Chapter 2

Electric Fields

An electric field exists in a region if electrical forces are exerted on charged bodies in that

region. The direction of an electric field at a point is the direction in which a small positive

charge would move (under the influence of the field) if placed at that point.

The electric field intensity or electric field strength (E) at a point is defined as the

force exerted by the field on a unit charge placed at that point. (Unit = NC−1 = V m−1)

It follows from the definition of electric field intensity that the force, F , exerted on a

charge, Q, at a point where the field intensity is E is given by

F = EQ (2.1)

2.0.6 Field Intensity Due to a Point Charge

Figure 2.1: Electric Field Strength/Intensity

In Figure 2.1, the force, on the test charge, Q0, due to the point charge, Q, in a medium of

permittivity, ε , is given by

9

10 CHAPTER 2. ELECTRIC FIELDS

F = k|Q| |Q0|

r2 (2.2)

By definition, electric field intensity is force per unit charge and therefore the field

intensity, E, at the site of Q0 is given by

E = F/Q0

Therefore,In Equ.

2.3 only

put the

mag-

nitude

of the

charge

and NOT

the sign.

E =1

4πε0

|Q|r2 (2.3)

2.0.7 Electric Field Lines

Electric field lines are imagined as lines pointing in the same direction as the electric field

vector at any point. These are their properties:

1. The electric field vector E is always tangent to the electric field line at any point.

2. The area density of lines through a plane area perpendicular to the field lines is pro-

portional to the strength of the field upon that area. Thus, E is large when the field

lines are close together.

When drawing electric field lines, follow these rules:

1. Lines begin at a positive charge and terminate at a negative charge, except when there

is an excess of charge, in which case they terminate at infinity.

2. The number of lines drawn must be proportional to the magnitude of the charge.

3. No lines can cross.

Note that these drawings are just visual representation; although the electric field is quan-

tized in charge, it is not quantized in position.

11

Figure 2.2: Field Lines

Figure 2.3: Electric field patterns

12 CHAPTER 2. ELECTRIC FIELDS

In summary

Our observations of electric fields caused by charges and their properties:

• There are two different kinds of charges, called positive and negative. Like charges

attract; opposite charges repel each other.

• The force of attraction between two charges varies with the inverse square of their

separation.

• Charge is conserved.

• Charge is quantized.

Chapter 3

Electric Potential

Information about the electric field may be given by stating the field strength at any point;

alternatively the potential can be quoted.

(A) Work and energy

Work is done when the point of application of a force (or a component of it) undergoes a

displacement in its own direction. The product of the force (or its component) F and the

displacement s is taken as a measure of the work done W , i.e. W = F × s. When F is in

newtons and s in metres, W is in newton-metres, or joules.

If a body A exerts a force on body B and work is done, a transfer of energy occurs which

is measured by the work done. So, if we raise a mass m through a vertical height h, the work

done W by the force we apply (i.e. by mg) is W = mgh (assuming the earth’s gravitational

field strength g is constant). The energy transfer is mgh and we consider that the system

gains and stores that amount of gravitational potential energy in its gravitational field. This

energy is obtained from the transfer of chemical energy by our muscular activity. When

the mass falls the system loses gravitational potential energy and, neglecting air resistance,

there is a transfer of kinetic energy to the mass equal to the work done by gravity.

13

14 CHAPTER 3. ELECTRIC POTENTIAL

(B) Meaning of potential

A charge in an electric field experiences a force and if it moves, work will, in general, be

done. If a positive charge is moved from A to B in a direction opposite to that of the field

E, Figure 3.1a, an external agent has to do work against the forces of the field and energy

has to be supplied. As a result, the system (of the charge in the field) gains an amount of

electrical potential energy equal to the work done. This is analogous to a mass being raised

in the earth’s gravitational field g, Figure 3.1b. When the charge is allowed to return from

B to A, work is done by the forces of the field and the electrical potential energy previously

gained by the system is lost. If, for example, the motion is in a vacuum, an equivalent

amount of kinetic energy is transferred to the charge.

Figure 3.1: Analogy between electrical and gravitational potential energy.

In general, the potential energy associated with a charge at a point in an electric field

depends on the location of the point and the magnitude of the charge (since the force acting

depends on the charge, i.e. F = QE). Therefore if we state the magnitude of the charge we

can describe an electric field in terms of the potential energies of that charge at different

points. A unit positive charge is chosen and the change in potential energy which occurs

when a charge is moved from one point to another is called the change of potential of the

field itself.

15

Hence the potential at point B in Figure 3.1 exceeds that at A by the energy needed to

take unit positive charge from A to B. To be strictly accurate, however, we should refer to

the energy needed per unit charge when a very small charge moves from one point to the

other since the introduction of a unit charge would in general modify the field.

If for theoretical purposes we select as the zero of potential the potential at an infinite

distance from any electric charges, potential can be defined as follows.

The potential at a point is a field is defined as the energy required to move unit

positive charge from infinity to the point.

It is always assumed that the charge does not affect the field. The choice of the zero of

potential is purely arbitrary and although infinity may be a few hundred metres in some

cases, in atomic physics where distances of 10−10m are involved it need only be a very

small distance away from the charge responsible for the field.

Potential is a property of a point in a field and is a scalar since it deals with a quantity

of work done or potential energy per unit charge. The symbol for potential is V and the

unit is the joule per coulomb (JC−1) or the volt (V ). Just as a mass moves from a point

of higher gravitational potential to one of lower potential (i.e. it falls towards the earth’s

surface), so a positive charge is urged by an electric field to move from a point of higher

electric potential to one of lower potential. Negative charges move in the opposite direction

if free to do so.

(C) Potential and field strength compared

When describing a field, potential is usually a more useful quantity than field strength

because, being a scalar, it can be added directly when more than one field is concerned.

Field strength is a vector and addition (by the parallelogram law) is more complex. Also,

it is often more important to know what energy changes occur (rather than what forces act)

when charges move in a field and these are readily calculated if potentials are known.

16 CHAPTER 3. ELECTRIC POTENTIAL

Potential due to a point charge

The potential at a point A in the field of, and distance r from, an isolated point charge +Q

situated at O in a medium of permittivity ε , Fig. 3.2, can be calculated. Imagine that a very

small point charge +Q0 is moved by an external agent from C, distance x from A, through

a very small distance δx to B without affecting the field due to +Q.

Figure 3.2:

Assuming the repulsive force on +Q0 due to the field remains constant over δx, the

work done δW by the external agent over δx against the force of the field is

δW = F (−δx) (3.1)

(The minus sign is necessary because the displacement δx is in the opposite direction

to that in which F acts.)

By Coulomb’s law

F =1

4πε

QQ0

x2 (3.2)

Therefore,

δW =1

4πε

QQ0

x2 (−δx) (3.3)

The total work done W in bringing the charge from +Q0 from infinity (where x = ∞) to

17

the point A, a distance r from O (where x = r).

W =−QQ0

4πε

∫ r

∞

(dxx2

)=−QQ0

4πε

[−1x

]r

∞

(3.4)

W =QQ0

4πε· 1

r(3.5)

The potential V at A is the energy needed to move unit positive charge from infinity to

A.

V =WQ0

(3.6)

So, the potential V at a distance r from a point charge Q in a medium of permittivity ε

is given by

V =1

4πε· Q

r(3.7)

Note that

• Potential is a scalar quantity and therefore the potential at a point due to a number of

point charges is the algebraic sum of the (seeparate) potentials due to each charge.

• The potential due to a positive charge is positive and that due to a negative charge is

negative.

18 CHAPTER 3. ELECTRIC POTENTIAL

Equipotentials

Figure 3.3: Equipotential surfaces (in two dimensions) for (a) a point charge and (b) parallelplates

All points in a field that have the same potential can be imagined as lying on a surface -

called an equipotential surface. When a charge moves on such a surface no energy transfer

occurs and no work is done. The force due to the field must therefore act at right angles to

the equipotential surface at any point and so equipotential surfaces and field lines always

intersect at right angles.

S. S. Calbio http://www.izifundo.weebly.com 2015

BISHOP ANSTEY HIGH SCHOOL & TRINITY COLLEGE EAST

SIXTH FORM

CXC CAPE PHYSICS, UNIT 2

Ms. S. S. CALBIO – Homework #2

Due: Tuesday 8th September 2015 @ 8:00am

Electric Field Strength/ Intensity & Potential

1) Find (a) the potential, (b) the electric field intensity at a point C on a line ABC in a

vacuum where AB=BC=5.0cm given that there are point charges of 6.0μC and -4.0 μC at A

and B respectively. State the direction of the field. (Assume 1/4πε0= 9 x 109mF-1)

2) ABC is an equilateral triangle of side 4.0cm in a vacuum. There are point charges of 8.0

μC at A and B. Find (a) the potential, (b) the electric field intensity at C. State the direction

of the field. (Assume 1/4πε0= 9 x 109mF-1)

3)

a. For each of the following, state whether it is a scalar or a vector and give an

appropriate unit:

i. Electric potential

ii. Electric field strength

b. Points A and B are 0.10m apart. A point charge of +3.0 x 10-9C is placed at A

and a point charge of -1.0 x 10-9 C is placed at B.

i. X is the point on the straight line through A and B, between A and B,

where the electric potential is zero. Calculate the distance AX.

ii. Show on a diagram the approximate position of a point, Y, on the

straight line through A and B where the electric field strength is zero.

Explain your reasoning, but no calculation is expected.