Ch. 37 Magnetic Properties of Matter – Formulae
1. Magnetization vector = Magnetic moment per unit volume
I = M/V
2. Magnetic intensity
H = B/µ0 - I .. (2)
Where
H = magnetic intensity
B - resultant magnetic field
I = intensity of magnetization
Magnetic intensity due to a magnetic pole of pole strength m at a distance r from it is
H = m/(4 πr²) …(5)
6. Magnetic susceptibility
I = χH … (6)
Χ is called the susceptibility of the material.
7. Permeability
B = µH … (7)
µ = µ0 (1+χ) is a constant and is called the permeability of the material.
µ0 is the permeability of vacuum.
µr = µ/µ0 = 1+ χ is called the relative permeability of the material.
9. Curie’s law
χ = c/T … (9)
where c = Curie’s constant
For ferromagnetic materials
Χ = c’/(T - Tc)
Where
Tc is the Curie point and
c’ = constant
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Saturday, September 27, 2008
Formula revision Electromagnetic induction - July-Dec Revision
1. Faraday’s law of electromagnetic induction
Є = -dФ/dt … (1)
Where
Є = emf produced
Ф = ∫B.dS = the flux of the magnetic field through the area.
2. i = Є/R = -(1/R) dФ/dt …(2)
where i = current in the circuit
R = resistance of the circuit
3. Є = vBl
Where
Є = emf produced
v = velocity of the conductor
B = magnetic field in which the conductor is moving
l = length of the conductor
4. Induced electric field
∫E.dl = -dФ/dt .. (4)
where
E = induced electric filed due to magnetic field B
5. Self induction
Magnetic field through the area bounded by a current-carrying loop is proportional to the current flowing through it.
Ф = Li … (5)
Where
Ф = ∫B.dS = the flux of the magnetic field through the area.
L = is a constant called the self-inductance of the loop.
i = current through the loop.
6. Self induced EMF
Є = -dФ/dt = -Ldi/dt ….(6)
7. Self inductance of a long solenoid
L = µ0n² πr² l … (7)
8. Growth of current through an LR circuit
i = i0(1 - e-tR/L) … (8)
= i0(1 - e-t/ τ ) … (9)
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
τ = L/R = time constant of the LR circuit
10. Decay of current in a LR circuit
i = i0(1 - e-tR/L) … (10)
= i0(1 - e-t/ τ ) … (11)
i = current in the circuit at time t
i0 = current in the circuit at time t = 0
R = resistance of the circuit
L = inductance of the circuit
τ = L/R = time constant of the LR circuit
12. Energy stored in an inductor
U = ½ Li² … (12)
13. Energy density
u = U/V = B²/2µ0
14. Mutual induction
Ф = Mi … (14)
Where
M = constant called mutual inductance of the given pair of circuits
Є = -Mdi/dt …. (15)
Є = -dФ/dt … (1)
Where
Є = emf produced
Ф = ∫B.dS = the flux of the magnetic field through the area.
2. i = Є/R = -(1/R) dФ/dt …(2)
where i = current in the circuit
R = resistance of the circuit
3. Є = vBl
Where
Є = emf produced
v = velocity of the conductor
B = magnetic field in which the conductor is moving
l = length of the conductor
4. Induced electric field
∫E.dl = -dФ/dt .. (4)
where
E = induced electric filed due to magnetic field B
5. Self induction
Magnetic field through the area bounded by a current-carrying loop is proportional to the current flowing through it.
Ф = Li … (5)
Where
Ф = ∫B.dS = the flux of the magnetic field through the area.
L = is a constant called the self-inductance of the loop.
i = current through the loop.
6. Self induced EMF
Є = -dФ/dt = -Ldi/dt ….(6)
7. Self inductance of a long solenoid
L = µ0n² πr² l … (7)
8. Growth of current through an LR circuit
i = i0(1 - e-tR/L) … (8)
= i0(1 - e-t/ τ ) … (9)
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
τ = L/R = time constant of the LR circuit
10. Decay of current in a LR circuit
i = i0(1 - e-tR/L) … (10)
= i0(1 - e-t/ τ ) … (11)
i = current in the circuit at time t
i0 = current in the circuit at time t = 0
R = resistance of the circuit
L = inductance of the circuit
τ = L/R = time constant of the LR circuit
12. Energy stored in an inductor
U = ½ Li² … (12)
13. Energy density
u = U/V = B²/2µ0
14. Mutual induction
Ф = Mi … (14)
Where
M = constant called mutual inductance of the given pair of circuits
Є = -Mdi/dt …. (15)
Tuesday, September 23, 2008
July Dec Revision - Kinetic theory of gases
Any sample of gas is made of molecules.
The observed behaviour of gas results from the behaviour of its large number of molecules.
Kinetic theory of gases attempts to develop a model of the molecular behaviour which should result in the observed behaviour an ideal gas.
Assumptions of kinetic theory of gases
1. All gases are made of molecules moving randomly in all directions
2. The size of molecule is much smaller than the average separation between the molecules.
3. The molecules exert no force on each other or on the walls of the container except during collision (no atraction force or repulsion force).
4. All collisions between two molecules or between a molecule and a wall are perfectly elastic. Also the time spent during a collision is negligibly small.
5. the molecules obey Newton's laws of motion.
6. When a gas is left for sufficient time in a closed container, it comes to a steady state. The density and the distribution of molecules with different velocities are independent of position, direction and time.
The assumptions are close to the real situations at low densities.
The molecular size is roughly 100 times smaller than the average separation between the molecules at 0.1 atm and room temperature.
The real molecules do exert electric forces on each other but these forces can be neglected as the average separation between molecules is large as compared to their size.
Pressure of an ideal gas
p = (1/3)ρ*Avg(v²) ........ (1)
where
ρ = density of gas = mass per unit area
Avg(v²) = average of the speeds of molecules squared
pV = (1/3)M*Avg(v²) ....... (2)
M = Mass of gas in the closed container
pV = (1/3)nm*Avg(v²) ........ (3)
n = number of molecules of gas in the container
m = mass of each molecule
RMS Speed: The square root of mean square speed is called root-mean-square speed or rms speed.
It is denoted by the symbol vrms
Avg(v²) = (vrms)²
The equation (1) can be written as
p = (1/3)ρ*(vrms)²
Then
vrms) = √[3p/ρ] = √[3pV/M]
Total translational kinetic energy of all the molecules of the gas is
K = Σ (1/2mv² = (1/2)M(vrms)² ... (4)
The average kinetic energy of a molecule = (1/2)m(vrms)²
Then from equation (2)
K = (3/2)pV
according to the kinetic theory of gases, the internal energy of an ideal gas is the same as the total translational kinetic energy of its molecules.
For different kinds of gases, it is not the rms speed but average kinetic energy of individual molecules that has a fixed value at a given temperature.
The heavier molecules move with smaller rms speed and the lighter molecules move with larger rms speed.
All gas laws can be deduced from kinetic theory of gases.
Ideal gas equation
pV = nRT
R = universal gas constant = 8.314 J/mol-L
The average speed of molecules is somewhat less than the rms speed.
Average speed = (Σv)/n = √[8kT/πm]
The observed behaviour of gas results from the behaviour of its large number of molecules.
Kinetic theory of gases attempts to develop a model of the molecular behaviour which should result in the observed behaviour an ideal gas.
Assumptions of kinetic theory of gases
1. All gases are made of molecules moving randomly in all directions
2. The size of molecule is much smaller than the average separation between the molecules.
3. The molecules exert no force on each other or on the walls of the container except during collision (no atraction force or repulsion force).
4. All collisions between two molecules or between a molecule and a wall are perfectly elastic. Also the time spent during a collision is negligibly small.
5. the molecules obey Newton's laws of motion.
6. When a gas is left for sufficient time in a closed container, it comes to a steady state. The density and the distribution of molecules with different velocities are independent of position, direction and time.
The assumptions are close to the real situations at low densities.
The molecular size is roughly 100 times smaller than the average separation between the molecules at 0.1 atm and room temperature.
The real molecules do exert electric forces on each other but these forces can be neglected as the average separation between molecules is large as compared to their size.
Pressure of an ideal gas
p = (1/3)ρ*Avg(v²) ........ (1)
where
ρ = density of gas = mass per unit area
Avg(v²) = average of the speeds of molecules squared
pV = (1/3)M*Avg(v²) ....... (2)
M = Mass of gas in the closed container
pV = (1/3)nm*Avg(v²) ........ (3)
n = number of molecules of gas in the container
m = mass of each molecule
RMS Speed: The square root of mean square speed is called root-mean-square speed or rms speed.
It is denoted by the symbol vrms
Avg(v²) = (vrms)²
The equation (1) can be written as
p = (1/3)ρ*(vrms)²
Then
vrms) = √[3p/ρ] = √[3pV/M]
Total translational kinetic energy of all the molecules of the gas is
K = Σ (1/2mv² = (1/2)M(vrms)² ... (4)
The average kinetic energy of a molecule = (1/2)m(vrms)²
Then from equation (2)
K = (3/2)pV
according to the kinetic theory of gases, the internal energy of an ideal gas is the same as the total translational kinetic energy of its molecules.
For different kinds of gases, it is not the rms speed but average kinetic energy of individual molecules that has a fixed value at a given temperature.
The heavier molecules move with smaller rms speed and the lighter molecules move with larger rms speed.
All gas laws can be deduced from kinetic theory of gases.
Ideal gas equation
pV = nRT
R = universal gas constant = 8.314 J/mol-L
The average speed of molecules is somewhat less than the rms speed.
Average speed = (Σv)/n = √[8kT/πm]
Friday, September 12, 2008
Friction - July Dec Revision
1. Normal force = Nf = Mg
Where
M = mass of the object
g = acceleration due to gravity
2. fk = µk Nf
where
fk = magnitude of kinetic friction
µk = coefficient of kinetic friction
3. fmax = µs Nf
where
fmax = maximum static friction
µs = coefficient of static friction
The actual static friction can be less than maximum static friction if the force applied is less than fmax .
4. On an adjustable inclined plane, a block is kept and the angle is gradually increased so that the block begins to move.
Then fmax = mg sin θ
Nf = mg cos θ
There coefficient of static friction µs
µs = fmax / Nf
= tan θ = h/d
where θ = angle of incline when the block starts moving
m = mass of the block placed on incline
h = height of the incline
d= length of the incline
5. T find kinetic friction the angle of the incline is slightly reduced and the block is made to move with uniform velocity
In this case
µk = coefficient of kinetic friction = tan θ’ = h’/d’
(h is going to decrease and d is going to increase)
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Where
M = mass of the object
g = acceleration due to gravity
2. fk = µk Nf
where
fk = magnitude of kinetic friction
µk = coefficient of kinetic friction
3. fmax = µs Nf
where
fmax = maximum static friction
µs = coefficient of static friction
The actual static friction can be less than maximum static friction if the force applied is less than fmax .
4. On an adjustable inclined plane, a block is kept and the angle is gradually increased so that the block begins to move.
Then fmax = mg sin θ
Nf = mg cos θ
There coefficient of static friction µs
µs = fmax / Nf
= tan θ = h/d
where θ = angle of incline when the block starts moving
m = mass of the block placed on incline
h = height of the incline
d= length of the incline
5. T find kinetic friction the angle of the incline is slightly reduced and the block is made to move with uniform velocity
In this case
µk = coefficient of kinetic friction = tan θ’ = h’/d’
(h is going to decrease and d is going to increase)
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Tuesday, September 2, 2008
Earth is not Strictly an Inertial Frame - July Dec Revision
In H C Verma,in Newton's Laws chapter, it was given.
The earth in not strictly an inertial frame. However we can say that the earth in an inertial frame of reference to a good approximation.
Thus for routine affairs, acceleration (a) = 0 if and only if external force (F) = 0 is true in the earth frame of reference.
This fact was identified and formualated by Newton as first law(Newton's first law).
Is Newton's first law a law?
If we restrict the statement to measurements made from earth frame, this becomes a law.
If we try to universalize the statement to different frames, it becomes a definition of inertial frame.
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The earth in not strictly an inertial frame. However we can say that the earth in an inertial frame of reference to a good approximation.
Thus for routine affairs, acceleration (a) = 0 if and only if external force (F) = 0 is true in the earth frame of reference.
This fact was identified and formualated by Newton as first law(Newton's first law).
Is Newton's first law a law?
If we restrict the statement to measurements made from earth frame, this becomes a law.
If we try to universalize the statement to different frames, it becomes a definition of inertial frame.
Join Orkut community IIT-JEE-Academy for interaction regarding various issues and doubts
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Newton’s Laws of Motion –Study Plan - Session 2
Day 2 Study Plan
5.2 Newton's second law
5.3 Working with Newton's laws
W.O.E. 3 and 4
5.2 Newton's second law
The acceleration of a particle as measured from an inertial frame is given by the (vector) sum of all the forces acting on the particle divided by its mass.
a = F/m or F = ma
Acceleration and force are measured at the same instant. If force becomes zero at an instant, acceleration also becomes zero at the same instant.
5.3 Working with Newton’s First and Second Law
1. Decide the System
We have to assume that forces are acting on a system and the system is at rest or in motion. In this context, the system may be a single particle, a block, a combination of two blocks one kept over the other or two blocks connected by a string etc. But there is a restriction for treating one as a system. All parts of the system should have identical acceleration.
Step 2. Identify the Forces
Once the system is decided, make a list of the forces acting on the system due to all the objects other than the system. Any force applied by the system should not be included in the list of the forces (material from the chapter on forces should help you in deciding various forces exerted by the system and forces exerted by the objects on the system).
Step 3. Make a Free Body Diagram
Represent the system by a point in a separate diagram and draw vectors representing the forces with this point as the common origin
Step 4: Choose axes and Write Equations.
Example 5.2
The example describes a block being pulled by a man with help of a string.
The system under analysis is the block.
It is accelerating in the horizontal direction. So there is net force in the horizontal direction.
It is not accelerating in the vertical direction. Hence net force is vertical direction is zero.
Worked out examples:
3,4,5,
Obective I
3,5, 6,9,10,
Attempt questions in Objective questions (OBJ II)
4
Exercises
1, 2, 3,
5.2 Newton's second law
5.3 Working with Newton's laws
W.O.E. 3 and 4
5.2 Newton's second law
The acceleration of a particle as measured from an inertial frame is given by the (vector) sum of all the forces acting on the particle divided by its mass.
a = F/m or F = ma
Acceleration and force are measured at the same instant. If force becomes zero at an instant, acceleration also becomes zero at the same instant.
5.3 Working with Newton’s First and Second Law
1. Decide the System
We have to assume that forces are acting on a system and the system is at rest or in motion. In this context, the system may be a single particle, a block, a combination of two blocks one kept over the other or two blocks connected by a string etc. But there is a restriction for treating one as a system. All parts of the system should have identical acceleration.
Step 2. Identify the Forces
Once the system is decided, make a list of the forces acting on the system due to all the objects other than the system. Any force applied by the system should not be included in the list of the forces (material from the chapter on forces should help you in deciding various forces exerted by the system and forces exerted by the objects on the system).
Step 3. Make a Free Body Diagram
Represent the system by a point in a separate diagram and draw vectors representing the forces with this point as the common origin
Step 4: Choose axes and Write Equations.
Example 5.2
The example describes a block being pulled by a man with help of a string.
The system under analysis is the block.
It is accelerating in the horizontal direction. So there is net force in the horizontal direction.
It is not accelerating in the vertical direction. Hence net force is vertical direction is zero.
Worked out examples:
3,4,5,
Obective I
3,5, 6,9,10,
Attempt questions in Objective questions (OBJ II)
4
Exercises
1, 2, 3,
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