PHYSICS ELECTROSTASTICS

ELECTROSTATICS

ELECTRIC CHARGE

Charge is the property associated with matter due to which is produces and experiences

electrical and magnetic effects. The study of electrical effects of charge at rest is called

electrostatics.

The strength of particle’s electric interaction with objects around it depends on its

electric charge, which can be either positive or negative. An object with equal amounts

of two kinds of charge is electrically neutral, whereas one with an imbalance is

electrically charged.

In the table given below, if a body in the first column is rubbed against a body in the

second column, the body in first column will acquire positive charge, while that in the

second column will acquire negative charge.

TABLE I

Sl. No First Column Second Column

1. Glass Rod Silk Rod

2. Flannes or cat skin Ebonite rod

3. Woollen cloth Amber

4. Woollen cloth Rubber shoes

5. Woollen cloth Plastics objects

Electric Charge: Electric charge can be written as ne where n is a positive or negative

integer and e is a constant of nature called the elementary charge (approximately 1.60 x

10-19C). Electric charge is conserved, the (algebraic) net charge of any isolated system

cannot be changed.

Regarding charge following points are worth nothing:

(a) Like charges repel each other and unlike charges attract each other.

(b) Charge is a scalar and can be of two types; positive or negative.

(c) Charge is quantized, i.e., the charge on anybody will be some integral multiple of

e, i.e.,

Q - ± ne. where n- 1, 2, 3……………

Charge on anybody can never be (1/3e), 1.5e etc.

(d) The electrostatic unit of charge is stat-coulomb and electromagnetic unit is ab-

coulomb in CGS system. But in SI system the unit of charge is coulomb, I coulomb

=1/10ab-coulomb = 3x109 stat-coulomb.

NOTE: Recently, it has been discovered that elementary particles such as proton or

neutron are composed of quarks having charge (±1/3)e and (±2/3)e. However, as quarks

do not exist in Free State, the quantum of charge is still e.

Example-1: How many electrons are there in one coulomb of negative charge?

Sol: The negative charge is due to presence of excess electrons, since they carry

negative charge. Because an electron has a charge whose magnitude is e = 1.6x10-19 C,

the number of electrons

N=q/e=1.0/1.6x10-19

N=6.25x1018

(5) Unit and dimensional formula

S.I. unit of charge is Ampere x sec = coulomb ©, smaller S.I. units are mC, µC.

C.G.S. unit of charge is Stat coulomb or e.s.u. Electromagnetic unit of charge is ab

coulomb

1C=3x109 stat coulomb = 1/10 ab coulomb.

Dimensional formula [Q]=[AT]

(6) Charge is

Transferable: It can be transferred from one body to

another

Associated with mass: Charge cannot exist without

mass but reverse is not true.

Conserved: It can neither be created nor be destroyed.

Invariant: Independent of velocity of charged particle.

→ →

(7) Electric charge produced electric field (E), magnetic field (B) and electromagnetic

radiations.

→ → →

+ v = 0 + v = constant + v ≠ constant

→ → → →→

E only E and B E, B and radiates energy

(8) Point charge: A finite size body may behave like a point charge if it produces an

inverse square electric field. For example an isolated charged sphere behave like a point

charge at very large distance as well as very small distance close to it’s surface.

(9) Charge on a conductor: Charge given to a conductor always resides on it’s outer

surface. This is way a solid and hollow conducting sphere of same outer radius will hold

maximum equal charge. If surface is uniform the charge distributes uniformly on the

surface and for irregular surface the distribution of charge, i.e., Charge density is not

uniform. It is maximum where the radius of curvature is minimum and vice versa. i.e.,

(1/R). This is why charge leaks from sharp points.

(10) Charge Distribution: It may be of two types

(i) Discrete distribution of charge: A System consisting of ultimate individual

charges

(ii) Continuous distribution of charge: An amount of charge distribute uniformly or

non-uniformly on a body. It is of following three types

(a) Line charge distribution: Charge on a line e.g. charged straight wire,

circular charged ring etc.

Charge = Linear charge density

Length

S.I. unit is C/M

Diamension is [L-1TA]

(b) Surface charge distribution: Charge distributed on surface e.g. plane sheet

of charge, conducting sphere, conducting cylinder of

=Charge = Surface charge density

Area

S.I. Unit is C/m2

(c) Volume charge density: Charge distributes through out the volume of the body

e.g. charge on a dielectric sphere

= charge= Volume charge density

Volume

S.I. Unit is C/m3

Method of Charging:

A body can be charged by following methods.

(1) By Friction: By rubbing two bodies together, both positive and negative charges in

equal amounts appear simultaneously due to transfer of electrons from one body to

the other.

(i) When a glass rod is rubbed with silk, the rod becomes positively charged

while the silk becomes negatively charged. The decrease in the mass of glass

rod is equal to the total mass of electrons lost by it.

(ii) Ebonite on rubbing with wool becomes negatively charged making the wool

positively charged.

(iii) Clouds also get charged by friction.

(iv) A comb moving through dry hair gets electrically charged. It starts attracting

small bits of paper.

(2) By electrostatic induction: If a charged body is brought near an uncharged body,

one side of neutral body (closer to charged body) becomes oppositely charged while the

other side becomes similarly charged.

Induced charge can be lesser or equal to inducing charge (but never greater) and its

maximum value is given by Q’=-Q[1-1/K]

Where Q is the inducing charge and K is the dielectric constant of the material of the

uncharged body. It is also known as specific inductive capacity (SIC) of the medium, or

relative permittivity Er of the medium (relative means with respect to free space)

Different dielectric constants

Medium K Medium K

Vaccum 1 Mica 6

Air 1.0003 Silicon 12

Paraffin Wax 2.1 Germanium 16

Rubber 3 Glycerin 50

Transformer oil 4.5 Water 80

Glass 5-10 Metal

(3) Charging by conduction: Take two conductors, one charged and other uncharged.

Bring the conductors in contact with each other. The charge (whether –ve or +ve) under

its own repulsion will spread over both the conductors. Thus the conductors will be

charged with the same sign. This is called as charging by conduction (through contact).

1 Coulomb = 3 X 109 esu of charge = emu of charge

(1 Faraday = 96500 coulomb, 1 Amp Hr = 3600 coulombs)

The esu of charge is also called Static coloumb (stat. coul.) or frankline (Fr).

= emu of charge = 3X 1010 = C

esu of charge

Frankline (i.e. esu of charge) is the smallest unit of charge, while faraday is largest.

[Remember – Faraday is unit of capacity]

Properties of charge –

1. Like charges repel and opposite charges attract.

Ex. – A (+)ve charge sphere will attract –

Sol. – (i) (-) ve charged

(ii) Neutral

2. Charge is a scalar and can be of two types only viz positive and negative. This is

because it adds algebraically and represents excess or deficiency of electrons.

3. Charge is transferable – If a charged body is put in contact with an uncharged

body, uncharged body becomes charged due to transfer of electrons from one

body to the other. If the charged body is positive and it will withdraw some

electrons from uncharged body and if negative will transfer some of its excess

electrons to the uncharged body.

The process of charge transfer is called ‘Conduction’ and in conduction –

1. The charged body loses some of its charge (which is equal to the charge gained

by uncharged body).

2. The charges on both the books are similar if initially one is charged and other

uncharged.

3. The charge gained by uncharged body is always lesser than initial charge present

on charged body, i.e. whole of the charge cannot be transferred by conduction

from one body to the other. Actually, the flow of charge stops when both acquire

same potential.

Exception:-

Ex: Can ever the whole charge of a body be transferred to the other? If yes how and if

not why?

Ans: Yes, if the charged body is enclosed by a conducting body and connected to it, the

whole charge will be transferred to the conducting body, as charge resides on the outer

surface of a conductor.

4. Charge is invariant – This means that charge like phase is independent of frame

of reference, i.e. charge on a body does not charge whatever be its speed. While

charge density or mass of a body depends on its speed and increases with

increase in speed.

5. Charge is always associated with Mass I.e. charge cannot exist without mass

though mass can exist without charge, so:

a. The particles such as photon or neutrino which have no (rest) mass can

never have a charge (as charge cannot exist without mass).

b. As charge cannot exist without mass, the presence of charge is a

convincing proof of existence of mass.

c. In charging, the mass of a body charges.

6. Charge is conserved – In isolated system, total charge does not charge with time,

though individual charges may charge i.e. charge can neither be created nor

destroyed. It therefore, follows that simultaneously equal quantities of positive

and negative charge can appear or disappear. This is what actually happens in

pair production and anniweatron.

Conservation of charge is also found to hold good in all types of reactions either

chemical, nuclear or decay.

In pair production and anniweatron neither mass nor energy is conserved

separately but (mass + energy) is conserved.

In pair production, presence of nucleus is a must to conserve momentum. In

absence of nucleus, both energy and momentum will not be conserved

simultaneously and the process cannot take place.

7. Accelerated charge radiates energy: Electromagnetic theory has established that

a charged particle at rest produces only electric field in the space surrounding it.

However, if the charged particle is in unaccelerated motion it produces both

electric and magnetic fields but doesnot radiate energy. And if the motion of

charged particle is accelerated it not only produces electric and magnetic fields

but also radiates energy in the space surrounding the charge in the form of

electromagnetic waves.

V V = Constant

E E & B

But no radiation

V ≠ Constant

E, B and radiates energy

8. Similar charges repel each other while dissimilar attract.

The True test of electrification is repulsion and not attraction as attraction may

also take place between a charged and an uncharged body and also between two

similarly charged bodies.

Quest: Can two similarly charged bodies ever attract each other?

Ans: Yes, when the charge on one body (Q) is much greater than that on the

other (q) and they are close enough to each other so that force of attraction

between (Q) and induced charge on the other exceeds the force of repulsion

between (Q) and (q).

If the charges are point, no induction will take place and hence, two similar

point charges can never attract each other.

9. Charge resides on the outer surface of a conductor because like charge repel and

try to get as far as possible from one another and stay at the farthest distance

from each other which is outer surface of the conductor. This is why a solid and a

hollow conductor sphere of same outer radius soap bubble expand on charging.

10. In case of conducting body no doubt charge resides on its outer surface, the

distribution of charge, i.e. charge density is not uniform. It is maximum where the

radius of curvature is minimum and vice-versa, i.e. σ α (1/R). This is why charge

leaks from sharp points.

Proof: a. As conductor is an equipotential surface, i.e. Vs = constant and incase of

spherical conductor

os =4rf0

1Rq

with q = 4rR2v

So, os = 4rf0

1R

4rR2v = Cons tan t

i.e.vR = Cons tan t or vaR1

b. Lighting rods are made up of conductors with one of their ends earthed while

the other sharp and protects a building rom lighting either by neutralizing or

conducting charge of the cloud to the ground.

11. A body can be charged by friction, induction or conduction.

In friction when two bodies are rubbed together, electrons are transferred from

one body to the other. As a result of this one body becomes positively charged

while the other negatively charged. E.g. when a glass rod is rubbed with silk, the

rod becomes positively charged while the silk negatively. However, ebonite on

rubbing with wool becomes negatively charged making the wool positively

charged. Clouds also become charged by friction. In charging by friction in

accordance with conservation of charge, both positive and negative charges in

equal amounts appear simultaneously due to transfer of electrons from one body

to the other.

In case of induction it is worth noting that –

1. Inducing body neither gains nor loses charge.

2. The nature of induced charge is always opposite to that of inducing charge is

always opposite to that of inducing charge.

3. Induced charge can be lesser or equal to inducing charge (but never greater)

and its maximum value is given by –

q’ = -q(1 – 1/K)

Where q is the inducing charge and K is the dielectric constant of the

material of the uncharged body.

4. For metals in electrostatics, k = 3, So, q’ = -q

i.e. in metals induced charge is equal and opposite to inducing charge.

5. Induction takes place only in bodies (either conducting or non-conducting)

and not in particles.

12. If a charged body is brought near a neutral body, the charged body will attract

opposite charge and repel similar charge present in the neutral body. As a result

of this one side of the neutral body becomes positive while the other negative

this process is called “Electrostatic Induction”.

13. Charge can be detected and measured with the help of gold leaf electroscope,

electrometer, voltammeter or ballistic – galvanometer. In case of gold leaf

electroscope –

a. If a charged body is brought near uncharged electroscope, charge on the disc

of electroscope will be opposite to that of body while leaves similar to that of

body and leaves while diverse.

b. If an uncharged electroscope is touched by a charged body, disc and leaves

both acquire charge similar to that of body and leaves will diverse.

c. If electroscope is charged by induction, disc and leaves both will acquire charge

opposite to that of inducting body and leaves will diverse. In fig © electroscope is

charged by induction using a positive charged body.

d. If a charged body is brought near a charged electroscope, the leaves will

further diverse if the charge on the body is similar to that on the electroscope

and will usually converge if opposite. This is how we determine the nature of

charge.

If the induction effect is strong enough leaves offer converging may again

diverse.

Ex: What is the difference between ‘charging by induction and charging by conduction’?

1. In induction the two bodies are ‘close to each other’, while in conduction touch

each other.

2. In induction charge on inducing body remains uncharged while in conduction

charge on charging body charges.

3. In induction induced charge is always opposite in nature to the ‘inducing charge’

while in conduction the charge on the two bodies is always of same nature.

4. In ‘induction’ induced charge can be equal in magnitude to inducing charge but in

conduction ‘charge transferred’ is usually lesser than initial charge present.

COULOMB’s LAW –

Coulomb found that force between two point charges at rest –

1. Varies directly as the magnitude of each charge, i.e. Fα q1 X q2

2. Varies inversely as the square of distance between them, i.e. Fα 1/r2

3. Depends on the nature of medium between the charges.

4. Is always along the line joining the charges.

5. Is attractive if charges are unlike and repulsive if like.

Fα q1 X q2

α 1/r2

Fair =4rf0

1r2

q1 q2

Fmed =4rf0 K

1r2

q1 q2

4rf0

1 =9X109

c2

Km 2

(SI Unit)

K = Constant = Characterizes the medium between the charges and is called

dielectric constant, specific inductive capacity (S.I.C) or relative permittivity and

for vaccum, free space or air its value is taken to be 1.

F12 =4rf0Kr3

q1q2r12|

=4rf1

r3

q1q2r 12

f=f0 K

K =f0

f =fr (relative permittivity or

dielectric cons tan t)

F12 ! q1 K q2

r

F12 = Force on q1 due to q2.

r12

S= Unit vector directed to q1 from q2

f0 =8.85 X 10-12 F/m

The equilibrium of a charged particle under the action of Colombian forces along

can never be stable. This statement is known as Earnshaw’s Theorem.

Ex: A copper atom consists of copper nucleus surrounded by 29 electrons. The atomic

weight of copper is 63.5g/mol. Let us now take two pieces of copper each weighing 10g.

Let us transfer one electron from one piece to another for every 1000 atoms in a piece.

What will be the Coulomb force between the two pieces after the transfer of electrons if

they are 2.10cm apart?

Sol.: No of atoms in 10gm of copper =63.5

6X 1023 X 10 = 9.45 X 1022

Total electron transferred =1000

1X 9.45 X 1022 =9.45 X 1019

q = ne = 9.45 X 1019 X 1.6 X 10-19 = 15.12c

Treating each piece of copper as point charge, electric force between them from

coulomb’s law when they are 10 cm apart

F =(10 X 10-2)2

9 X 109 X (15.2)2

=2.08 X 10143

Ex. (a) Two similar point charges q1 and q2 are placed at a distance r apart in

air. If a dielectric slab of thickness t and dielectric const. K is put

between the charges, calculate the coulomb force of repulsion.

(b) If the thickness of the slab covers half the distance between the

charges, the coulombs force repulsive is reduced 4 : 9. Calculate the in

the ratio dielectric constant of the slab.

Sol. (a)

1 2 1 22 2

0 0

q q q q1 1

4 r ' 4 Kr

r ' r K

If there exists a slab of thickness t and dielectric constant K, the effective

air separation between the charges will be:

F =

1 22

0

q q1

4 r t t K

(b) 0

F 4

F 9

4

9 =

1 22

1 202

0

q q

r rr K

1 2 2

q q14

4 r

K = 4

PRINCIPLE OF SUPERPOSITION OF COULOMB'S LAW:

The resultant force on a test charge is a vector sum of forces due to individual

charges.

res 1 2 3 4F F F F F

Equilibrium of Charge Particle:

If the net force acting on the charge particle is zero that we say that, the charge

particle is in equilibrium.

Equilibrium of a charge

Case (Q1, Q2)

when Q1 and Q2 both are of similar nature

Let |Q1| < |Q2|

For q to be in equilibrium,

Fq = 0,

1 222

KqQ KQ q

x r x

1 2Q Q

x r x

1

1 2

Qx r

Q Q

nearer to charge of smaller magnitude.

Case 2:

Q1 and Q2

when Q1 and Q2 are of opposite nature.

F1 = 12

KQ q

x

F2 =

22

KQ q

r x

1 222

KQ q KQ q

x r x

2

12

2

Q x

Q r x

1

2

Q x

r xQ

2 1 1x Q r Q x Q

1x r Q

x =

1

1 2

r Q

Q Q

From charge of smaller magnitude.

Ex. Two point charges +q and +4q are placed at a distance and apart. Find the

magnitude, sign and location of a third charge which makes the system in

equilibrium.

Force on charge q due to 4q (repulsive) (in the direction of BA).

In order to make A in equilibrium, a negative charge (let q1) be placed between

A and B at a distance x from A.

For the equilibrium of A

12 2

0 0

q q1 q 4q 1

4 l 4 x

q4x2 = l2 q1

Considering the equilibrium of C,

1 12 2

0 0

q 4q q q1 1

4 4 xl x

4x2 = (l – x)2

2x = + (l – x)

x = l

3

2

1 2

l4q

4q3q

l 9

Ex. A pith ball of mass 9 × 10–5 kg carries a charge of 5 µc. What must be the

magnitude and sign of the charge on a pith ball B held 2 cm directly above the

pith ball A, such that the pith ball A remain's stationary?

FAB = m1g

=

1 22

0

q q1

4 AB

92 = 7.84 µC.

Ex. Three identical spheres each having a charge q and radius R, are kept in such

away that each touches the other two. Find the magnitude of the electric

force on any sphere due to other two.

FAB =

2

2

0

1 qBA

4 2R

2

AC 2

0

1 qF CA

4 2R

A ABF 3 F

=

2

0

1 3 9

4 4 R

Ex. Two equally charged identical metal spheres A and B repel each other with a

force 2 × 10–5 N. Another identical uncharged sphere C is touched to B and

then placed at the mid point between A and B. What is the net electric force

on C? [R 1981]

F = 2

2

K q

r

= 2 × 10–5 N

when sphere C touches B, the charge on B, q will distribute equally on B and C

as spheres are identical conductors.

For conductors in contact

V1 = V2

1 2

1 2

q q

r r

1 2r r

1 2q q

Also

q1 + q2 = q

1 2

qq q

2

So sphere C will experience a force

FCA =

2

qKq2

2Fr

2

along AB due to charge on A.

FCB =

2

q qK2 2

Fr

2

along BA due to charge on B.

So the net force on C due to charges on A and B

FC = FCA – FCB

= 2F – F = F

along AB

Ex. Three identical spheres each having a charge q and radius R, are kept in such a

way that each touches the other two. Find the magnitude of the electric force

on any sphere due to other two.

Sol. As for external points a charged sphere behaves as if the whole of its charge

where concentrated at its centre.

Force on A due to B

FAB =

2

0

1 q q

4 2R

=

2

2

2

1 q

4 4R

along BA

Force on A due to C

FAC =

2

0

1 q q

4 2R

=

2

2

0

1 q

4 4R

along CA

FAB = FAC = F

FA = 2 2F F 2FFcos60

= 3 F

FA =

2

0

1 3 q

4 4 R

Ex. Five point charges, each of value +q are placed on five vertices of a regular

hexagon of side LM. What is the magnitude of the force on a point charge of

value –q coulomb placed at the centre of a hexagon?

If there had been a sixth charge +q at the remaining vertex of hexagon force

due to all the six charges on –q at O will be zero (as the forces due to individual

charges will balance each other), i.e.,

RF 0

Now if f is the force due to sixth charge and F due to remaining five charges,

F F 0

i.e., F f

or f = F

=

2

0

1 q q

4 L

F = f

=

2

0

1 q

4 L

Ex. An α-particle passes rapidly, through the exact centre of a hydrogen molecule

moving on a line perpendicular to the inter-nuclear axis. The distance

-particle experience the

maximum force and what is it?

Sol.

FR = 2F cos Ө

along BA

2

2 20

1 2eF4 x a

cos Ө =

1

2 2 2

x

x a

with a = b

2

FR =

2

12 22 20 2

1 2e x24 x a

x a

i.e., FR =

2

32 2 2

0

e x

x a

For FR to be max,

RdF 0dx

x = a

2

= b

2 2 Ans.

Fmax. =

2

2

0

8e

3 3 b

Ex. A point charge q is situated at a distance d from one end of a thin non

conducting rod of length L having a charge Q (uniformly distributed along its

length). Find the magnitude of electric force between the two.

dF = 2

0

1 qdQ

4 x

dQ = QdxL

dF = 2

0

1 qQdx

4 Lx

F =

d L

2

0 d

1 qQ dx

4 L x

=

d L

0 d

1 qQ 1

4 L x

=

0

1 qQ 1 1

4 L d d L

F = 0

1 qQ

4 d d L

Ex. Two identical charged spheres are suspended by strings of equal length the

strings make an angle of 30° with each other. When suspended in a liquid of

density 0.8 gm/cc, the angle remains the same. What is the dielectric constant

of the liquid? (Density of the material of sphere is 1.6 gm/cc).

T cos Ө = mg ….(1)

T sin Ө = F …(2)

tan Ө = F

mg ….(3)

When the balls are suspended in a liquid of density σ and dielectric constant K,

the electric force will become

1

K times, i.e.,

F' =

F

K while

weight mg' = mg – th

= mg – vσ g

mg' =

mg 1

V =

m

tan Ө' = F '

mg'

=

F

Kmg 1

'

K =

1.62

1.6 0.8

Ex. (a) Two similar helium filled spherical balloons tied to a 5 gm weight with

strings and each carrying an electric charge q float in equilibrium as

shown in fig. Find the magnitude of q in eqn assuming that the charge

on each balloon acts as it were concentrated at its centre.

(b) Find the volume of each balloon. Neglect the weight of the unfilled

balloons and assume that the density of air = 0.00129 gm/cc and

density of helium inside the balloon = 0.0002 gm/cc).

Equilibrium of weight

….(1)

Equilibrium of a balloon

F = T sin Ө

Th – mg = T cos Ө

i.e. Vσ g – Vƿ g = T cos Ө …(3)

F = mg

tan2

and mg

Vg2

…(4)

F = 2

q q

x in eqn. units

q = 2 mgx tan

2

and V =

m

2

= 1665 eqn of charge

V =

5

2 0.00129 0.0002

= 55 10

218

= 2294 cc

TRANSLATORY EQUILIBRIUM

When several forces act on a body simultaneously in such a way that the

resultant force on the body is zero, i.e.,

F 0

With iF F

The body is said to be in translatory equilibrium.

1. As if a vector is zero all its components must vanish, i.e. in equilibrium

as–

iF F 0

xF 0,

yF 0

and zF 0

So in equilibrium forces along x-axis must balance each other and same

is true for other directions.

2. As for a body

F 0

means ma 0

or dV

0dt

V const or zero

i.e., if a body is in translatory equilibrium it will be either at rest or in

uniform motion.

If it is at rest, the equilibrium is called static, otherwise dynamic.

3. If the forces are conservative, then as for conservative force

dVF

dr and for equilibrium (F = 0)

So F = dV

0dr

i.e., dV

0dr

i.e., in conservative fields at equilibrium potential energy is optimum,

i.e., in equilibrium potential energy is maximum or minimum or

constant.

4. Dynamic equilibrium types:

Types of dynamic equilibrium.

(a) Stable equilibrium:

If on slight displacement from equilibrium position a body has

tendency to regain its original position, it is said to be in stable

equilibrium. In case of stable equilibrium potential energy is

minimum

2

2

d Vve

dr and so centre of gravity is lowest.

(b) Unstable equilibrium:

If on slight displacement from equilibrium position body moves in

the direction of displacement, the equilibrium is said to be

unstable. In this situation potential energy of the body is

maximum

2

2

d Vnegative

dr and so centre of gravity is

highest.

Examples–

(c) Neutral equilibrium:

If on slight displacement from equilibrium position a body has no

tendency to come back to original position or to move in the

direction of displacement, it is said to be in neutral equilibrium. In

this situation potential energy of the body is constant

2

2

d V0

dr and so centre of gravity remains at constant height.

5. In case of stable equilibrium lesser the potential energy or lower the

centre of gravity, i.e., greater the base area more stable is the

equilibrium.

6. If we plot graphs between F Uand ,r r

at equilibrium F will be zero

while U will be optimum (max or min or constant). If

U = min

i.e., 2

2

d U

dr

= (+) ve, equilibrium is stable.

U = max

i.e., 2

2

d U

dr

= negative, equilibrium is unstable.

and U = constt.

i.e., 2

2

d U

dr = 0,

Equilibrium is neutral.

ELECTRIC FIELD AND POTENTIAL

* The space surrounding an electric charge q in which another charge q0

experiences a (electrostatic) force of attraction, or repulsion, is called the

electric field of the charge q.

* q ԑ Source charge

Point charge a group of point charges continuous distribution of charges

q0 ԑ test charge must be vanishingly small so that it does not modify the

Electric field of the source charge.

Intensity (or strength) of Electric field–

0

FEq

The intensity of electric field at a point in an electric field is the ratio of the

force acting on the test-charge placed at that point to the magnitude of the

test-charge.

If the intensity of electric field E at a point in an electric field be known, then

we can determine the force F acting on a charge q placed at that point by the

following eqn.

F qE

3 1E MLT A

gl gl

gl2 2

21mv2

v 2gl

v

l

= 2gl

l

= 2g

l

2g

l Ans.

Example-6:

An inclined plane making an angle 30° with the horizontal is placed in a

kg and charge 0.01 C is allowed to slide down from rest from a height of 1 m.

If the coefficient of friction is 0.2, find the time it will take the particle to

reach the bottom.

Sol. The different forces on the particle are shown in figure.

From,

N = mg cos 30° + qԑ cos 60°

Friction

f = µN

= µ mg cos 30° + µ ԑ cos 60°

Now the total force F acting along the inclined plane is

F = mg sin 30° – µN – qԑ cos 30°

or F = mg sin 30° – mg cos 30° – µqԑ cos 60° – qԑ cos 30°

Thus acceleration is

or a = F

m

= g sin 30° – µg cos 30°

q qcos 60 cos 30

m m

or a = 9.8 × 0.5 – 0.2 × 9.8

3 0.2 0.01 100 0.01 100 30.5

2 1 1 2

Now, distance travelled in time t is

s = 210 at2

or t =

2 2

a

[As s = 1

2sin 30

]

or =

4

2.237

= 1.345 sec.

Example-7:

In space horizontal Electric field

mgE

q exist as shown in figure and a

mass m attached at the end of a light rod. If mass m is released from the

position shown in figure find the angular velocity of the rod when it passes

through the bottom most position

(A) g

l (B)

2g

l

(C) 3g

l (D)

5g

l

Sol. According to work energy theorem:

w = ∆T

WE + Wg = 21mv 02

….(1)

WE = qE l sinӨ ,

Wg = mg (l – l cos Ө)

– l cos Ө)

= 21mv

2 from eqn. (1)

mg l sin Ө + mg l – mg l cos Ө

= 21mv

2

mgQE

q

Electric Lines of Force:

Faraday gave a new approach for representation of electric field in the form of electric

lines of force. Electric lines of force are graphical representation of

electric field. “An electric line of force is an imaginary line or curve drawn

through a region of space so that its tangent at any point is in the

direction of the electric field vector at that point.”

This model of electric field has the following characteristics:

(i) Electric lines of force are originated from positive charge and terminal into

negative charge.

(ii) The number of electric lines of force originates from a point charge q is q/ԑ0.

Electric lines of force may be fraction.

(iii) The number of lines per unit area that pass through a surface perpendicular to

the electric field lines is proportional to the strength of field in that region.

(iv) No electric lines of force cross each other. If two electric lines of force cross

each other, it means electric field has two directions at the point of

cross. This is not physically possible.

(v) Electric lines of force for two equal positive point charges are said to

have rotational symmetry about the axis joining the charges.

(vi) Electric lines of force for point positive charge and a nearby negative

point charge that are equal in magnitude are said to have rotational

symmetry about an axis passing through both charges in the plane of the

page.

(vii) Electric dipole Electric lines of force due to infinitely large sheet of

positive charge is normal to the sheet.

(viii) No electrostatic lines of force are present inside a conductor. Also

electric lines of force are perpendicular to the surface of conductor. For

example if a conducting sphere is placed in a region where uniform

electric field is present, then induced charges are developed on the

sphere.

(ix) If a charged particle is released from rest in region where only uniform

electric field is present, then charged particle move along an electric line

of force. But if charged particle has initial velocity, then the charged

particle may or may not follow the electric lines of force.

(x) Electric lines of force inside the parallel plate capacitor is uniform. It

shows that field inside the parallel plate capacitor is uniform. But at the

edge of plates, electric lines of force are curved. It shows electric lines of

force at the edge of plates is non-uniform This is known as fringing

effect.

If the size of plates are infinitely large, then fringing effect can be

neglected.

(xi) If a metallic plate is introduced between plates of a charged capacitor,

then electric lines of force can be discontinuous.

(xii) If a dielectric plate is introduced between plates of a charged capacitor,

then number of lines of forces in dielectric is lesser than that in case of

vacuum space.

(xiii) Electrostatics electric lines of force can never be closed loops, as a line

can never start and end on the same charge. Also if a line of force is a

closed curve, work done round a closed path will not be zero and electric

field will not remain conservative.

(xiv) Lines of force have tendency to contract longitudinally like a stretched

elastic string producing attraction between opposite charges and repel

each other laterally resulting in, repulsion between similar charges and

edge-effect (curving of lines of force near the edges of a charged

conductor).

ELECTRIC POTENTIAL AND ELECTRIC POTENTIAL DIFFERENCE:

Electric Potential:

"Electric potential at any point in a electric field is equal to the ratio of the work

done in bringing a test charge from infinity to that point, to the value of test

charge."

Suppose, W be the work required in bringing a test charge q0 from infinity to a

point b against the repulsive force F acting on it, then potential at the point b is

bb

0

WV

q

Since, W and q0 both are scalar quantities; the potential is also a scalar quantity.

Electric Potential Difference:

The potential difference between two points in an electric field is equal to the

ratio of work done in moving a test charge from one point to the other, to the

value of test charge. Suppose W work be done in bringing a small test charge q0

from the point a to a point b against the repulsive force acting on it, then

potential difference between the points is

a bb a

0

WV V

q

Obviously, potential difference is also a scalar quantity.

IMPORTANT FEATURES

1. Electric potential due to a point charge q:

From the definition of potential,

0

UV

q

=

0

0

0

qq1.

4 r

q

or V = 0

1 q.

4 r

Here, r is the distance from the point charge q to be point at which the

potential is evaluated.

If q is positive, the potential that it produces is positive at all points; if q

is negative, it produces a potential that is negative everywhere. In either

case, V is equal to zero at r =3 .

2. Electric potential due to a system of charges:

Just as the electric field due to a collection of point charges is the vector

sum of the fields produced by each charge, the electric potential due to a

collection of point charges is the scalar sum of the potentials due to each

charge.

i

i0 i

q1V

4 r

3. In the equation

i

i0 i

q1V ,

4 r if the whole charge is at equal

distance r0 from the point where V is to be evaluated, then we can write,

V =

net

0 0

q1.

4 r

Where qnet is the algebraic sum of all the charges of which the system is

made.

Example-11:

In a regular polygon of n sides, each corner is at a distance r from the centre.

Identical charges are placed at (n – 1) corners. At the centre, the intensity is E

and the potential is V. The ratio V

E has magnitude.

(A) rn (B) r(n – 1)

(C) n 1

r (D)

r n 1

n

Sol. E = 2

0

q

4 r

and

0

n 1 qv

4 r

0

2

0

n 1 q

4 rv

qE

4 r

= (n – r)

TABLE : Electric Potential of Various Systems

S.No. First Column Second Column

1. Isolated charge

0

qV

4 r

2. A ring of charge

2 20

q qE4 R x

3. A disc of charge

2 2

0

E R x x2

4. Infinite sheet of charge

Not defined

5. Infinitely long line of charge Not defined

6. Finite line of charge

0

sec tanV ln

4 sec tan

7. Charged spherical shell

(a) Inside 0 < r < R

0

qV

4 R

(b) Outside r > R

0

qV

4 r

8. Solid sphere of charge

(a) Inside 0 < r < R

2 2

2

0

R rE 36 R

(b) Outside r > R

0

qV

4 r