Concept review Ch. 29 Electric Field and Potential

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

E_r = [1/4πε₀](2p cos θ)/r³

E_θ = [1/4πε₀](p sin θ)/ r³

Resultant electric field at P = E= √ (E_r²+ E_θ²)

= [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 = E_θ/ E_r = 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.

Study guide H C Verma JEE Physics Ch. 29 ELECTRIC FIELD AND POTENTIAL

Syllabus

Coulomb’s law; Electric field and potential; Electrical potential energy of a system of point charges and of electrical dipoles in a uniform electrostatic field; Electric field lines; Flux of electric field;
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Verma - Topics

29.1 What is electric charge?
29.2 Coulomb's law
29.3 electric field
29.4 Lines of electric force
29.5 Electric potential energy
29.6 Electric potential
29.7 Electric potential due to a point charge
29.8 Relation between electric field and potential
29.9 Electric dipole
29.10 Torque on an electric dipole placed in an lectric field
29.11 Potential energy of a diple placed in a uniform electric field
29.12 Conductors, insulators nad semiconductors
29.13 The electric field inside a conductor

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Study Plan

Day 1

29.1 What is electric charge?
29.2 Coulomb's law

Day 2
29.3 electric field
Ex. 29.1, 29.2
29.4 Lines of electric force
Worked out examples 1,2

Day 3

29.5 Electric potential energy
Ex. 29.3
29.6 Electric potential
Ex. 29.4
29.7 Electric potential due to a point charge
Ex. 29.4
WOE 3 to 5

Day 4

29.8 Relation between electric field and potential
Ex. 29.5
WOE 5 to 10


Day 5
29.9 Electric dipole
29.10 Torque on an electric dipole placed in an lectric field
29.11 Potential energy of a diple placed in a uniform electric field


Day 6
29.12 Conductors, insulators nad semiconductors
29.13 The electric field inside a conductor
WOE 11 to 15

Day 7
WOE 17 to 19

Day 8

Exercises 1 to 10

Day 9
Exercises 11 to 20

Day 10
Exercises 21 to 30

Day 11
Exercises 31 to 35

Day 12
Exercises 36 to 40

Day 13
Exercises 41 to 45

Day 14
Exercises 46 to 50

Day 15
Exercises 51 to 55

Day 16
Exercises 56 to 60

Day 17
Exercises 61 to 65

Day 18
Exercises 66 to 70

Day 19
Exercises 71 to 75

Day 20
Exercises Objectve I 1 to 9 and Objective II 1 to 8

Special task: Questions for short answer, concept review and formual review





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Introduction

The credit for the discovery of electricity goes to Thales of Miletus, regarded as one of the seven wise men of Greece. Around 600 B.C., Thales discovered that when amber is rubbed with wool, it acquires the property of attracting light objects such as straw, wood shavings etc. The investigation of this phenomenon remained in abeyance for over 2000 years.

Near the end of the 16th century A.D., william Gilbert, a physician to Queen Elizabeth I, found that like amber, there are several othe substances which, when rubbed with suitable substances, exhibit the property of attracting light objects. For example, a glass rod rubbed with silk attracts light objects such as paper, cork etc. Similarly an ebonite rod when rubbed with fur attracts light objects. As this phenomenon was first observed with amber and amber is called electron in Greek, Gilbert name the phenomenon as electricity.

He proposed that a body upon rubbing becomes eletrified or charged and acquires the property of attracting light objects. Such a body is said to have acquired an electric charge.

Subsequently, it was observed that (i) two ebonite rods rubbed with fur repel each other and; (ii) two glass rods rubbed with silk repel each other, while (iii) an ebonite rod rubbed with fur attracts a glass rod rubbed with silk.
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Concepts covered

The chapter may seem to be a difficult chapter for the first reading.

29.1 What is electric charge is a simple one.

29.2 Coulomb's law - this topic is also straight forward. Charges have forces of attraction or repulsion among them.

Through experimental data, Coulomb proposed the law that gave the formual for finding the force exerted by a charged particle on the other charged particle.

F = kq1q2/r² ... (29.1)

q1, q2 are the charges on particles
r is the separation or distance between them and
k is a constant.

k in SI units is 8.98755 * 10^9 N-m²/C²

29.3 Electric Field

A charge produces an electric field in the space around it and this electric field exerts a force on any other charge placed it.

The field takes finite time to propagate. If a charge is displaced from its position, the field at a distance r will change after a time t = r/c, where c is the speed of light.

Intensity of electric field: is defined in this way. If a charge q is brought into the electric field and if it experiences an electric Vector F (Vr(F)), we define the intensity of electric field at the given point where the charge q is there as

Vr(E) = Vr(F)/q

Vr(E) = Vector of electric field

29.4 Lines of force - concept is similar to magnetic lines of force

29.5 Electric potential energy - this topic may require some time to think over and grasp.

29.6 Electric Potential - Electric field is a vector quantity. Electric potential is a scalar quantity. This concept follows from the concept of electric potential energy.

29.7 This topics gives expressesions for electric potential for a point charge and a system having number of point charges. Follows from the topic above

29.8 This section gives the relationship between the scalar quantity potential and vector quantity field. spend some time to grasp the concepts once again.

29.9 Electric dipole: You know by now dipole. Some of the chemical compounds have dipole. In this section, electric dipole moment is defined. It is a vector quantity.

Vr(p) = q*Vr(d)
p = electric dipole moment
q = charge one positive and one negative
d = distance between charges which is small. vector gives the direction from negative charge to positive charge.

There is expression for electric potential due to a dipole. If you are already comfortable with the concept of electric potential (scalar quantity) this topic will be easy for you.

Then follows discussion of electric field (vector quantity)

29.10 gives the expression for torque experienced by dipole when placed in an electric field.

29.11 discusses potential energy of the dipole whne placed in a uniform electric field.

29.12 describes why some materials are conductors, insulators and semiconductors. The conductors have large number of free electrons.

29.13 informs us that inside a conductor the electric field zero. The electrons shift to one side of the conductor and therefore once face of it becomes negatively charged and the other face positively charged.

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Audiovisual lectures

Lesson 30: Electric Charges and Coulomb's Law www.curriki.org/nroc/Introductory_Physics_2/lesson30/Container.html


Lesson 31: Electric Fields
www.curriki.org/nroc/Introductory_Physics_2/lesson31/Container.html

Lesson 32: Electric Potential
www.curriki.org/nroc/Introductory_Physics_2/lesson32/Container.html

Chapter 11: Conductors and Capacitors
Lesson 33: Electrostatics with Conductors
www.curriki.org/nroc/Introductory_Physics_2/lesson33/Container.html



Websites

http://www.colorado.edu/physics/phys1120/phys1120_fa07/notes/notes/Knight25_coul_lect.pdf

http://www.colorado.edu/physics/phys2020/phys2020_fa01///notes/lecture_notes/CH16/ch16.pdf
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JEE Question 2007 Paper II

Positive and negative point charges of equal magnitude are kept at (0,0,a/2)and (0,0,-a/2) respectively. The work done by the electric field when another positive point charge is moved form (-a,0,0) to (0,a,0) is

(A) positive
(B) negative
(C) zero
(D) depends on the path connecting the initial and final positions

Correct Choice: C
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For more questions and answers on the topics related to electricity visit
http://iit-jee-physics-atps.blogspot.com/2008/03/jee-past-objective-questions.html
Posting started on 23.3.2008