platinum-rhodium, platinum, nickel, constantan, bismuth.
At the cold junction current is from the metal coming earliner in the series to the metal coming latter in the series. Furthre apart two metals lie in the series, larger is the emf produced.
Neutral and inversion temperature
The temperature of the hot junction at which the thermo emf is maximum is called the neutral temperature and the temperature at which the thermo emf changes its sign is called the inversion temperature.
If θc, θn, θi are temperature of the cold junction, neutral temperature and inversion temperature respectively
c
θn – θc = θi - θn
Sign convention in thermocouples
The thermo emf developed in a thermocouple of metals A and B denoted ЄAB is taken to be positive if the direction of the current is from A to B at the hot junction.
Thermo emf depends on temperature with the relation
ЄAB = aABθ + ½ bsub>ABθ²
Where aAB and bAB are constants for a pair of metals A and B.
This gives dЄAB/dθ = aAB + bsub>ABθ
The quantity dЄAB/dθ is called thermoelectric power at temperature θ.
The emf is maximum when dЄAB/dθ = 0, or θ = - aAB/ bAB. This is the neutral temperature.
At θ = - 2aAB/ bAB. The emf becomes zero. This is the inversion temperature.
Law of intermediate Metal
Suppose ЄAB, ЄAC, ЄBC are emfs from thermocoupes AB,AC, and BC.
If hot junctions and cold junctions of the three thermocouples are at the same temperature,
Then,
ЄAB = ЄAC - ЄBC
Law of intermediate temperature
Let Є θ1, θ2, represent the thermo-emf of a given thermocouple when the temperatures of junctions are maintained at θ1and θ2. Then
Є θ1, θ2 = Є θ1, θ3 + Є θ3, θ2
Thomson effect
When current is passed through a metallic wire with non uniform temperature, it is observed that heat is absorbed in certain segments, and produced in certain segments. This heat is over and above i²Rt heat. This effect is given the name Thomson effect.
If a charge ΔQ is passed through a small section of the wire having a temperature difference ΔT between the ends, the Thomson heat is
ΔH = σ(ΔQ)( ΔT)
Where σ is a constant for a given metal at a given temperature.
The quantity ΔH/ΔQ = σΔT is called the Thomson emf.
σ is called the Thomson coefficient. It is taken as positive if heat is absorbed when a current is passed from the low temperature end to the high temperature end.
Copper, silver zinc, antimony, cadmium etc. have positive σ.
Iron, cobalt, nickel, bismuth, platinum etc. have negative σ. Negative sigma means, in these metal wires, heat is absorbed when current is passed from higher temperature end to the lower temperature end.
In lead, sigma is almost zero.
Explanation for Seebeck, Peltier and Thomson Effects
The density of free electrons is different in different metals. When two metals are joined to form a junction, the electrons tend to diffuse from the side with the higher concentration to the side with lower concentration. This produces a charge difference and hence an emf across the junction. This emf is the Peltier emf. When current is forced through this junction, positive or negative work is done the charge carriers, and due to the work thermal energy is produced or absorbed.
Similarly, when temperature of a metal piece is not uniform, density of free electrons varies inside the metal. The free electrons tend diffuse from the higher concentration regions to the lower concentration regions. This gives rise charge difference and hence an emf between the hot and cold parts of the metal. Hence when current is forced through this wire, positive or negative work is done and thermal energy is produced or absorbed. This is the Thomson effect.
In a thermocouple there are two junctions. If both a kept at the same temperature, Peltier emfs produced balance each other. If the junctions are at different temperatures, the emfs developed are different and there is a net emf in the loop. Also due to non uniform temperatures in the loop, Thomson effect is also present and Thomson emf is also produced. The actual emf produced in a thermocouple loop is the algebraic sum of the net Peltier effect and the net Thomspon effect.
So emf in a thermocouple loop = ЄAB =
(ΠAB)T - (Π AB ) T0 +(T – T0) (σA - σA)
(More explanation is needed for the formula.
Electrolysis
(Not there in JEE Physics syllabus)
When a current passes through an electrolyte, chemical changes occur in the electrolyte and substances are liberated at the electrodes. This process is called electrolysis. A conducting liquid is called electrolyte. The vessel in which electrolyte is there and in which electrolysis takes place is called electrolytic cell.
When AgNO3 solution is put in an electrolytic cell, a fraction of the dissolved AgNO3 molecules are separated into two parts, Ag+
and NO3- ions. Each ion has electric charge positive or negative.
WWhen a battery is connected to the electrodes placed in the electrolytic cell, cations ( Ag+) move toward the cathode and anions move toward the anode. The ions give up the charge at the electrodes. Ag+ becomes Ag and gets deposited on the cathode.
NO3- ions go to anode and gives its extra electron to it. NO3 is formed and it reacts with the silver anode an forms AgNO3 and gets dissolved in the solution. This way, silver is contnously removed from the anode and deposited on the cathode with the concentration of the electrolyte remaining unchanged.
Electrolysis Faraday’s laws of electrolysis
1. The mass of a substance liberated at an electrode is proportional to the charge passing through the electrolyte.
2. the mass of a substance liberated at an electrode by a given maoutn of charge is proportional to the chemical equivalent of the substance.
Hence m α Q or m α it
M = Zit
Where Z is a constant for the substance being liberated. The constant Z is called the electrochemical equivalent (ECE) of a substance.
The SI unit of ECE is kilogram/coulomb written as kg/C
The chemical equivalent of a substance s equal to its relative atomic mass divided by its valency.
For silver relative atomic mass is 108 and its valency is 1 (Check silver’s atomic number and valency). Therefore, chemical equivalent of silver is 108 kg/C.
Work done by the electric field on the free electrons in time t is
W = potential difference * charge
= V(it)
= (iR)it = i²Rt
Joules laws
1. The heat produced in a given resistor in a given time is proportional to the square of the current in it, i.e.,
H α i²
2. The heat produced in a given resistor by a given current is proportional to the time for which the current exists in it, i.e.,
H α t
3. The heat produced in a given resistor by a given current in a given time is proportional to its resistance.
H α R
These three are known as Joule’s laws.
Verification of Joule’s laws
Done by Joules calorimeter.
K-oil is taken in a copper calorimeter, current i1 is sent for a time t and the rise in temperature (Δ θ1) is noted. Later on the oil is allowed to cool to the room temperature and current i2 is sent for a time of ‘t’ same as last time. The rise in temperature (θ2) is noted. (Δ θ2)
It can be found that
Δθ1/ Δθ2 = i1²/i2²
This shows that Δθ α i²
Verification of the law H α t is made by taking reading of rise in temperature at regular time intervals. It can be found that temperature rise in uniform in every time period. It increases by equal amount in equal time periods.
Verification of H α R is done by taking two joule calorimeters. Different resistances R1 and R2 are dipped in the K-oil of different calorimeters and the system is connected to a battery. The initial temperatures are noted and after some time the final temperatures arenoted. It can be seen that
Δθ1/ Δθ2 = R1/R2.
Seebeck effect
If the junctions of two metallic strips (joined at the ends to form a loop) are kept at different temeratures , there is an electric current in the loop.
This effect is called Seebeck effect and the emf developed is called the Seebeck emf and thermo emf.
Peltier effect
This effect is reverse of Seebeck effect.
When the two junctions of a thermocouple (two different metallic strips joined at the ends to form a loop) are kept at the same temperature and electric current passed through them, it was observed that heat was produced at one junction and it was absorbed at the other junction. So one junction became hot and the other junction became cooler.
Thermoelectric series
Metals are arranged in a series called thermoelectric series, which may be used to predict the direction of current in a thermocouple in the temperature range 0°C to 100°C.
Antimony, nichrome, iron, zinc, copper, gold, silver, lead, aluminium, mercury, platinum-rhodium, platinum, nickel, constantan, bismuth.
At the cold junction current is from the metal coming earliner in the series to the metal coming latter in the series. Furthre apart two metals lie in the series, larger is the emf produced.
Neutral and inversion temperature
The temperature of the hot junction at which the thermo emf is maximum is called the neutral temperature and the temperature at which the thermo emf changes its sign is called the inversion temperature.
If θc, θn, θi are temperature of the cold junction, neutral temperature and inversion temperature respectively
c
θn – θc = θi - θn
Sign convention in thermocouples
The thermo emf developed in a thermocouple of metals A and B denoted ЄAB is taken to be positive if the direction of the current is from A to B at the hot junction.
Thermo emf depends on temperature with the relation
ЄAB = aABθ + ½ bsub>ABθ²
Where aAB and bAB are constants for a pair of metals A and B.
This gives dЄAB/dθ = aAB + bsub>ABθ
The quantity dЄAB/dθ is called thermoelectric power at temperature θ.
The emf is maximum when dЄAB/dθ = 0, or θ = - aAB/ bAB. This is the neutral temperature.
At θ = - 2aAB/ bAB. The emf becomes zero. This is the inversion temperature.
Law of intermediate Metal
Suppose ЄAB, ЄAC, ЄBC are emfs from thermocoupes AB,AC, and BC.
If hot junctions and cold junctions of the three thermocouples are at the same temperature,
Then,
ЄAB = ЄAC - ЄBC
Law of intermediate temperature
Let Є θ1, θ2, represent the thermo-emf of a given thermocouple when the temperatures of junctions are maintained at θ1and θ2. Then
Є θ1, θ2 = Є θ1, θ3 + Є θ3, θ2
Thomson effect
When current is passed through a metallic wire with non uniform temperature, it is observed that heat is absorbed in certain segments, and produced in certain segments. This heat is over and above i²Rt heat. This effect is given the name Thomson effect.
If a charge ΔQ is passed through a small section of the wire having a temperature difference ΔT between the ends, the Thomson heat is
ΔH = σ(ΔQ)( ΔT)
Where σ is a constant for a given metal at a given temperature.
The quantity ΔH/ΔQ = σΔT is called the Thomson emf.
σ is called the Thomson coefficient. It is taken as positive if heat is absorbed when a current is passed from the low temperature end to the high temperature end.
Copper, silver zinc, antimony, cadmium etc. have positive σ.
Iron, cobalt, nickel, bismuth, platinum etc. have negative σ. Negative sigma means, in these metal wires, heat is absorbed when current is passed from higher temperature end to the lower temperature end.
In lead, sigma is almost zero.
Explanation for Seebeck, Peltier and Thomson Effects
The density of free electrons is different in different metals. When two metals are joined to form a junction, the electrons tend to diffuse from the side with the higher concentration to the side with lower concentration. This produces a charge difference and hence an emf across the junction. This emf is the Peltier emf. When current is forced through this junction, positive or negative work is done the charge carriers, and due to the work thermal energy is produced or absorbed.
Similarly, when temperature of a metal piece is not uniform, density of free electrons varies inside the metal. The free electrons tend diffuse from the higher concentration regions to the lower concentration regions. This gives rise charge difference and hence an emf between the hot and cold parts of the metal. Hence when current is forced through this wire, positive or negative work is done and thermal energy is produced or absorbed. This is the Thomson effect.
In a thermocouple there are two junctions. If both a kept at the same temperature, Peltier emfs produced balance each other. If the junctions are at different temperatures, the emfs developed are different and there is a net emf in the loop. Also due to non uniform temperatures in the loop, Thomson effect is also present and Thomson emf is also produced. The actual emf produced in a thermocouple loop is the algebraic sum of the net Peltier effect and the net Thomspon effect.
So emf in a thermocouple loop = ЄAB =
(ΠAB)T - (Π AB ) T0 +(T – T0) (σA - σA)
(More explanation is needed for the formula.
Electrolysis
(Not there in JEE Physics syllabus)
When a current passes through an electrolyte, chemical changes occur in the electrolyte and substances are liberated at the electrodes. This process is called electrolysis. A conducting liquid is called electrolyte. The vessel in which electrolyte is there and in which electrolysis takes place is called electrolytic cell.
When AgNO3 solution is put in an electrolytic cell, a fraction of the dissolved AgNO3 molecules are separated into two parts, Ag+
and NO3- ions. Each ion has electric charge positive or negative.
WWhen a battery is connected to the electrodes placed in the electrolytic cell, cations ( Ag+) move toward the cathode and anions move toward the anode. The ions give up the charge at the electrodes. Ag+ becomes Ag and gets deposited on the cathode.
NO3- ions go to anode and gives its extra electron to it. NO3 is formed and it reacts with the silver anode an forms AgNO3 and gets dissolved in the solution. This way, silver is contnously removed from the anode and deposited on the cathode with the concentration of the electrolyte remaining unchanged.
Electrolysis Faraday’s laws of electrolysis
1. The mass of a substance liberated at an electrode is proportional to the charge passing through the electrolyte.
2. the mass of a substance liberated at an electrode by a given maoutn of charge is proportional to the chemical equivalent of the substance.
Hence m α Q or m α it
M = Zit
Where Z is a constant for the substance being liberated. The constant Z is called the electrochemical equivalent (ECE) of a substance.
The SI unit of ECE is kilogram/coulomb written as kg/C
The chemical equivalent of a substance s equal to its relative atomic mass divided by its valency.
For silver relative atomic mass is 108 and its valency is 1 (Check silver’s atomic number and valency). Therefore, chemical equivalent of silver is 108 kg/C.
Concept Review Ch.34 Magnetic Field
Definition of magnetic field
Magnetic field exerts force on a charge particle
Some facts about the magnetic force
a) From a point P, a charge particle can move in any direction or along any line. Along one of these possible lines, if the charge is moving, there is no magnetic force. Magnetic force is defined to be acting along this line.
b) The magnitude of the magnetic force is proportional to the product of speed of the charged particle v and sinθ, θ being the angle the speed makes with the line along which magnetic field is acting. Hence magnetic force is proportional to |v*sinθ|
c) The direction of the magnetic force is perpendicular to the direction of the magnetic field as well as to direction of the velocity.
d) The magnetic force is also proportional to the magnitude of charge q.
e) Its direction is different and opposite for positive and negative charges.
Magnetic field can be defined mathematically as
Vr(F) = qVr(v) × Vr(B) (34.1)
Equation uniquely determines the direction of magnetic field B from the rules of the vector product.
Units of magnetic field
The SI unit of magnetic field is newton/ampere-metre. It is written as Tesla.
Tesla is newton/ampere-metre. Tesla is also defined as weber/m².
Another unit in common use is gauss .
1 T = 104 gauss
We have magnetic field of the order of 10-5 near the earth's surface.
superconducting magnets can create a magnetic field of the order of 10 T.
Earlier, the concept of magnetic field was referred to as magnetic induction.
Electromagnetic field
Electric field and magnetic field are not basically independent. They are two aspects of same entity electromagnetic field. Whether th electromagnetic field will show up as an electric field or a magnetic field or a combination depends on the frame from which we are looking at the field.
Motion of a Charged particle in a uniform magnetic field.
Magentic force on a charged particle is perpendicular to its velocity. Hence there will not any change in its speed or kinetic energy.
The magnetic force will deflect the particle without changing speed and in a uniform field, the particle will move along a circle perpendicular to the magnetic field. The conclusion is that, the magnetic force provides centripetal force.
If r be the radius of the circle,
qvB = mv²/r (LHS is the expression for magnetic force and RHS is expression mass * acceleration)
r = mv/qB ...(34.2)
The time taken to complete the circle is
T = 2πr/v = 2πm/qB ... (34.3)
The time period or time taken to complete one circle is independent of speed.
But the radius (34.2) depends on v. Hence if speed increases, the radius is larger.
Frequency of revolutions is
ν = 1/T = qB/2πm ... (34.4)
This frequency is called cyclotron frequency.
If the velocity of charge is not perpendicular to the magnetic field, the resultant path will be a helix.
The radius of the path will be determined by velocity component which is perpendicular to the magnetic field.
Magnetic Force on a current carrying wire
In a current carrying wire, electrons, which are charge carrying particles are moving and hence in a magnetic field, a current carrying conductor would experince magnetic force.
Vr(F) = iVR(l)×Vr(B) ...(34.6)
Vr is used to denote vector.
The quanity iVr(l) denotes current element of length of l.
Torque on a current loop
If there is a rectangular loop carrying current i in a uniform magnetic field B
then net torque acting on the loop is
Г = iABsinθ
Where i = current in the loop
A = area
B = magnetic field
θ = the angle of inclination of the loop with the plane perpendicular to the plane of magnetic field.
We can also define
Vr(Г) = iVr(A)× Vr(B) ...(34.7)
iVr(A) can be termed as Vr(μ) the magnetic dipole moment of simply magentic moment of the current loop.
-------------------
In the material below vectors shown in bold letters.
Definition of magnetic field
Magnetic field exerts force on a moving charged particle
Some facts about the magnetic force
a) From a point P, a charged particle can move in any direction or along any line. Along one of these possible lines, if the charge is moving, there is no magnetic force. Magnetic force is defined to be acting along this line.
b) The magnitude of the magnetic force is proportional to the product of speed of the charged particle v and sinθ, θ being the angle the speed makes with the line along which magnetic field is acting. Hence magnetic force is proportional to |v*sinθ|
c) The direction of the magnetic force is perpendicular to the direction of the magnetic field as well as to direction of the velocity.
d) The magnetic force is also proportional to the magnitude of charge q.
e) Its direction is different and opposite for positive and negative charges.
Magnetic field can be defined mathematically as
F = qv × B
Equation uniquely determines the direction of magnetic field B from the rules of the vector product.
Units of magnetic field
The SI unit of magnetic field is newton/ampere-metre. It is written as Tesla.
Tesla is newton/ampere-metre. Tesla is also defined as weber/m².
Another unit in common use is gauss .
1 T = 104 gauss
We have magnetic field of the order of 10-5 near the earth's surface.
Superconducting magnets can create a magnetic field of the order of 10 T.
Earlier, the concept of magnetic field was referred to as magnetic induction.
Electromagnetic field
Electric field and magnetic field are not basically independent. They are two aspects of same entity electromagnetic field. Whether the electromagnetic field will show up as an electric field or a magnetic field or a combination depends on the frame from which we are looking at the field.
Motion of a Charged particle in a uniform magnetic field.
Magnetic force on a charged particle is perpendicular to its velocity. Hence there will not any change in its speed or kinetic energy.
The magnetic force will deflect the particle without changing speed and in a uniform field, the particle will move along a circle perpendicular to the magnetic field. The conclusion is that, the magnetic force provides centripetal force.
If r be the radius of the circle,
qvB = mv²/r (LHS is the expression for magnetic force and RHS is expression mass * acceleration)
r = mv/qB
The time taken to complete the circle is
T = 2πr/v = 2πm/qB
The time period or time taken to complete one circle is independent of speed.
But the radius depends on v. Hence if speed increases, the radius is larger.
Frequency of revolutions is
ν = 1/T = qB/2πm ... (34.4)
This frequency is called cyclotron frequency.
If the velocity of charge is not perpendicular to the magnetic field, the resultant path will be a helix.
The radius of the path will be determined by velocity component which is perpendicular to the magnetic field.
Magnetic Force on a current carrying wire
In a current carrying wire, electrons, which are charge carrying particles are moving and hence in a magnetic field, a current carrying conductor would experience magnetic force.
If a straight wire of length l carry8ng a current i is placed in a uniform magnetic field B, the force on it is
F = il×B
The quantity il denotes current element of length of l.
Torque on a current loop
If there is a rectangular loop carrying current i in a uniform magnetic field B
then net torque acting on the loop is
Г = iABsin θ
Where i = current in the loop
A = area
B = magnetic field
θ = the angle of inclination of the loop with the plane perpendicular to the plane of magnetic field.
We can also define
Г = iA× B
iA can be termed as μ the magnetic dipole moment or simply magnetic moment of the current loop.
If there are n turns in the loop, each turn experiences a torque.
The net torque is
Г = niA× B
μ = niA
Magnetic field exerts force on a charge particle
Some facts about the magnetic force
a) From a point P, a charge particle can move in any direction or along any line. Along one of these possible lines, if the charge is moving, there is no magnetic force. Magnetic force is defined to be acting along this line.
b) The magnitude of the magnetic force is proportional to the product of speed of the charged particle v and sinθ, θ being the angle the speed makes with the line along which magnetic field is acting. Hence magnetic force is proportional to |v*sinθ|
c) The direction of the magnetic force is perpendicular to the direction of the magnetic field as well as to direction of the velocity.
d) The magnetic force is also proportional to the magnitude of charge q.
e) Its direction is different and opposite for positive and negative charges.
Magnetic field can be defined mathematically as
Vr(F) = qVr(v) × Vr(B) (34.1)
Equation uniquely determines the direction of magnetic field B from the rules of the vector product.
Units of magnetic field
The SI unit of magnetic field is newton/ampere-metre. It is written as Tesla.
Tesla is newton/ampere-metre. Tesla is also defined as weber/m².
Another unit in common use is gauss .
1 T = 104 gauss
We have magnetic field of the order of 10-5 near the earth's surface.
superconducting magnets can create a magnetic field of the order of 10 T.
Earlier, the concept of magnetic field was referred to as magnetic induction.
Electromagnetic field
Electric field and magnetic field are not basically independent. They are two aspects of same entity electromagnetic field. Whether th electromagnetic field will show up as an electric field or a magnetic field or a combination depends on the frame from which we are looking at the field.
Motion of a Charged particle in a uniform magnetic field.
Magentic force on a charged particle is perpendicular to its velocity. Hence there will not any change in its speed or kinetic energy.
The magnetic force will deflect the particle without changing speed and in a uniform field, the particle will move along a circle perpendicular to the magnetic field. The conclusion is that, the magnetic force provides centripetal force.
If r be the radius of the circle,
qvB = mv²/r (LHS is the expression for magnetic force and RHS is expression mass * acceleration)
r = mv/qB ...(34.2)
The time taken to complete the circle is
T = 2πr/v = 2πm/qB ... (34.3)
The time period or time taken to complete one circle is independent of speed.
But the radius (34.2) depends on v. Hence if speed increases, the radius is larger.
Frequency of revolutions is
ν = 1/T = qB/2πm ... (34.4)
This frequency is called cyclotron frequency.
If the velocity of charge is not perpendicular to the magnetic field, the resultant path will be a helix.
The radius of the path will be determined by velocity component which is perpendicular to the magnetic field.
Magnetic Force on a current carrying wire
In a current carrying wire, electrons, which are charge carrying particles are moving and hence in a magnetic field, a current carrying conductor would experince magnetic force.
Vr(F) = iVR(l)×Vr(B) ...(34.6)
Vr is used to denote vector.
The quanity iVr(l) denotes current element of length of l.
Torque on a current loop
If there is a rectangular loop carrying current i in a uniform magnetic field B
then net torque acting on the loop is
Г = iABsinθ
Where i = current in the loop
A = area
B = magnetic field
θ = the angle of inclination of the loop with the plane perpendicular to the plane of magnetic field.
We can also define
Vr(Г) = iVr(A)× Vr(B) ...(34.7)
iVr(A) can be termed as Vr(μ) the magnetic dipole moment of simply magentic moment of the current loop.
-------------------
In the material below vectors shown in bold letters.
Definition of magnetic field
Magnetic field exerts force on a moving charged particle
Some facts about the magnetic force
a) From a point P, a charged particle can move in any direction or along any line. Along one of these possible lines, if the charge is moving, there is no magnetic force. Magnetic force is defined to be acting along this line.
b) The magnitude of the magnetic force is proportional to the product of speed of the charged particle v and sinθ, θ being the angle the speed makes with the line along which magnetic field is acting. Hence magnetic force is proportional to |v*sinθ|
c) The direction of the magnetic force is perpendicular to the direction of the magnetic field as well as to direction of the velocity.
d) The magnetic force is also proportional to the magnitude of charge q.
e) Its direction is different and opposite for positive and negative charges.
Magnetic field can be defined mathematically as
F = qv × B
Equation uniquely determines the direction of magnetic field B from the rules of the vector product.
Units of magnetic field
The SI unit of magnetic field is newton/ampere-metre. It is written as Tesla.
Tesla is newton/ampere-metre. Tesla is also defined as weber/m².
Another unit in common use is gauss .
1 T = 104 gauss
We have magnetic field of the order of 10-5 near the earth's surface.
Superconducting magnets can create a magnetic field of the order of 10 T.
Earlier, the concept of magnetic field was referred to as magnetic induction.
Electromagnetic field
Electric field and magnetic field are not basically independent. They are two aspects of same entity electromagnetic field. Whether the electromagnetic field will show up as an electric field or a magnetic field or a combination depends on the frame from which we are looking at the field.
Motion of a Charged particle in a uniform magnetic field.
Magnetic force on a charged particle is perpendicular to its velocity. Hence there will not any change in its speed or kinetic energy.
The magnetic force will deflect the particle without changing speed and in a uniform field, the particle will move along a circle perpendicular to the magnetic field. The conclusion is that, the magnetic force provides centripetal force.
If r be the radius of the circle,
qvB = mv²/r (LHS is the expression for magnetic force and RHS is expression mass * acceleration)
r = mv/qB
The time taken to complete the circle is
T = 2πr/v = 2πm/qB
The time period or time taken to complete one circle is independent of speed.
But the radius depends on v. Hence if speed increases, the radius is larger.
Frequency of revolutions is
ν = 1/T = qB/2πm ... (34.4)
This frequency is called cyclotron frequency.
If the velocity of charge is not perpendicular to the magnetic field, the resultant path will be a helix.
The radius of the path will be determined by velocity component which is perpendicular to the magnetic field.
Magnetic Force on a current carrying wire
In a current carrying wire, electrons, which are charge carrying particles are moving and hence in a magnetic field, a current carrying conductor would experience magnetic force.
If a straight wire of length l carry8ng a current i is placed in a uniform magnetic field B, the force on it is
F = il×B
The quantity il denotes current element of length of l.
Torque on a current loop
If there is a rectangular loop carrying current i in a uniform magnetic field B
then net torque acting on the loop is
Г = iABsin θ
Where i = current in the loop
A = area
B = magnetic field
θ = the angle of inclination of the loop with the plane perpendicular to the plane of magnetic field.
We can also define
Г = iA× B
iA can be termed as μ the magnetic dipole moment or simply magnetic moment of the current loop.
If there are n turns in the loop, each turn experiences a torque.
The net torque is
Г = niA× B
μ = niA
Concept Review Ch.35 Magnetic Field Due to A Current
A magnetic field can be produced by moving charges or electric currents(electric current is moving charged particles - electrons).
Biot-Savart law
Biot-Savart law is the basic equation that gives us the magnetic field due to electric current in a conductor.
The magnetic field due to a current element dl at a point P with a distance r from dl is
Vector of dB = Vr(dB) = [1/4πε0c²]i[Vr(dl)×Vr(r)]/r³ ...(35.1)
where c = speed of light
i = current
Vr(dl) = length vector of the current element
Vr(r) = ector joining the current element to the point P where we are finding magnetic field
[1/ε0c²] is written as μ0 and is called the permeability of vacuum.
The value of μ0 is 4π*10-7
Vector of dB = Vr(dB) = [μ0i/4π][Vr(dl)×Vr(r)]/r³ ... (35.2)
The magnitude of the magnetic field is
dB = [μ0idl sinθ /4πr²]
Where θ is the angle between current element and the vector joining current element and the point P.
The direction of the field is perpendicular to the plane containing the current element and the point.
Magnetic field due to current in a straight wire at a point P with a distance d from it.
B = [μ0i/4πd][cosθ1 - cosθ2]
θ1 and θ2 are the values of θ corresponding to the lower end and the upper end respectively of the straigth wire.
See the Fig 35.3 to remember the method of measuring θ.
If point P is on a perpendicular bisector of the wire.
θ1 and θ2 are equal. If the length of the wire is α and distance of the point P is d from the wire
cosθ1 = α/SQRT(α²+ 4d²)
cosθ2 = - α/SQRT(α²+ 4d²)
θ1 is anlge withlower end of the wire.
θ2 is anle with the upper end of the wire.
B = [μ0iα]/[2πdSQRT(α²+ 4d²)]
If the straight wire is a very long one
θ1 = 0 and θ2 is equal to π. So
B = μ0i/2πd ... (35.6)
Magnetic field lines
Magnetic field lines are similar to electric field lines. A tangent to a magnetic field line gives the direction of the magnetic field existing at that point. For a long straight wire, the field lines are circles with their centres on the wire.
Force between parallel wires carryng current
If the two wires are treated as long straight wires carrying current i1 (W1) and i2 (W2)
Field on an element on the wire carrying current i2 is
B = μ0i1/2πd
Magnetic force on the element
dF = i2dlμ0i1/2πd
So the force per unit length of the wire
W2 due to the wire W1 is
dF/dl = i2μ0i1/2πd
= μ0i1i2/2πd ...(35.7)
Same amount of force is applied by W2 on unit length of W1.
If both the wires carry current in same direction they attract each other. If they carry current in opposite directions, they repel each other.
Definition of ampere
If two parallel, long wires, kept 1 m apart in vacuum, carry equal currents in the same directin and there is a force of attraction of 2*10-7 newton per metre of each wire, the current in each wire is said to be 1 ampere.
Field due to a circular current
Field at the centre
Radius of circular loop = a
current in the loop = i
B = μ0i/2a
Field at an axial point due to a circular conductor
B = μ0ia²/2(a²+d²)^(3/2)
where
a = radius of the circular conductor
d is the distance of the point from the centre of the circular conductor
If the field is far away from the centre
d>>a
B = 2μ0πia²/4πd³
As πia² is magnetic dipole moment of circular conductor (μ)
B = 2μ0μ/4πd³ ... (35.9)
Ampere's law
The circulation of Line integral of Vr(B).Vr(dl)of the resultant magnetic field along a closed, plane curve is equal to μ0 times the total current crossing the area bounded by the closed curve provided the electric field inside the loop remains constant.
Line integral of Vr(B).Vr(dl) = μ0i ... (35.10)
You can derive the magnetic field at a point due to current in a long straight wire
using Ampere's law and verify that the formula is the same as the one derived by using Biot Savart Law.
Ampere's law can be derived from Biot-Savart law and Biot-Savart law from Ampere's law.
Ampere's law is useful under certain symmetrical conditions.
Finding magnetic field at a point due to a long, straight current using Ampere's law
Solenoid
A solenoid is an insulated wire wound closely in the form of a helix. The length of the solenoid is large compared to its radius of its loop.
The magnetic field inside a very tightly wound long solenoid is uniform everywhere and it zero outside it.
B = μ0ni ...(35.11)
n= number of turns per unit length
Toroid
If a nonconducting ring is taken and a conducting wire is wound closely around it we get a toroid.
In a toroid magnetic field is
B = μ0Ni/2πr
Where N = number of turns in the toroid
r = is the distance of point P (where we have to find the magnetic field) from the centre of the toroid.
Biot-Savart law
Biot-Savart law is the basic equation that gives us the magnetic field due to electric current in a conductor.
The magnetic field due to a current element dl at a point P with a distance r from dl is
Vector of dB = Vr(dB) = [1/4πε0c²]i[Vr(dl)×Vr(r)]/r³ ...(35.1)
where c = speed of light
i = current
Vr(dl) = length vector of the current element
Vr(r) = ector joining the current element to the point P where we are finding magnetic field
[1/ε0c²] is written as μ0 and is called the permeability of vacuum.
The value of μ0 is 4π*10-7
Vector of dB = Vr(dB) = [μ0i/4π][Vr(dl)×Vr(r)]/r³ ... (35.2)
The magnitude of the magnetic field is
dB = [μ0idl sinθ /4πr²]
Where θ is the angle between current element and the vector joining current element and the point P.
The direction of the field is perpendicular to the plane containing the current element and the point.
Magnetic field due to current in a straight wire at a point P with a distance d from it.
B = [μ0i/4πd][cosθ1 - cosθ2]
θ1 and θ2 are the values of θ corresponding to the lower end and the upper end respectively of the straigth wire.
See the Fig 35.3 to remember the method of measuring θ.
If point P is on a perpendicular bisector of the wire.
θ1 and θ2 are equal. If the length of the wire is α and distance of the point P is d from the wire
cosθ1 = α/SQRT(α²+ 4d²)
cosθ2 = - α/SQRT(α²+ 4d²)
θ1 is anlge withlower end of the wire.
θ2 is anle with the upper end of the wire.
B = [μ0iα]/[2πdSQRT(α²+ 4d²)]
If the straight wire is a very long one
θ1 = 0 and θ2 is equal to π. So
B = μ0i/2πd ... (35.6)
Magnetic field lines
Magnetic field lines are similar to electric field lines. A tangent to a magnetic field line gives the direction of the magnetic field existing at that point. For a long straight wire, the field lines are circles with their centres on the wire.
Force between parallel wires carryng current
If the two wires are treated as long straight wires carrying current i1 (W1) and i2 (W2)
Field on an element on the wire carrying current i2 is
B = μ0i1/2πd
Magnetic force on the element
dF = i2dlμ0i1/2πd
So the force per unit length of the wire
W2 due to the wire W1 is
dF/dl = i2μ0i1/2πd
= μ0i1i2/2πd ...(35.7)
Same amount of force is applied by W2 on unit length of W1.
If both the wires carry current in same direction they attract each other. If they carry current in opposite directions, they repel each other.
Definition of ampere
If two parallel, long wires, kept 1 m apart in vacuum, carry equal currents in the same directin and there is a force of attraction of 2*10-7 newton per metre of each wire, the current in each wire is said to be 1 ampere.
Field due to a circular current
Field at the centre
Radius of circular loop = a
current in the loop = i
B = μ0i/2a
Field at an axial point due to a circular conductor
B = μ0ia²/2(a²+d²)^(3/2)
where
a = radius of the circular conductor
d is the distance of the point from the centre of the circular conductor
If the field is far away from the centre
d>>a
B = 2μ0πia²/4πd³
As πia² is magnetic dipole moment of circular conductor (μ)
B = 2μ0μ/4πd³ ... (35.9)
Ampere's law
The circulation of Line integral of Vr(B).Vr(dl)of the resultant magnetic field along a closed, plane curve is equal to μ0 times the total current crossing the area bounded by the closed curve provided the electric field inside the loop remains constant.
Line integral of Vr(B).Vr(dl) = μ0i ... (35.10)
You can derive the magnetic field at a point due to current in a long straight wire
using Ampere's law and verify that the formula is the same as the one derived by using Biot Savart Law.
Ampere's law can be derived from Biot-Savart law and Biot-Savart law from Ampere's law.
Ampere's law is useful under certain symmetrical conditions.
Finding magnetic field at a point due to a long, straight current using Ampere's law
Solenoid
A solenoid is an insulated wire wound closely in the form of a helix. The length of the solenoid is large compared to its radius of its loop.
The magnetic field inside a very tightly wound long solenoid is uniform everywhere and it zero outside it.
B = μ0ni ...(35.11)
n= number of turns per unit length
Toroid
If a nonconducting ring is taken and a conducting wire is wound closely around it we get a toroid.
In a toroid magnetic field is
B = μ0Ni/2πr
Where N = number of turns in the toroid
r = is the distance of point P (where we have to find the magnetic field) from the centre of the toroid.
Concept Review Ch.36 Permanent Magnets
Model of magnetic charge
a. There are two types of magnetic charges, positive magnetic charge and negative magnetic charge.
A magentic charge when placed in a magnetic field experiences a force.
The force on a postive magnetic charge is along the field and the force on a negative magnetic charge is opposite the field.
b. A magnetic charge produces a magnetic field.
The field is radially outward if the charge if positive and is inward if it is negative.
C. A magnetic dipole is formed when a negative magnetic charge -m and a positive magentic charge +m are placed at a small separation d.
md is called magnetic dipole moment and its direction is from -m to +m.
The line joining -m to +m is called the axis of the dipole.
d. a magnetic dipole of dipole moment md can be replaced by a current loop or area carrying a current i.
md = iA
How natural magnets exist?
Matter is made of atoms and each atom contains electrons circulating around the nucleus. The circulation of electron around a nucleus can be assumed as small circular current loop. We already mentioned in the model of magnetism that a current loop is equivalent to a magnetic dipole. In permanent magnets, these small circular loops are arranged nearly parallel to each other along the length of the magnet. At any point inside the magnet, the net current is zeo because the currents from the adjacent loops cancel each other over any cross section or face. However on the surface, such cancellation of current does not take place and hence there is a net current. Due to such a surface current, a magnet in the form of a cylinder is equivalent to a closely wound, current carrying solenoid and hence produces a magnetic field similar to the solenoid.
Magnetic length of the magnent
Magnetic poles appear slightly inside the bar. The distance between the locations of the assumed poles is called the magnetic length of the magnet. the distance between the ends is called the geometrical length.
It is found from observation that magnetic length/geometrical length = 0.84.
Magnetic moment of a bar magnet
It is denoted by M. Magnetic length of a bar magnet is denoted by 2l.If m is the pole strength and 2l is magnetic length of a bar magnet, its magnetic moment is
M = 2ml
Torque on a bar magnet placed in a magnetic field
Г = MB sin θ
θ is the angle between the magnetic field assumed to be along Y axis and the bar magnet. B is the magnetic field and M is the magnetic moment of the bar magnet.
Potential energy of a bar magnet placed in a magnetic field
U(θ) = -MB Cos θ = -M.B
When potential energy is assumed to be zero when θ = 90° (Magnetic field assumed to be along Y axis)
Magnetic field due to a bar magnet
End on position
A position on the magnetic axis of a bar magnet is called an end-on position. The magnetic field at a point in end-on position at a distance d from the centre of the magnet is:
B = (µ0/4 π)[(2Md)/(d² – l²)²] …(6)
If d is very large compared to l, then
B = (μ0/4π)(2M/d³)
Broadside on position
A position on a perpendicular bisector of the bar magnet is called broadside-on position. The magnetic field at a point in broadside-on position at a distance d from the centre of the magnet is:
B = (µ0/4 π)[m 2l/(d² + l²)3/2 ] …(8)
= (µ0/4 π)[M/(d² + l²)3/2 ]
If d is very large compared to l,
B = (μ0/4π)(M/d³)
Magnetic scalar potential
Change in potential is defined as
V(r2) – V(r1) = -r1∫r2 B.dr
Normaly, the potential at infinity ( a point far away from all sources of magnetic field) is taken to be zero.
Magnetic scalar potential due to a magnetic dipole
Magnetic scalar potential at a point P which is at a distance r from the mid point of the magnetic dipole, and the angle between the dipole axis and the line joining the mid point of the dipole to the point P is θ
V = (µ0/4 π)[Mcos θ /r²)
Where
M = 2ml =magnetic moment of the dipole
Magnetic field due to a dipole
Magnetic field at P =
(µ0/4 π)[M /r²)√(1 +3 cos² θ)] ..
Terrestrial magnetism
(There are questions in past papers about the magnetic needle placed at various places on the earth)
Earth has a magnetic dipole of dipole moment about 8.0*10^23 /T located at its centre. The axis of this dipole makes an angle of about 11.5 with the earth’s axis of rotation.
The dipole axis cuts the earth’s surface at two points, one near the geographical north pole and the other near the geographical south pole. The first of these points is called geomagnetic north pole and the other is called geomagnetic south pole.
If a bar magnet is freely suspended at a point near the earth’s surface, it will stay along the magnetic field there, the north pole pointing towards the direction of magnetic field. At the geomagnetic poles, the magnetic field is vertical. If the bar magnet is suspended near the geomagnetic north pole, it will become vertical with its north pole towards the earth’s surface. Similarly, a bar magnet freely suspended near the geomagnetic south pole, will become vertical with its south pole pointing towards the earth’s surface. Hence at geomagnetic poles a freely suspended bar magnet becomes vertical and this property can be definition of geomagnetic poles.
In the assumed magnetic dipole inside the earth, the south pole of the dipole will be towards the geomagnetic north pole and the north pole will be towards the geomagnetic south pole. That is why north pole of the freely suspended bar magnets points towards geomagnetic north pole, at that place. Similarly south pole of the freely suspended bar magnet will be towards the south pole at that place.
Earth’s magnetic field changes in both in direction and magnitude as time passes. Noticeable changes occur over 10 years long periods. Even reversals in direction take place. In the past 7.6*10^7 years 171 reversals have taken place. The latest reversal is believed to have taken place around 10,000 years ago.
Elements of the Earth’s magnetic field
a. declination b. inclination or dip and c. horizontal component of the field
Declination
A plane passing through the geographical poles (that is axis of rotation of the earth) and a given point P on the earth’s surface is called the geographical meridian at the point P. The plane passing through the geomagnetic poles (dipole axis of the earth) and the point P is called the magnetic meridian at the point P.
The angle between the magnetic meridian and the geographical meridian at a point is called the declination at that point.
Navigators use a magnetic compass needle to locate direction. The needle stays in equilibrium when it is in magnetic meridian. For finding the true north, navigators have use declination and arrive at the direction of true north.
Inclination or dip
The angle made by the earth’s magnetic field with the horizontal direction in the magnetic meridian, is called the inclination of dip at that point.
Magnetic meridian contains the dipole axis of earth.
In the magnetic northern hemisphere, the vertical component of the earth’s magnetic field points downwards, The north pole of a freely suspended magnet, therefore dips in northern hemisphere. The freely suspended magnet will not be horizontal.
The knowledge of declination and inclination completely specifies the direction of the earth’s magnetic field with respect to earth’s axis of rotation.
Horizontal component of earth’s magnetic field
It is the horizontal component of the earth’s magnetic field in the magnetic meridian at a point. If B is magnetic field and δ is dip at a point
Horizontal component BH = B cos δ
We can express it as B = BH/ cos δ
Determination of dip at a place
Dip at a place is measured by an apparatus known as dip circle.
Neutral point
If the horizontal component of the magnetic field due to a magnet is equal and opposite to the earth’s horizontal magnetic field at a point, the net horizontal field is zero at such a point. If a compass needle is placed at such a point, it can stay in any position. Such a point where horizontal component of earth’s magnetic field is cancelled by the horizontal field of a magnet is called neutral point.
Tangent Galvanometer
Tangent galvanometer is used to measure current. In this instrument a magnetic needed is placed at the centre of circular coil mounted in a vertical plane. When current i is passed through this apparatus, the magnetic field produced at the centre of the coil is
B = µ0in/2r
Where i = current through the coil
N = no of turns in the coil.
r = radius of the coil..
As the field is perpendicular to the plane of the coil, its direction is horizontal and perpendicular to the magnetic meridian and hence perpendicular to the horizontal component BH of the earth’s magnetic field. Hence the resultant horizontal field is
Br = √(B² + BH²)
The resultant will make angle θ with BH
tan θ = B/ BH
Hence B = BH tan θ = µ0in/2r
i = 2r BH tan θ/µ0n = Ktan θ
where K = 2r BH/µ0n is a constant for the give galvanometer at a given place and it is called reduction factor of the galvanometer.
The reduction factor is measured by passed known current and measuring the deflection (θ) of the needle in the galvanometer.
Sensitivity of the galvanometer
Good sensitivity means that the change in deflection is large for a given fractional change in current.
We have i = K tan θ, we get dθ = ½ sin 2θ (di/i)
For good sensitivity, θ = 45°.
Hence when θ = 45°, the galvanometer is most sensitive.
Tangent Law of perpendicular fieds
When the compass needle is palced in the earth’s magnetic field it stays along the horizontal component BH of the field. The magnetic forces m BH and -m BH on the poles will be in a straight line and do not produce any torque. But if an external horizontal magnetic field B perpendicular to BH is introced, the needle deflects from its position. This concept is used in the tangent galvanometer.
The resulting force is
Br = √(B² + BH²)
The resultant will make angle θ with BH
tan θ = B/ BH
Hence B = BH tan θ
This is known as the tangent law of perpendicular fields.
Moving Coil galvanometer
In the magnetic field produced by two poles of a strong permanent magnet, a soft iron core over which rectangular insulated wire is wound is placed. The soft iron core is fixed to a torsion head through a fine strip of phosphor bronze on bottom side also it fixed through a spring made of phosphor bronze. When the current is passed through coil there is a torque on it and deflection takes place. The torque is equated to the torque produced in the phosphor bronze strip due to its twisting and current value is determined.
Moving coil galvanometer
A rectangular insulated conductor coil of several turns is wound over a soft iron core. This core with the coil is attached to a torsion head at the upper end through a strip of phosphor bronze and at the lower end to a spring made of phosphor bronze. The coil with the core is suspended between two pole pieces of a strong permanent magnet. When current is passed through the coil, as the coil is in the magnetic field B of the permanent magnet, a torque niAB acts on the coil and due to this torque the core with the coil deflected.. A small mirror is fixed to the phosphor bronze strip and a lamp scale arrangement is used to measure the deflection.
niAB = kθ
i = kθ/nAB
The constant k/nAB is called the galvanometer constant. The constant is found by passing a known current and measuring the deflection.
The sensitivity of a moving coil galvanometer is defined as θ/i. For large sensitivity, the filed B should be large. Soft iron core increases magnetic field.
Deflection magnetometer.
The instrument has a small compass needle pivoted at the centre of a graduated circular scale. This arrangement is kept in a wooden frame having two long arms having metre scales. We can find M/ BH ratio using this apparatus. M is the magnetic moment of a magnet and BH is the horizontal component of the earth’s magnetic field. This measurement can be made in Tan A position or Tan B position of the magnetometer.
Tan A position
The arms of the magnetometer are kept along the magnetic east-west direction. This direction is perpendicular to the direction of the needle. The magnet is kept on one of the arms parallel to its length. Hence the compass needle is in end-on position of the bar magnet. (End on position is a position on the magnetic axis of a bar magnet)
The magnetic field due to the bar magnet at the site of the needle is
B = (µ0/4 π)[(2Md)/(d² – l²)²]
Where l = length of the bar magnet and d = distance of the centre of the bar magnet from the centre of the compass needle.
From tangent law B = BH tan θ
Hence M/ BHt = (4π/µ0) [d² – l²)²/2d] tan θ
Tan-B position
In this position, the arms of the magnetometer are kept in the magnetic north-south direction. The bar magnet is placed on one of the arms symmetrically at right angles to the wooden arm.
In this position the compass centre point is in broadside-on position of the bar magnet.
Therefore the magnetic field due to the bar magnet at the compass centre point due to the magnet is
B = (µ0/4 π)[M/(d² + l²)3/2 ]
Using the tangent law B = BH tan θ
Hence M/ BH = (4π/µ0)[(d² + l²)3/2 ]tan θ
Applications of the deflection magnetometer
1. Comparison of the magnetic moments two magnets. If the magnetic moments of the two magnets are M1 and M2, then we can find M1/ BH and M2/ BH and then can find M1/M2.
Null method of finding M1/M2
In this method, the two magnets are placed at the opposite ends of the wooden legs of the deflection magnetometer and the distances of the magnets are adjusted on wooden arms so that deflection of the needle is zero. That means magnetic fields produced by the magnets cancel each other.
In Tan A position
B = (µ0/4 π)[(2M1d1)/(d1² – l1²)²] = (µ0/4 π)[(2M2d2)/(d2² – l2²)²]
M1/M2 = (d2/d1)[(d1² – l1²)²/)/(d2² – l2²)²]
In tan B position
B = (µ0/4 π)[M1/(d1² + l1²)3/2 ] = (µ0/4 π)[M2/(d2² + l2²)3/2 ]
M1/M2 = (d1² + l1²)3/2 /(d2² + l2²)3/2
2. The horizontal components of earth’s magnetic field at two places can be done. M/ BH1 and M/ BH2 help in this.
3. Verification of inverse law. The magnetic field due to a magnetic pole is inversely proportional to the distance of the point from the pole.
From the M/ BH expression we can find an expression for cot θ
And a graph is plotted between [d² – l²)²/2d] and cot θ for Tan A position readings. We get a straight line
Similarly a graph is plotted between [(d² + l²)3/2 ] and cot θ for readings in Tan B position. We get a straight line.
Oscillation magnetometer
Using oscillation magnetometer, the oscillations of bar magnet and the taken are determined.
T = time per one osciallation = 2 π√(I/MBH)
That gives MBH = 4π²I/T²
I moment of inertia of the bar magnet can be determined from I = W(a²+b²)/12 where W = mass of the magnet, a = geometric length, b = breadth of the magnet.
Determination of M and BH
We can find M/ BH using a deflection magnetometer and M BH using an oscillation magnetometer.
If M/ BH = X abnd M BH= Y
Then M² = XY
And M = √(XY)
BH = √(Y/X)
Gauss’s law for Magnetism
Magnetic field due to magnetic charge is
B = (µ0/4 π)(m/r²)
From the above expression we can derive Gauss’s law for magnetism as
∮B.ds = µ0minside
where ∮B.ds is the magnetic flux and minside is the “net pole strength inside the closed surface. But net magnetic stranegh enclosed by any closer surface is zero.
Hence ∮B.ds = 0
a. There are two types of magnetic charges, positive magnetic charge and negative magnetic charge.
A magentic charge when placed in a magnetic field experiences a force.
The force on a postive magnetic charge is along the field and the force on a negative magnetic charge is opposite the field.
b. A magnetic charge produces a magnetic field.
The field is radially outward if the charge if positive and is inward if it is negative.
C. A magnetic dipole is formed when a negative magnetic charge -m and a positive magentic charge +m are placed at a small separation d.
md is called magnetic dipole moment and its direction is from -m to +m.
The line joining -m to +m is called the axis of the dipole.
d. a magnetic dipole of dipole moment md can be replaced by a current loop or area carrying a current i.
md = iA
How natural magnets exist?
Matter is made of atoms and each atom contains electrons circulating around the nucleus. The circulation of electron around a nucleus can be assumed as small circular current loop. We already mentioned in the model of magnetism that a current loop is equivalent to a magnetic dipole. In permanent magnets, these small circular loops are arranged nearly parallel to each other along the length of the magnet. At any point inside the magnet, the net current is zeo because the currents from the adjacent loops cancel each other over any cross section or face. However on the surface, such cancellation of current does not take place and hence there is a net current. Due to such a surface current, a magnet in the form of a cylinder is equivalent to a closely wound, current carrying solenoid and hence produces a magnetic field similar to the solenoid.
Magnetic length of the magnent
Magnetic poles appear slightly inside the bar. The distance between the locations of the assumed poles is called the magnetic length of the magnet. the distance between the ends is called the geometrical length.
It is found from observation that magnetic length/geometrical length = 0.84.
Magnetic moment of a bar magnet
It is denoted by M. Magnetic length of a bar magnet is denoted by 2l.If m is the pole strength and 2l is magnetic length of a bar magnet, its magnetic moment is
M = 2ml
Torque on a bar magnet placed in a magnetic field
Г = MB sin θ
θ is the angle between the magnetic field assumed to be along Y axis and the bar magnet. B is the magnetic field and M is the magnetic moment of the bar magnet.
Potential energy of a bar magnet placed in a magnetic field
U(θ) = -MB Cos θ = -M.B
When potential energy is assumed to be zero when θ = 90° (Magnetic field assumed to be along Y axis)
Magnetic field due to a bar magnet
End on position
A position on the magnetic axis of a bar magnet is called an end-on position. The magnetic field at a point in end-on position at a distance d from the centre of the magnet is:
B = (µ0/4 π)[(2Md)/(d² – l²)²] …(6)
If d is very large compared to l, then
B = (μ0/4π)(2M/d³)
Broadside on position
A position on a perpendicular bisector of the bar magnet is called broadside-on position. The magnetic field at a point in broadside-on position at a distance d from the centre of the magnet is:
B = (µ0/4 π)[m 2l/(d² + l²)3/2 ] …(8)
= (µ0/4 π)[M/(d² + l²)3/2 ]
If d is very large compared to l,
B = (μ0/4π)(M/d³)
Magnetic scalar potential
Change in potential is defined as
V(r2) – V(r1) = -r1∫r2 B.dr
Normaly, the potential at infinity ( a point far away from all sources of magnetic field) is taken to be zero.
Magnetic scalar potential due to a magnetic dipole
Magnetic scalar potential at a point P which is at a distance r from the mid point of the magnetic dipole, and the angle between the dipole axis and the line joining the mid point of the dipole to the point P is θ
V = (µ0/4 π)[Mcos θ /r²)
Where
M = 2ml =magnetic moment of the dipole
Magnetic field due to a dipole
Magnetic field at P =
(µ0/4 π)[M /r²)√(1 +3 cos² θ)] ..
Terrestrial magnetism
(There are questions in past papers about the magnetic needle placed at various places on the earth)
Earth has a magnetic dipole of dipole moment about 8.0*10^23 /T located at its centre. The axis of this dipole makes an angle of about 11.5 with the earth’s axis of rotation.
The dipole axis cuts the earth’s surface at two points, one near the geographical north pole and the other near the geographical south pole. The first of these points is called geomagnetic north pole and the other is called geomagnetic south pole.
If a bar magnet is freely suspended at a point near the earth’s surface, it will stay along the magnetic field there, the north pole pointing towards the direction of magnetic field. At the geomagnetic poles, the magnetic field is vertical. If the bar magnet is suspended near the geomagnetic north pole, it will become vertical with its north pole towards the earth’s surface. Similarly, a bar magnet freely suspended near the geomagnetic south pole, will become vertical with its south pole pointing towards the earth’s surface. Hence at geomagnetic poles a freely suspended bar magnet becomes vertical and this property can be definition of geomagnetic poles.
In the assumed magnetic dipole inside the earth, the south pole of the dipole will be towards the geomagnetic north pole and the north pole will be towards the geomagnetic south pole. That is why north pole of the freely suspended bar magnets points towards geomagnetic north pole, at that place. Similarly south pole of the freely suspended bar magnet will be towards the south pole at that place.
Earth’s magnetic field changes in both in direction and magnitude as time passes. Noticeable changes occur over 10 years long periods. Even reversals in direction take place. In the past 7.6*10^7 years 171 reversals have taken place. The latest reversal is believed to have taken place around 10,000 years ago.
Elements of the Earth’s magnetic field
a. declination b. inclination or dip and c. horizontal component of the field
Declination
A plane passing through the geographical poles (that is axis of rotation of the earth) and a given point P on the earth’s surface is called the geographical meridian at the point P. The plane passing through the geomagnetic poles (dipole axis of the earth) and the point P is called the magnetic meridian at the point P.
The angle between the magnetic meridian and the geographical meridian at a point is called the declination at that point.
Navigators use a magnetic compass needle to locate direction. The needle stays in equilibrium when it is in magnetic meridian. For finding the true north, navigators have use declination and arrive at the direction of true north.
Inclination or dip
The angle made by the earth’s magnetic field with the horizontal direction in the magnetic meridian, is called the inclination of dip at that point.
Magnetic meridian contains the dipole axis of earth.
In the magnetic northern hemisphere, the vertical component of the earth’s magnetic field points downwards, The north pole of a freely suspended magnet, therefore dips in northern hemisphere. The freely suspended magnet will not be horizontal.
The knowledge of declination and inclination completely specifies the direction of the earth’s magnetic field with respect to earth’s axis of rotation.
Horizontal component of earth’s magnetic field
It is the horizontal component of the earth’s magnetic field in the magnetic meridian at a point. If B is magnetic field and δ is dip at a point
Horizontal component BH = B cos δ
We can express it as B = BH/ cos δ
Determination of dip at a place
Dip at a place is measured by an apparatus known as dip circle.
Neutral point
If the horizontal component of the magnetic field due to a magnet is equal and opposite to the earth’s horizontal magnetic field at a point, the net horizontal field is zero at such a point. If a compass needle is placed at such a point, it can stay in any position. Such a point where horizontal component of earth’s magnetic field is cancelled by the horizontal field of a magnet is called neutral point.
Tangent Galvanometer
Tangent galvanometer is used to measure current. In this instrument a magnetic needed is placed at the centre of circular coil mounted in a vertical plane. When current i is passed through this apparatus, the magnetic field produced at the centre of the coil is
B = µ0in/2r
Where i = current through the coil
N = no of turns in the coil.
r = radius of the coil..
As the field is perpendicular to the plane of the coil, its direction is horizontal and perpendicular to the magnetic meridian and hence perpendicular to the horizontal component BH of the earth’s magnetic field. Hence the resultant horizontal field is
Br = √(B² + BH²)
The resultant will make angle θ with BH
tan θ = B/ BH
Hence B = BH tan θ = µ0in/2r
i = 2r BH tan θ/µ0n = Ktan θ
where K = 2r BH/µ0n is a constant for the give galvanometer at a given place and it is called reduction factor of the galvanometer.
The reduction factor is measured by passed known current and measuring the deflection (θ) of the needle in the galvanometer.
Sensitivity of the galvanometer
Good sensitivity means that the change in deflection is large for a given fractional change in current.
We have i = K tan θ, we get dθ = ½ sin 2θ (di/i)
For good sensitivity, θ = 45°.
Hence when θ = 45°, the galvanometer is most sensitive.
Tangent Law of perpendicular fieds
When the compass needle is palced in the earth’s magnetic field it stays along the horizontal component BH of the field. The magnetic forces m BH and -m BH on the poles will be in a straight line and do not produce any torque. But if an external horizontal magnetic field B perpendicular to BH is introced, the needle deflects from its position. This concept is used in the tangent galvanometer.
The resulting force is
Br = √(B² + BH²)
The resultant will make angle θ with BH
tan θ = B/ BH
Hence B = BH tan θ
This is known as the tangent law of perpendicular fields.
Moving Coil galvanometer
In the magnetic field produced by two poles of a strong permanent magnet, a soft iron core over which rectangular insulated wire is wound is placed. The soft iron core is fixed to a torsion head through a fine strip of phosphor bronze on bottom side also it fixed through a spring made of phosphor bronze. When the current is passed through coil there is a torque on it and deflection takes place. The torque is equated to the torque produced in the phosphor bronze strip due to its twisting and current value is determined.
Moving coil galvanometer
A rectangular insulated conductor coil of several turns is wound over a soft iron core. This core with the coil is attached to a torsion head at the upper end through a strip of phosphor bronze and at the lower end to a spring made of phosphor bronze. The coil with the core is suspended between two pole pieces of a strong permanent magnet. When current is passed through the coil, as the coil is in the magnetic field B of the permanent magnet, a torque niAB acts on the coil and due to this torque the core with the coil deflected.. A small mirror is fixed to the phosphor bronze strip and a lamp scale arrangement is used to measure the deflection.
niAB = kθ
i = kθ/nAB
The constant k/nAB is called the galvanometer constant. The constant is found by passing a known current and measuring the deflection.
The sensitivity of a moving coil galvanometer is defined as θ/i. For large sensitivity, the filed B should be large. Soft iron core increases magnetic field.
Deflection magnetometer.
The instrument has a small compass needle pivoted at the centre of a graduated circular scale. This arrangement is kept in a wooden frame having two long arms having metre scales. We can find M/ BH ratio using this apparatus. M is the magnetic moment of a magnet and BH is the horizontal component of the earth’s magnetic field. This measurement can be made in Tan A position or Tan B position of the magnetometer.
Tan A position
The arms of the magnetometer are kept along the magnetic east-west direction. This direction is perpendicular to the direction of the needle. The magnet is kept on one of the arms parallel to its length. Hence the compass needle is in end-on position of the bar magnet. (End on position is a position on the magnetic axis of a bar magnet)
The magnetic field due to the bar magnet at the site of the needle is
B = (µ0/4 π)[(2Md)/(d² – l²)²]
Where l = length of the bar magnet and d = distance of the centre of the bar magnet from the centre of the compass needle.
From tangent law B = BH tan θ
Hence M/ BHt = (4π/µ0) [d² – l²)²/2d] tan θ
Tan-B position
In this position, the arms of the magnetometer are kept in the magnetic north-south direction. The bar magnet is placed on one of the arms symmetrically at right angles to the wooden arm.
In this position the compass centre point is in broadside-on position of the bar magnet.
Therefore the magnetic field due to the bar magnet at the compass centre point due to the magnet is
B = (µ0/4 π)[M/(d² + l²)3/2 ]
Using the tangent law B = BH tan θ
Hence M/ BH = (4π/µ0)[(d² + l²)3/2 ]tan θ
Applications of the deflection magnetometer
1. Comparison of the magnetic moments two magnets. If the magnetic moments of the two magnets are M1 and M2, then we can find M1/ BH and M2/ BH and then can find M1/M2.
Null method of finding M1/M2
In this method, the two magnets are placed at the opposite ends of the wooden legs of the deflection magnetometer and the distances of the magnets are adjusted on wooden arms so that deflection of the needle is zero. That means magnetic fields produced by the magnets cancel each other.
In Tan A position
B = (µ0/4 π)[(2M1d1)/(d1² – l1²)²] = (µ0/4 π)[(2M2d2)/(d2² – l2²)²]
M1/M2 = (d2/d1)[(d1² – l1²)²/)/(d2² – l2²)²]
In tan B position
B = (µ0/4 π)[M1/(d1² + l1²)3/2 ] = (µ0/4 π)[M2/(d2² + l2²)3/2 ]
M1/M2 = (d1² + l1²)3/2 /(d2² + l2²)3/2
2. The horizontal components of earth’s magnetic field at two places can be done. M/ BH1 and M/ BH2 help in this.
3. Verification of inverse law. The magnetic field due to a magnetic pole is inversely proportional to the distance of the point from the pole.
From the M/ BH expression we can find an expression for cot θ
And a graph is plotted between [d² – l²)²/2d] and cot θ for Tan A position readings. We get a straight line
Similarly a graph is plotted between [(d² + l²)3/2 ] and cot θ for readings in Tan B position. We get a straight line.
Oscillation magnetometer
Using oscillation magnetometer, the oscillations of bar magnet and the taken are determined.
T = time per one osciallation = 2 π√(I/MBH)
That gives MBH = 4π²I/T²
I moment of inertia of the bar magnet can be determined from I = W(a²+b²)/12 where W = mass of the magnet, a = geometric length, b = breadth of the magnet.
Determination of M and BH
We can find M/ BH using a deflection magnetometer and M BH using an oscillation magnetometer.
If M/ BH = X abnd M BH= Y
Then M² = XY
And M = √(XY)
BH = √(Y/X)
Gauss’s law for Magnetism
Magnetic field due to magnetic charge is
B = (µ0/4 π)(m/r²)
From the above expression we can derive Gauss’s law for magnetism as
∮B.ds = µ0minside
where ∮B.ds is the magnetic flux and minside is the “net pole strength inside the closed surface. But net magnetic stranegh enclosed by any closer surface is zero.
Hence ∮B.ds = 0
Concept Review Ch.37 Magnetic Properties of Matter
Revision Points
(Revision points help you during the revision process. After you have read the text once or twice, and understood its contents well, revision points help you to study the material quickly and also help you to recollect more details described in the text book)
A current loop has magnetic dipole movement. Hence each electron in an atom has a magnetic moment due to its orbital motion. Besides this, each electron at rest also has a permanent angular momentum, which is called spin angular momentum (The concept was explained in chemistry atomic structure chapter as well as in the Bohr model chapter of physics). This magnetic moment has a fixed magnitude μs = 9.285*10^-24 J/T.
The resultant magnetic moment of an atom is the vector sum of magnetic moment due orbital movement and spin angular momentum.
The magnetic moments of the electrons of an atom have tendency to cancel in pairs. For example, the magnetic moments of a helium atom cancel each other.
In some atoms such a cancellation is not there and the magnetic moment of an atom is not zero. Such atoms can be represented by a magnetic dipole.
In general magnetic moments of atoms are randomly oriented and there is no net magnetic moment in any volume of material that contains several thousand atoms. However, when material is kept in an external magnetic field, torques act on the atomic dipoles and these torques align them parallel to the field. The degree of alignment increases if the strength of the applied field is increased and also if the temperature is decreased. With sufficiently strong field, the alignment is near perfect and we say the material is magnetically saturated.
When the atomic dipoles are aligned, partially or fully, there is a net magnetic moment in the direction of the field in any small volume of the material.
Magnetization vector I is defined as the magnetic moment per unit volume. It is also called the intensity of magnetization of simply magnetization.
There fore I = M/V
Where M = magnetic moment in units ampere-metre²
V = volume
Units of I are ampere/metre
Bar magnet case:
In case of a bar magnet with pole strength of m, length 2l and area of cross section A, magnetic moment is 2ml.
Hence I = M/V = 2ml/A(2l) = m/A
So we see that in the case of bar magnet, intensity of magnetization turns out to be pole strength per unit face area.
Paramagnetism
The tendency to increase the magnetic field due to magnetization of material is called paramagnetism and materials which exhibit this property are called paramagnetic materials.
Ferromagnetic materials
In some materials, the permanent atomic magnetic moments have strong tendency to align themselves without any external field. Because of this tendency, even if a small magnetic field is applied, it gives rise to large magnetization. These materials are called ferromagnetic materials.
Diamagnetism
In many materials, individual atoms do not have net magnetic dipole moment. When such materials are placed in a magnetic field, dipole moments are induced in the atoms by the applied field. The field so induced opposes the original field. Hence the resultant field will be smaller. This phenomenon is called diamagnetism.
All materials are diamagnetic. But in some materials which are paramagnetic and ferromagnetic, diamagnetism will not be shown.
Magnetic Intensity
When a magnetic field is applied to a material, the material gets magnetized. The actual magnetic field inside the material is the sum of the applied magnetic field and the magnetic field due to magnetization of the material.
If B is the resulting magnetic field it will be equal to μ0(H + I)
Where H is magnetizing field intensity, I = Intensity of magnetization of the material.
Units of H are units of I, that ampere/metre
If no material is there B = μ0H
Magnetic susceptibility
For paramagnetic and diamagnetic substances, the intensity of magnetization of a material is directly proportional to the magnetic intensity.
I = χH
The proportionality constant χ is called the susceptibility of the material. Materials with positive χ values paramagnetic and materials with negative χ value are diamagnetic materials.
Permeability
B = μ0(H + I) = μ0(H + χH)
= μ0(1 + χ)H
We can write it as B = μH
μ is called permeability of the material. The permeability of vacuum is μ0 as χ = 0.
μr = μ/μ0 = 1 + χ is called the relative permeability,
Take a solenoid. The magnetic field inside is B0.
When a material is inserted in the solenoid, the magnetic field becomes B and B/ B0 and this will be μr of the material.
Curie’s law
As the temperature is increased, the randomization of individual atomic magnetic moments increases and hence magnetization of a given material for a given applied magnetic intensity decreases. This means that χ decreases as T increases.
Curie’ law states that in the region away from saturation, the susceptibility (χ) of a paramagnetic substance is inversely proportional to the absolute temperature.
Χ = c/T
Where c is a constant called Curie constant.
Curie point: Ferromagnetic material becomes paramagnetic at a certain temperature. This temperature is called Curie point or Curie temperature (Tc).
The susceptibility of such materials after the Curie point follows the Curie’s law with the formula χ = c’/(T - Tc). c' is the constant.
Properties of Dia-, Para-, and Ferromagnetic substances
1. The lines of magnetic field become denser in a paramagnetic or ferromagnetic materials and less dense in diamagnetic materials.
2. The magnetic susceptibility is a small but positive quantity for paramagnetic substances. It is of the order of several thousand for ferromagnetic materials. For diamagnetic material it is small negative quantity.
3. Ferromagnetism is normally found in solids only.
4. A paramagnetic substance is weakly attracted by a magnet. A ferromagnetic substance is strongly attracted by the magnet. Diamagnetic substance is weakly repelled by the magnet.
Hysteresis
A ferromagnetic material when it is magnetized once will not come back to zero when the current becomes zero in the current carrying coil. To reduce it to zero current in the opposite direction must be passed.
As H is increased and then decreased to its original value, the magnetization inside a ferromagnetic material does not return to its original value. This fact is called hysteresis.
Soft Iron and Steel
Soft iron is easily magnetized by a magnetizing field but only a small magnetization is retained when the field is removed. The loss of energy, as the material goes through periodic variations in magnetizing fields is small. Soft iron is suitable for making electromagnets and cores inside current carrying coils to increase the magnetic field.
Steel is more suitable for making permanent magnets. Large field is required to magnetize but the field is retained to a large extent and it is not easily destroyed by stray reverse fields. The coercive force is large.
(Revision points help you during the revision process. After you have read the text once or twice, and understood its contents well, revision points help you to study the material quickly and also help you to recollect more details described in the text book)
A current loop has magnetic dipole movement. Hence each electron in an atom has a magnetic moment due to its orbital motion. Besides this, each electron at rest also has a permanent angular momentum, which is called spin angular momentum (The concept was explained in chemistry atomic structure chapter as well as in the Bohr model chapter of physics). This magnetic moment has a fixed magnitude μs = 9.285*10^-24 J/T.
The resultant magnetic moment of an atom is the vector sum of magnetic moment due orbital movement and spin angular momentum.
The magnetic moments of the electrons of an atom have tendency to cancel in pairs. For example, the magnetic moments of a helium atom cancel each other.
In some atoms such a cancellation is not there and the magnetic moment of an atom is not zero. Such atoms can be represented by a magnetic dipole.
In general magnetic moments of atoms are randomly oriented and there is no net magnetic moment in any volume of material that contains several thousand atoms. However, when material is kept in an external magnetic field, torques act on the atomic dipoles and these torques align them parallel to the field. The degree of alignment increases if the strength of the applied field is increased and also if the temperature is decreased. With sufficiently strong field, the alignment is near perfect and we say the material is magnetically saturated.
When the atomic dipoles are aligned, partially or fully, there is a net magnetic moment in the direction of the field in any small volume of the material.
Magnetization vector I is defined as the magnetic moment per unit volume. It is also called the intensity of magnetization of simply magnetization.
There fore I = M/V
Where M = magnetic moment in units ampere-metre²
V = volume
Units of I are ampere/metre
Bar magnet case:
In case of a bar magnet with pole strength of m, length 2l and area of cross section A, magnetic moment is 2ml.
Hence I = M/V = 2ml/A(2l) = m/A
So we see that in the case of bar magnet, intensity of magnetization turns out to be pole strength per unit face area.
Paramagnetism
The tendency to increase the magnetic field due to magnetization of material is called paramagnetism and materials which exhibit this property are called paramagnetic materials.
Ferromagnetic materials
In some materials, the permanent atomic magnetic moments have strong tendency to align themselves without any external field. Because of this tendency, even if a small magnetic field is applied, it gives rise to large magnetization. These materials are called ferromagnetic materials.
Diamagnetism
In many materials, individual atoms do not have net magnetic dipole moment. When such materials are placed in a magnetic field, dipole moments are induced in the atoms by the applied field. The field so induced opposes the original field. Hence the resultant field will be smaller. This phenomenon is called diamagnetism.
All materials are diamagnetic. But in some materials which are paramagnetic and ferromagnetic, diamagnetism will not be shown.
Magnetic Intensity
When a magnetic field is applied to a material, the material gets magnetized. The actual magnetic field inside the material is the sum of the applied magnetic field and the magnetic field due to magnetization of the material.
If B is the resulting magnetic field it will be equal to μ0(H + I)
Where H is magnetizing field intensity, I = Intensity of magnetization of the material.
Units of H are units of I, that ampere/metre
If no material is there B = μ0H
Magnetic susceptibility
For paramagnetic and diamagnetic substances, the intensity of magnetization of a material is directly proportional to the magnetic intensity.
I = χH
The proportionality constant χ is called the susceptibility of the material. Materials with positive χ values paramagnetic and materials with negative χ value are diamagnetic materials.
Permeability
B = μ0(H + I) = μ0(H + χH)
= μ0(1 + χ)H
We can write it as B = μH
μ is called permeability of the material. The permeability of vacuum is μ0 as χ = 0.
μr = μ/μ0 = 1 + χ is called the relative permeability,
Take a solenoid. The magnetic field inside is B0.
When a material is inserted in the solenoid, the magnetic field becomes B and B/ B0 and this will be μr of the material.
Curie’s law
As the temperature is increased, the randomization of individual atomic magnetic moments increases and hence magnetization of a given material for a given applied magnetic intensity decreases. This means that χ decreases as T increases.
Curie’ law states that in the region away from saturation, the susceptibility (χ) of a paramagnetic substance is inversely proportional to the absolute temperature.
Χ = c/T
Where c is a constant called Curie constant.
Curie point: Ferromagnetic material becomes paramagnetic at a certain temperature. This temperature is called Curie point or Curie temperature (Tc).
The susceptibility of such materials after the Curie point follows the Curie’s law with the formula χ = c’/(T - Tc). c' is the constant.
Properties of Dia-, Para-, and Ferromagnetic substances
1. The lines of magnetic field become denser in a paramagnetic or ferromagnetic materials and less dense in diamagnetic materials.
2. The magnetic susceptibility is a small but positive quantity for paramagnetic substances. It is of the order of several thousand for ferromagnetic materials. For diamagnetic material it is small negative quantity.
3. Ferromagnetism is normally found in solids only.
4. A paramagnetic substance is weakly attracted by a magnet. A ferromagnetic substance is strongly attracted by the magnet. Diamagnetic substance is weakly repelled by the magnet.
Hysteresis
A ferromagnetic material when it is magnetized once will not come back to zero when the current becomes zero in the current carrying coil. To reduce it to zero current in the opposite direction must be passed.
As H is increased and then decreased to its original value, the magnetization inside a ferromagnetic material does not return to its original value. This fact is called hysteresis.
Soft Iron and Steel
Soft iron is easily magnetized by a magnetizing field but only a small magnetization is retained when the field is removed. The loss of energy, as the material goes through periodic variations in magnetizing fields is small. Soft iron is suitable for making electromagnets and cores inside current carrying coils to increase the magnetic field.
Steel is more suitable for making permanent magnets. Large field is required to magnetize but the field is retained to a large extent and it is not easily destroyed by stray reverse fields. The coercive force is large.
Concept Review Ch.38 Electromagnetic Induction
Electromagnetic Induction
Faraday's law of electromagentic induction
Whenver the flux of magnetic field through the area bounded by a closed conducting loop changes, an emf is produced in the loop.
The emf is given by
ε = -dф/dt ... (38.1)
Where ф = ∫Vr(B).Vr(dS) is the flux of the magnetic field through the area.
The SI unit lf magnetic flux is called weber which is equivalent to tesla-metre.
Lenz's law
The direction of the induced current is such that it opposes the change that has induced it.
Origin of EMF
An electric current is established in a conducting wire when an electric field exists in it.
What is an electric field: A charge produces something called an electric field in the space around it and this electric field exerts a force on any charge (except the source charge itself) placed in it.
The flow of charge or movement of charge in response to an electric tends to destroy the field and some external mechanism is needed to maintain the electric field in the wire.
It is the work done per unit charge by this external mechanism that we call emf.
What is the mechanism that produces emf in induced emf?
The flux ∫B.dS can be changed by
a. keeping the magnetic field constant as time passes and moving whole or part of loop.
b. Keeping the loop at rest nad changing the magnetic field
c. Combination of both
Motional EMF
Example of a rod of length l moving through a constant magnetic field.
The free electrons in the rod move with the velocity v with which the rod is moving. As charge is moving in a magnetic field, the magnetic field exerts an average force Fb = qv×B on each free electron. They move in the direction of force and negative charge gets accumulated at one and positive charge gets accumulated at the other end. This charge exerts a force on the free electrons when the force exerted by the magnetic field and the electric field due to charges at the two ends are equal, then there is no resultant force on free electrons.
The potential difference between the ends is then vBl where l is the length of the rod. It is the magnetic force on the mviong free electrons that maintains the potential difference V = vBl and hence produces an emf
Є = vBl.
Induced electric field
If the magnetic field changes with time, it is found that induced current starts in closed loop. But as electrons in the conductor are not moving(random motion of electrons is disregarded) magnetic field cannot exert force. These electrons at rest may be forced to move only by an electric field and hence the conclusion that an electric field appears at t = 0. The presence of a conducting loop is not necessary to have an induced electric field. As long as magnetic field (B) is chaging the induced electric field is present.
The electric field which is responsible for the current is produced by the changing magnetic field and is called the induced electric field. This electric field is nonelectrostatic and nonconservative in nature. The lines of induced electric field are closed curve. There are no starting and terminating points of the field lines.
If E be the induced electric field, the force on a charge placed in the field is qE. The work done per unit charge as the charge moves through dl is ∫E.dl
The emf developed in the loop is therefore
Є = ∫E.dl
According to Faraday’s law Є = -dΦdt = ∫E.dl
Eddy Current
When a solid plate of metal is moving in a region having a magnetic field, current may be induced some circular paths in the surface of the metal. There is no definite conducting loop but the system itself results in locating some loops and current termed as eddy current flows through it. The eddy current flow results in thermal energy and this thermal energy comes at the cost of the kinetic energy of the plate and the plate slows down. This slowing down is called electromagnetic damping. To reduce it, slots are cut in the plate and such an arrangement reduces possible paths of the eddy current considerably.
Self induction
When a current is established in a closed conducting loop, it produces a magnetic field. This magnetic field has its flux through an area bounded by the loop. If the current changes with time, the flux through the loop changes and hence an emf is induced in the loop. This process is called self induction as the emf is induced in the loop by changing the current in the same loop.
Self induced emf = -Ldi/dt
Where L is called self-inductance of the loop and is a constant depending on the geometrical construction of the loop.
Self inductance of a long solenoid
Self inductance of a long solenoid is
L = µ0n² πr² l
Where L = Self inductance
n = number of turns per unit length of the solenoid
r = radius of the solenoid
l = length of the solenoid
So the self-inductance depends only on geometrical factors.
A coil or a solenoid made from a thick wire has negligible resistance but a considerable self-inductance. Such an element is called an ideal inductor and is indicated by a coil symbol.
The self induced emf in a coil opposes the change in the current that has induced it in accordance with the Lenz’s law. If the current in the coil is increasing, induced current will be opposite to the original current. If the current in the coil is decreasing, the induced current will be along the original current trying to stop its decrease.
Growth of Current in an LR circuit
LR circuit has a resistance and an inductance.
Applied emf is Є
Self induced emf is –L(di/dt)
Therefore according to Kirchoff’s loop law Є – L(di/dt) = Ri
That means L(di/dt) = Є – Ri
By solving the differential equation and substituting the initial conditions that at t =0, i = 0 we get the expression for i,
i = i0(1 - e-tR/L)
= i0(1 - e-t/ τ )
where
i = current in the circuit at time t
i0 = Є/R
Є = applied emf
R = resistance of the circuit
L = inductance of the circuit
Writing L/r = τ,
i = i0(1 - e-t/ τ )
τ = L/R = time constant of the LR circuit
L/R has dimensions of time and is called the time constant of the LR circuit.
The current in the circuit gradually rises from t = 0 and attains the maximum value i0 after a long time.
At t = τ, the current is
i = i0(I – 1/e) = 0.63 i0.
The time constant indicates how fast will the current grow. If the time constant is small, the growth happens quickly or it steep. While in principle, it may take infinite time for the current to attain its maximum value, in practice in a small number ot time constants the current reaches almost its maximum value.
Decay of Current in an LR circuit
By a special arrangement after the current stabilizes in a LR circuit, the battery is disconnected and the circuit is completed without the battery. Hence the current which at time t = 0; is i0 starts decreasing with time. Now only induced emf is in the cicuit.
Hence –L(di/dt) = Ri
Solving the differential equation we get
i = i0(1 - e-tR/L)
= i0(1 - e-t/ τ )
where
τ = L/R = time constant of the LR circuit
Current does not fall to zero immediately, it gradually decreases.
At t = τ,
i = i0/e = 0.37 i0
The current reduces to 37% of the initial current in one time constant. If the time constant is small, the fall or decay will be steep.
Energy stored in an inductor
In a capacitor, when it is charged, electric field builds up between its plates and energy is stored in it. Similarly in the case of inductor, due to flow of current through it, magnetic field builds up in it and magnetic energy is strored in it.
The energy stored in the inductor carrying a current i, is
U = ½ Li²
Energy density in magnetic field
For a solenoid L = µ0n² πr² l
The magnetic energy therefore is
U = ½ Li² = µ0n² πr² li²
For the solenoid with radius r, length l and turns n per unit length, carrying current i, magnetic field within it is
B = µ0ni
Hence U can be written as B²V/2µ0
V = the volume enclosed by the solenoid = πr²l
As the field is uniform throughout the volume of the solenoid and zero outside, the energy per unit volume, that is the energy density is
u = U/V = B²/2µ0
Mutual Induction
Mutual induction is induction is due to two closed circuits, with current flowing in one circuit. The current flowing one circuit produces a magnetic field and this field has a flux through the area bounded by the other circuit.
We can write Ф = Mi
Where M is a constant and it is called mutual inductance of the pair of circuits, as current in either of the circuits will give the same flux. If the current i in one circuit changes with time, the flux also changes and an emf is induced in the second circuit. This phenomenon is called mutual induction.
Є = -dФ/dt = -M(di/dt)
Induction Coil
An induction coil is used to produce a large emf from a source of low emf. Ruhmkorff’s induction coil is described in the book. There is a make or break arrangement so that current increases in the primary coil and then suddenly decreases. The change in flux due to the change in current in the primary coil induces a large emf in the secondary coil which is in the coaxial position with the primary coil which itself is wound on a soft core.
From 12 V, emf of the order 50,000 V which can operate a discharge tube can be generated with this arrangement.
There is a capacitor in the primary coil circuit to avoid the sparks when current suddenly drops to zero in the primary circuit.
Faraday's law of electromagentic induction
Whenver the flux of magnetic field through the area bounded by a closed conducting loop changes, an emf is produced in the loop.
The emf is given by
ε = -dф/dt ... (38.1)
Where ф = ∫Vr(B).Vr(dS) is the flux of the magnetic field through the area.
The SI unit lf magnetic flux is called weber which is equivalent to tesla-metre.
Lenz's law
The direction of the induced current is such that it opposes the change that has induced it.
Origin of EMF
An electric current is established in a conducting wire when an electric field exists in it.
What is an electric field: A charge produces something called an electric field in the space around it and this electric field exerts a force on any charge (except the source charge itself) placed in it.
The flow of charge or movement of charge in response to an electric tends to destroy the field and some external mechanism is needed to maintain the electric field in the wire.
It is the work done per unit charge by this external mechanism that we call emf.
What is the mechanism that produces emf in induced emf?
The flux ∫B.dS can be changed by
a. keeping the magnetic field constant as time passes and moving whole or part of loop.
b. Keeping the loop at rest nad changing the magnetic field
c. Combination of both
Motional EMF
Example of a rod of length l moving through a constant magnetic field.
The free electrons in the rod move with the velocity v with which the rod is moving. As charge is moving in a magnetic field, the magnetic field exerts an average force Fb = qv×B on each free electron. They move in the direction of force and negative charge gets accumulated at one and positive charge gets accumulated at the other end. This charge exerts a force on the free electrons when the force exerted by the magnetic field and the electric field due to charges at the two ends are equal, then there is no resultant force on free electrons.
The potential difference between the ends is then vBl where l is the length of the rod. It is the magnetic force on the mviong free electrons that maintains the potential difference V = vBl and hence produces an emf
Є = vBl.
Induced electric field
If the magnetic field changes with time, it is found that induced current starts in closed loop. But as electrons in the conductor are not moving(random motion of electrons is disregarded) magnetic field cannot exert force. These electrons at rest may be forced to move only by an electric field and hence the conclusion that an electric field appears at t = 0. The presence of a conducting loop is not necessary to have an induced electric field. As long as magnetic field (B) is chaging the induced electric field is present.
The electric field which is responsible for the current is produced by the changing magnetic field and is called the induced electric field. This electric field is nonelectrostatic and nonconservative in nature. The lines of induced electric field are closed curve. There are no starting and terminating points of the field lines.
If E be the induced electric field, the force on a charge placed in the field is qE. The work done per unit charge as the charge moves through dl is ∫E.dl
The emf developed in the loop is therefore
Є = ∫E.dl
According to Faraday’s law Є = -dΦdt = ∫E.dl
Eddy Current
When a solid plate of metal is moving in a region having a magnetic field, current may be induced some circular paths in the surface of the metal. There is no definite conducting loop but the system itself results in locating some loops and current termed as eddy current flows through it. The eddy current flow results in thermal energy and this thermal energy comes at the cost of the kinetic energy of the plate and the plate slows down. This slowing down is called electromagnetic damping. To reduce it, slots are cut in the plate and such an arrangement reduces possible paths of the eddy current considerably.
Self induction
When a current is established in a closed conducting loop, it produces a magnetic field. This magnetic field has its flux through an area bounded by the loop. If the current changes with time, the flux through the loop changes and hence an emf is induced in the loop. This process is called self induction as the emf is induced in the loop by changing the current in the same loop.
Self induced emf = -Ldi/dt
Where L is called self-inductance of the loop and is a constant depending on the geometrical construction of the loop.
Self inductance of a long solenoid
Self inductance of a long solenoid is
L = µ0n² πr² l
Where L = Self inductance
n = number of turns per unit length of the solenoid
r = radius of the solenoid
l = length of the solenoid
So the self-inductance depends only on geometrical factors.
A coil or a solenoid made from a thick wire has negligible resistance but a considerable self-inductance. Such an element is called an ideal inductor and is indicated by a coil symbol.
The self induced emf in a coil opposes the change in the current that has induced it in accordance with the Lenz’s law. If the current in the coil is increasing, induced current will be opposite to the original current. If the current in the coil is decreasing, the induced current will be along the original current trying to stop its decrease.
Growth of Current in an LR circuit
LR circuit has a resistance and an inductance.
Applied emf is Є
Self induced emf is –L(di/dt)
Therefore according to Kirchoff’s loop law Є – L(di/dt) = Ri
That means L(di/dt) = Є – Ri
By solving the differential equation and substituting the initial conditions that at t =0, i = 0 we get the expression for i,
i = i0(1 - e-tR/L)
= i0(1 - e-t/ τ )
where
i = current in the circuit at time t
i0 = Є/R
Є = applied emf
R = resistance of the circuit
L = inductance of the circuit
Writing L/r = τ,
i = i0(1 - e-t/ τ )
τ = L/R = time constant of the LR circuit
L/R has dimensions of time and is called the time constant of the LR circuit.
The current in the circuit gradually rises from t = 0 and attains the maximum value i0 after a long time.
At t = τ, the current is
i = i0(I – 1/e) = 0.63 i0.
The time constant indicates how fast will the current grow. If the time constant is small, the growth happens quickly or it steep. While in principle, it may take infinite time for the current to attain its maximum value, in practice in a small number ot time constants the current reaches almost its maximum value.
Decay of Current in an LR circuit
By a special arrangement after the current stabilizes in a LR circuit, the battery is disconnected and the circuit is completed without the battery. Hence the current which at time t = 0; is i0 starts decreasing with time. Now only induced emf is in the cicuit.
Hence –L(di/dt) = Ri
Solving the differential equation we get
i = i0(1 - e-tR/L)
= i0(1 - e-t/ τ )
where
τ = L/R = time constant of the LR circuit
Current does not fall to zero immediately, it gradually decreases.
At t = τ,
i = i0/e = 0.37 i0
The current reduces to 37% of the initial current in one time constant. If the time constant is small, the fall or decay will be steep.
Energy stored in an inductor
In a capacitor, when it is charged, electric field builds up between its plates and energy is stored in it. Similarly in the case of inductor, due to flow of current through it, magnetic field builds up in it and magnetic energy is strored in it.
The energy stored in the inductor carrying a current i, is
U = ½ Li²
Energy density in magnetic field
For a solenoid L = µ0n² πr² l
The magnetic energy therefore is
U = ½ Li² = µ0n² πr² li²
For the solenoid with radius r, length l and turns n per unit length, carrying current i, magnetic field within it is
B = µ0ni
Hence U can be written as B²V/2µ0
V = the volume enclosed by the solenoid = πr²l
As the field is uniform throughout the volume of the solenoid and zero outside, the energy per unit volume, that is the energy density is
u = U/V = B²/2µ0
Mutual Induction
Mutual induction is induction is due to two closed circuits, with current flowing in one circuit. The current flowing one circuit produces a magnetic field and this field has a flux through the area bounded by the other circuit.
We can write Ф = Mi
Where M is a constant and it is called mutual inductance of the pair of circuits, as current in either of the circuits will give the same flux. If the current i in one circuit changes with time, the flux also changes and an emf is induced in the second circuit. This phenomenon is called mutual induction.
Є = -dФ/dt = -M(di/dt)
Induction Coil
An induction coil is used to produce a large emf from a source of low emf. Ruhmkorff’s induction coil is described in the book. There is a make or break arrangement so that current increases in the primary coil and then suddenly decreases. The change in flux due to the change in current in the primary coil induces a large emf in the secondary coil which is in the coaxial position with the primary coil which itself is wound on a soft core.
From 12 V, emf of the order 50,000 V which can operate a discharge tube can be generated with this arrangement.
There is a capacitor in the primary coil circuit to avoid the sparks when current suddenly drops to zero in the primary circuit.
Concept Review Ch.39 Alternating Current
Alternating Current
The current produced by a battery is always in one direction only.
But the electricity that we use in our house is generally alternating current.
If the direction of the current in a resistor or in any other element changes alternately, the current is called an alternating current.
The equation for current, that varies sinusoidally with time, is given by
i = i0sin(ωt+φ)
Where i = current and t = time.
i0 is the maximum value of the current and it is called peak current or the current amplitude.
When we plot the current on graph, the current repeats its value after each time interval T = 2π/ω
Production or Generation of Alternating Current (AC)
Alternating current or AC is produced using an AC generator also called AC dynamo.
It consists of three important parts:
1. a magnet
2. an armature with slip rings and
3. brushes
Magnet may be a permanent magnet or an electromagnet whose poles face each other producing a strong uniform magnetic field between them. It is attached to the body and is called stator.
Armature: It is a coil wound over a soft iron core which is attached or mounted to a shaft that rotates. On this shaft termed as a rotor, at one end there are two rings. These rings are called slip rings. The coil terminal points are connected to the two slip rings.
Brushes: There are two brushes attached to the stator that maintain contact continuously with slip rings and the brushes are connected to the external circuit.
As the armature is rotated in the magnetic field emf is generated in the coil.
Instantaneous and RMS current
i = i0sin(ωt+φ) ... (39.1) defines the instantaneous current at any instant t.
We can define average current as
ī (i bar) = ∫idt/∫dt = 1/T∫idt (integration limits are 0 to t)
Mean square current is average of square of instantaneous current. The square root of mean square current is called root mean square current or rms current. This is also known as the virtual current.
The rms current or the virtual current corresponding to the sinusoidal current i = i0sin(ωt+φ) is
irms = i0/√2
Alternating voltage
An alternating voltage (potential difference) may be written as
V = V0sin(ωt+φ)
This gives instantaneous voltage. The mean voltage V(bar) is zero over a complete cycle. The mean square voltage over a cycle is V0²/2 and the root mean square voltage (rms voltage or virtual voltage) is V0/√2.
Significance of RMS values
A constant voltage Vrms applied across a resistor produces the same thermal energy as that produced by the voltage V = V0sin(ωt+φ).
Phase factor
Current and emf are in general not in phase in an AC circuit.
If the emf is Є = Є0sin ωt
The current will be i = i0sin(ωt+φ)
The constant term φ is called the phase factor.
For purely resistive circuit, φ = 0;
For a purely capacitive circuit, φ = π/2 and
For a purely inductive circuit φ = - π/2.
Choke Coil
Choke coils are used with fluorescent mercury-tube fittings (tube lights) in houses. Choke coil is a coil having large inductance but a small resistance. The choke coil is used to reduce the voltage across the tube or the resistor.
An additional resistor could be used for the same purpose, but a resistor would consume power. The inductance will not consume power.
Hot-wire instruments
To measure alternating currents or voltages, we have to use a property so that the deflection of the moving part depends on i² and not on i. Hot wire instruments work on this principle.
Hot-wire ammeter
A platinum-iridium wire is used as a hot wire, whose rise in temperature is proportional to i².
Hot-wire voltmeter
The construction is similar to ammeter except that a shunt is connected in parallel to hot wire in ammeter and whereas a high resistance is connected in series with the hot wire in voltmeter.
DC dynamo
DC dynamo supplies current in one direction only in the circuit connected to it. A system split cylinders on the armature is used for this purpose and the split cylinders with brushes is called a slip ring commutator.
In this system, the ends of the armature are connected separately to the split cylinder halves C1 and C2. The carbon brushes (B1 and B2) press against two halves. In the rotation for half the time B1 is connected to C1 and for half the time it is connected to B2. Therefore in the external circuit current goes in the same direction. But its magnitude varies. To reduce variation in the current another coil perpendicular to the first one is added to the system in such a way that emf from this coil is maximum when emf from the first coil is minimum and vice versa. The sum of the two contains less oscillations and the current is more nearly constant. The coils can be increased further to reduce the variation.
DC Motor
A motor converts electrical energy into mechanical energy. The motor construction is similar to the DC dynamo. But power is supplied to the armature from a DC dynamo. Because the armature is in magnetic field, torque acts on it and it rotates. As the armature is on a shaft to which load is attached, work is done. As the coil rotates an induced emf e is produced opposite to applied emf Є. If the resistance of the circuit is R, the current at any instant is given by i = (Є-e)/R.
Transformer
A transformer changes the AC voltage applied to the primary coil to a higher or lower value at the output end of the secondary coil. It consists of two coils wound separately on a laminated soft-iron core. The source of alternating voltage is connected to one of the coils named as primary coil. An induced emf appears across the ends of the secondary.
If there N1 turns in the primary and N2 turns in the secondary, and if an alternating emf Є1 is applied across primary, the emf at out end of secondary will be
Є2 = -N2 Є1/N1
The minus sign shows that Є2 is 180° out of phase with Є1.
Current i2 = -N1*i1/N2
The minus sine shows that i2 is 180° out of phase with i1.
Step-up and step down transformer
If N2>N1, the secondary emf Є2 is larger in magnitude than the primary emf Є1. This type of transformer is called a step-up transformer. But the secondary current is less than the primary current. The primary coil has to sustain the high current and hence it is made with thick wire.
If N2 is less than N1, the secondary emf is smaller and it called step down transformer. In this case current is more in secondary, and this coil is to be made from thick wire.
Efficiency of transformer
There is some loss energy in the transformer due to resistances, hysteresis in the core, eddy currents in the core etc. but efficiencies up to 99% are easily achieved. Efficiency is defined as output/input in terms of power.
Transmission of Power
At the electricity generation plant power is stepped up to 66 kV and fed to the transmission lines. In a town or city, the voltage is stepped down to the required value such as 220V. Why is it done that way? As we saw above in step down transformer current is more in secondary and less in primary. As voltage is more in the primary current is less and we lose less in i²Rt losses if ‘i' is less.
The current produced by a battery is always in one direction only.
But the electricity that we use in our house is generally alternating current.
If the direction of the current in a resistor or in any other element changes alternately, the current is called an alternating current.
The equation for current, that varies sinusoidally with time, is given by
i = i0sin(ωt+φ)
Where i = current and t = time.
i0 is the maximum value of the current and it is called peak current or the current amplitude.
When we plot the current on graph, the current repeats its value after each time interval T = 2π/ω
Production or Generation of Alternating Current (AC)
Alternating current or AC is produced using an AC generator also called AC dynamo.
It consists of three important parts:
1. a magnet
2. an armature with slip rings and
3. brushes
Magnet may be a permanent magnet or an electromagnet whose poles face each other producing a strong uniform magnetic field between them. It is attached to the body and is called stator.
Armature: It is a coil wound over a soft iron core which is attached or mounted to a shaft that rotates. On this shaft termed as a rotor, at one end there are two rings. These rings are called slip rings. The coil terminal points are connected to the two slip rings.
Brushes: There are two brushes attached to the stator that maintain contact continuously with slip rings and the brushes are connected to the external circuit.
As the armature is rotated in the magnetic field emf is generated in the coil.
Instantaneous and RMS current
i = i0sin(ωt+φ) ... (39.1) defines the instantaneous current at any instant t.
We can define average current as
ī (i bar) = ∫idt/∫dt = 1/T∫idt (integration limits are 0 to t)
Mean square current is average of square of instantaneous current. The square root of mean square current is called root mean square current or rms current. This is also known as the virtual current.
The rms current or the virtual current corresponding to the sinusoidal current i = i0sin(ωt+φ) is
irms = i0/√2
Alternating voltage
An alternating voltage (potential difference) may be written as
V = V0sin(ωt+φ)
This gives instantaneous voltage. The mean voltage V(bar) is zero over a complete cycle. The mean square voltage over a cycle is V0²/2 and the root mean square voltage (rms voltage or virtual voltage) is V0/√2.
Significance of RMS values
A constant voltage Vrms applied across a resistor produces the same thermal energy as that produced by the voltage V = V0sin(ωt+φ).
Phase factor
Current and emf are in general not in phase in an AC circuit.
If the emf is Є = Є0sin ωt
The current will be i = i0sin(ωt+φ)
The constant term φ is called the phase factor.
For purely resistive circuit, φ = 0;
For a purely capacitive circuit, φ = π/2 and
For a purely inductive circuit φ = - π/2.
Choke Coil
Choke coils are used with fluorescent mercury-tube fittings (tube lights) in houses. Choke coil is a coil having large inductance but a small resistance. The choke coil is used to reduce the voltage across the tube or the resistor.
An additional resistor could be used for the same purpose, but a resistor would consume power. The inductance will not consume power.
Hot-wire instruments
To measure alternating currents or voltages, we have to use a property so that the deflection of the moving part depends on i² and not on i. Hot wire instruments work on this principle.
Hot-wire ammeter
A platinum-iridium wire is used as a hot wire, whose rise in temperature is proportional to i².
Hot-wire voltmeter
The construction is similar to ammeter except that a shunt is connected in parallel to hot wire in ammeter and whereas a high resistance is connected in series with the hot wire in voltmeter.
DC dynamo
DC dynamo supplies current in one direction only in the circuit connected to it. A system split cylinders on the armature is used for this purpose and the split cylinders with brushes is called a slip ring commutator.
In this system, the ends of the armature are connected separately to the split cylinder halves C1 and C2. The carbon brushes (B1 and B2) press against two halves. In the rotation for half the time B1 is connected to C1 and for half the time it is connected to B2. Therefore in the external circuit current goes in the same direction. But its magnitude varies. To reduce variation in the current another coil perpendicular to the first one is added to the system in such a way that emf from this coil is maximum when emf from the first coil is minimum and vice versa. The sum of the two contains less oscillations and the current is more nearly constant. The coils can be increased further to reduce the variation.
DC Motor
A motor converts electrical energy into mechanical energy. The motor construction is similar to the DC dynamo. But power is supplied to the armature from a DC dynamo. Because the armature is in magnetic field, torque acts on it and it rotates. As the armature is on a shaft to which load is attached, work is done. As the coil rotates an induced emf e is produced opposite to applied emf Є. If the resistance of the circuit is R, the current at any instant is given by i = (Є-e)/R.
Transformer
A transformer changes the AC voltage applied to the primary coil to a higher or lower value at the output end of the secondary coil. It consists of two coils wound separately on a laminated soft-iron core. The source of alternating voltage is connected to one of the coils named as primary coil. An induced emf appears across the ends of the secondary.
If there N1 turns in the primary and N2 turns in the secondary, and if an alternating emf Є1 is applied across primary, the emf at out end of secondary will be
Є2 = -N2 Є1/N1
The minus sign shows that Є2 is 180° out of phase with Є1.
Current i2 = -N1*i1/N2
The minus sine shows that i2 is 180° out of phase with i1.
Step-up and step down transformer
If N2>N1, the secondary emf Є2 is larger in magnitude than the primary emf Є1. This type of transformer is called a step-up transformer. But the secondary current is less than the primary current. The primary coil has to sustain the high current and hence it is made with thick wire.
If N2 is less than N1, the secondary emf is smaller and it called step down transformer. In this case current is more in secondary, and this coil is to be made from thick wire.
Efficiency of transformer
There is some loss energy in the transformer due to resistances, hysteresis in the core, eddy currents in the core etc. but efficiencies up to 99% are easily achieved. Efficiency is defined as output/input in terms of power.
Transmission of Power
At the electricity generation plant power is stepped up to 66 kV and fed to the transmission lines. In a town or city, the voltage is stepped down to the required value such as 220V. Why is it done that way? As we saw above in step down transformer current is more in secondary and less in primary. As voltage is more in the primary current is less and we lose less in i²Rt losses if ‘i' is less.
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