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ELECTROSTATICS

Electrostatics is the branch of electromagnetism where we study chargesat rest. It actually means that either they are at rest or moving with constantvelocity.

Electric Charge

It is the inherent property of certain fundamental particles. It accompaniesthem whereever they exist. Commonly known charged particles are protonand electron. The charge of a proton is taken as positive and that of electronis taken as negative. It is represented by symbol e.

e = 1.6 × 10–19 coulomb

Charge of proton = +e

Charge of electron = –e

Positive and negative sign were arbitrarily assigned by Benzamin Franklin.This does not mean that charge of proton is greater than charge of electron.

Properties of Electric Charge

(1) Charges interact with each other i.e., they exert force on each other.Like charges do not like (repel) each other while unlike charges like eachother (attract).

(2) Charge is of two kind : Positive and negative.

(3) Total charge of an isolated system is conserved (Consevation ofcharge)

(4) Charge is quantised

(5) Charge can be transferred : Charge can be transferred from one bodyto other. This occurs due to transfer of electrons from one body to other.One of the common example of transfer of charge is charging by friction.

Electrostatics

C H A PT E RAIEEE Syllabus

Electric charges: Conservation of charge, Coulomb’s law-forces between

two point charges, forces between multiple charges; superposition principle

and continuous charge distribution. Electric field: Electric field due to a point

charge, Electric field lines, Electric dipole, Electric field due to a dipole, Torque

on a dipole in a uniform electric field. Electric flux, Gauss’s law and its

applications to find field due to infinitely long, uniformly charged straight

wire, uniformly charged infinite plane sheet and uniformly charged thin

spherical shell. Electric potential and its calculation for a point charge,

electric dipole and system of charges; Equipotential surfaces, Electrical

potential energy of a system of two point charges in an electrostatic field.

Conductor and insulator, Dielectric and electric polarisation, Capacitor,

Combination of capacitor in series, in parallel, Capacitance of parallel plate

capacitor and without dielectric medium between the plates, energy stored

in capacitor.

THIS CHAPTER

COVERS : Electric Charge and its

Properties

Coulomb’s Law

Electric Field

Electric Lines of Force

Electric Field due to

Electric Dipole

Electric Dipole in

Uniform Electric Field

Electric Flux

Gauss’ Law and its

Applications

Electric Potential and

electrostatic Potential

Energy

Electric Capacitor

Parallel Plate

Capacitor with

Dielectric and

Conducting Slab

Energy Stored in the

Capacitor

Capacitors in Series

and Parallel

Combinations

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AIEEE/State CETs Electrostatics


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Frictional Electricity : When two bodies are rubbed with each other, they are found to attract each other.This is so because, on rubbing, transfer of electrons takes place from one body to other. One of themacquires a positive charge and other acquires a negative charge.

A BRubbing

e–

Transfer

Neutral Neutral

(6) Charge is invariant : Charge of a particle is independent of speed.

(7) Charge cannot exist without mass, while mass can exist without charge.

INTERACTION BETWEEN CHARGES

Coulomb’s Law

It gives an expression for the force between two charged particles or particles like objects.

221

01212 4

1||||

r

qqFF

q1 q2

F21F12 r

where, 229

0

/CNm1099.84

1

0 = 8.85 × 10–12 C2/m2N. This is called absolute permittivity of free space.

Important Points :

1. If q1q2 > 0, force is repulsive.

2. If q1q2 < 0, force is attractive.

3. This force is central and conservative.

4. This force is between two charges and is independent of the presence of other charges i.e., if some othercharges are present in the region, the force between two given charges remains same.

Coulomb’s Law in Vector Form

ix

qqF ˆ

4

12

21

012

, i

x

qqF ˆ

4

12

21

012

F12

q1 q2

F21

x-axis

x

ELECTRIC FIELD

This space around a charge distribution, in which the charge can exert force on other charges is called electricfield.

Electric Field Intensity

We define electric field intensity at a point as the force experienced per unit charge when a very small positivetest charge is placed at that point.

q

FE

q

0Limit

Units : SI units of electric field intensity are (i) N/C (ii) volt/metre

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Electrostatics AIEEE/State CETs


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Electric Field due to a Point Charge (Q) :

ir

qQF ˆ

4

12

0

, ir

Q

q

FE ˆ

4

12

0

+Q P q( )

x-axisr F E,

ir

QE ˆ

4

12

0

Application

Direction of Electric Field at Various Points (when charge Q is placed at origin) :

–y-axis

–x-axis

jy

QE ˆ

4

12

0

ix

QE ˆ

4

12

0

ix

QE ˆ

4

12

0

x-axis

y-axis

rr

QE ˆ

4

12

0

22

ˆˆˆ

yx

jyixr

jy

QE ˆ

4

12

0

Qx x

y

y

r

, where

Electric Field Intensity at O in Each Case Shown Below is zero

(1)O+Q +Q

rr(2)

aa

O

a+Q +Q

+Q

(3)

+Q +Q

+Q +Q

a a

a

a

O

(4) +Q +Q

+Q+Q

+Q+Q

O

a

a

a

a

a

a

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(5) +Q

+Q

+Q

+Q+Q

O

a

a a

a

a

ELECTRIC LINES OF FORCE

Invented by Faraday to visualise electric field in a region.

They are imaginary lines drawn such that

(1) If they are straight, they give the direction of electric field.

(2) If they are curved, then tangent drawn at any point gives the direction of electric field.

(3) Number of field lines crossing a cross-section is proportional to strength of electric field present.

Electric Lines of Force due to Various Configurations

(1) Isolated point charge (+) (2) Isolated point Charge (–)

q –q

(3) Electric dipole (4) Two identical charges

–q +q+q +q

Properties

From above examples, a few properties of electric lines of force can be seen.

(1) They come out of a positive charge or infinity and terminate at negative charge or at infinity.

(2) In free space, electric lines of force are continuous curves i.e., do not have sudden breaks.

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(3) Two field lines do not intersect each other, as at point of interaction, we will get two different directionsof electric field which is not possible.

(4) They contract longitudinally on account of attraction between opposite charges.

(5) They exert lateral pressure on each other on account of repulsion between like charges.

Following pattern of lines of force are not possible

1. +q 2. 3. 4.

ELECTRIC DIPOLE

An arrangement of two equal and opposite charges separated by some distance.

–q +q

p

2a

Dipole Moment

Dipole moment is a vector quantity directed from negative to positive charge. It is represented by p .

Its magnitude is p = (2a) × q

Units : C-m [coulomb-metre]

The most practical example of an electric dipole is a water molecule.

105° Hydrogen

Oxygen

Hydrogen

p

p

Ideal Dipole

An ideal dipole is a short dipole with large value of q and negligible value of 2a.

In c.g.s. system, units of dipole moment is Debye.

1 Debye = 10–18 esu-cm

For an electron and a proton separated by 1Å.

–e +e

1Å p = 1.6 × 10–19 C × 10–10 m.

p = 1.6 × 10–29 C-m

p = 4.8 × 10–19 esu-cm = 4.8 Debye

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Electric Field due to an Electric Dipole

1. For a point P on axial line

2220

axial)(4

2

ar

rpE

For an ideal dipole (r2 – a2 r2)

3

0

axial4

2

r

pE

2. For a point Q on equatorial line

2/3220

equatorial)(4 ar

pE

Q

r

Eequatorial

Eaxial

+q–q

(– , 0)a ( , 0)a P

r

OFor an ideal dipole (r2 + a2 r2)

3

0

equatorial4 r

pE

3. For an ideal dipole 2

axialequatorial

E

E

4. Electric Field at any point in the plane of a short dipole

P is a point in x-y plane at a distance r from the centre of dipole, such that OP makes an angle with dipolemoment.

p

cos

p sinx-axis

y-axis

r

P

Enet3

0eq

4

sin

r

pE

30

ax4

cos2

r

pE

pO

P

Eeq

axE

Enet

(a)

23

0net cos31

4

1

r

pE

(b) tan2

1tan

ax

eq

E

E tan

2

1tan

(c) The net electric field makes angle + with dipole moment.

(d) When pE + = 90° 2tan 1

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Electric Dipole Placed in a Uniform Electric Field (Torque on dipole in uniform electric field)

Case 1 : Ep || Case 2 : )(|| Ep

E

–qE qEp

–q +q E–qE

p

–q+q q

(a) Net force = 0 EqEq (a) Net force = 0 EqEq

(b) Net torque = Zero (b) Net torque = Zero

Case 3 : Ep Case 4 : p makes an angle with E

E

–qE2a

–q

+q qEE

–qE

2a

–q

qE+q

2 sina

(a) Net force = Zero (a) Net force = Zero

(b) = qE × 2a = pE (b) Ep or = p E sin

In vector form Ep

Potential Energy of Dipole

1. The external work required to change the orientation from 1 to 2 is Wext = – pE[cos2 – cos1]

2. Change in potential energy of dipole is U2 – U1 = –pE[cos2 – cos1]

3. Potential energy of dipole is U = –pE cos

ELECTRIC FLUX

It is defined as the number of field lines that pass through a surface in a direction normal to the surface.

Mathematically, AE . (If E is uniform)

In general, AdE .

Units : C

m-N 2

or, V-m

Important cases :

(1) AE || (2) AE (3) AE and make angle

AE

= EA

A

E

= 0

AE

= cosEA

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R

A

h

E

E

2. REAEBase

lateral = –E × R2 ( field lines enter through curved surface)

(5)

AR

E

R

Base = 0

curved = 0 (Total flux that enters = Total flux that leave)

2

entered2

1φ πRE

2

2

leavingR

E

(6)

E

R

A O2

curved

2base

RE

RE

GAUSS LAW

0

.

encqdAE

Illustration :

(1)q

R

Sphere0

sphere

q

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(2)

= 0

(3) q

1 20

2sphere1sphere

q

(4)

Sphere 1Sphere 2

Surface 3

–q +q

Electric flux through sphere 1: 0

1

q

,

Electric flux through sphere 2: 0

2

q

Electric flux through surface 3: 3 = 0

Application of Gauss Law

(1) Field Due a Point Charge

The field due to a point charge is spherically symmetric. So if we draw a gaussian sphere around thecharge, the strength of electric field will be same every where. Using above formula

204

1

r

qE

E E

EE

dA

Gaussian sphere

Er

q

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(2) Field Due to a Uniformly Charged Spherical Shell (r > R)

204 r

QE

(outside)

Q

O

R

r

E

E

E

EdA

Gaussian sphere

A charged spherical shell behaves as if whole charge is concentrated at the centre of shell.

At any point inside the shell, if we draw a gaussian sphere, the charge enclosed = zero

0. AdE E = 0 (inside)

E = 0

Gaussian sphere

r

O

R

If we draw a graph showing variation of electric field with distance from centre, it will be like this.

Er

R

E 1r

2

r

(3) Expression for electric field at any point inside the sphere due to non-conducting solid sphere

having uniform volume charge distribution (sphere of charge)

304 R

qrE

. In vector form

304 R

rqE

Rr

GaussainSurfaceIf we put

3

3

4Rq ,

03

r

E

Similarly, fields due to other bodies can be derived.

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Electric Flux

Some frequently asked cases :

1.

a/2

a

a

q

0square 6

q

2. q

0face each

0cube

6

q

q

3.

q

A

B C

D

E

F

cube =q

ABCD240

=q

80

ABEF = 0

4. q

0

cube 2

q

5.

q

0

cube 4

q

Important results for fields due to different bodies (derived by Gauss Law)

1. Point charge 2

:r

kQQ

2. Shell of charge with charge Q and radius 2:

r

kQR (outside) zero (inside)

3. Sphere of charge with charge Q and radius 2:

R

kQrR (inside) 2r

kQ (outside)

4. Infinite line of charge with linear charge density r

k

2:

5. Infinite plane surface of charge with charge density 02

:

6. Infinite conducting sheet of charge with charge density 0

:

.

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Electric Potential Difference (V)

1. It is the work done against electric field in moving a unit positive charge from one point to other. That is

V2 – V1 =

2

1

.drE

.

2. V for two points at a distance r1 and r2 from a point charge Q

V2 – V1 = V = KQ

12

11

rr

3. Kinetic energy gained by ‘q’ when moved across V is U = q.V.

4. V between two points in electric field does not depend on path.

ELECTRIC POTENTIAL (V)

1. V at a point is work done against electric field in moving a unit positive test charge from infinity to that

point,

r

drEV .

.

2. Potential due to a point charge Q at a distance r is r

KQV .

3. Potential due to dipole at distance r at angle 2

cos

r

KpV

4. Potential due to system of charge

3

3

2

2

1

1

r

Kq

r

Kq

r

KqVP . q1

q2

q3

r1

r2

r3

PIf V and E are functions of x, then

2

1

12

x

x

dxEVV .

Relation between Electric Field and Potential

1. In general,

(a) V2 – V1 = – 2

1

.

r

r

drE

(b) V = –

r

drE.

2.x

VEx

,

x

VEy

,

z

VEz

.

3. If V is a function of single variable r, dr

dVE .

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Equipotential Surface

An equipotential surface is a surface with a constant value of potential at all points on the surface.

Electric lines of force are always perpendicular to equipotential surface.

Shape of equipotential surface

Point charge Concentric spheres

Line charge Co-axial cylinders

Uniform field Plane parallel to each other

Electric Potential Energy

1. For a two point charge system

r

q1 q2 r

qKqU 21

2. For a three point charge system

q3

r31 r23

q2q1r12

31

13

23

32

12

21

04

1

r

qq

r

qq

r

qqU

CONDUCTORS

Conductor contain large amount of mobile charge carriers.

Properties :

1. Inside a conductor, electrostatic field is zero.

2. At the surface of charged conductor, electrostatic field must be perpendicular to the surface at every point.

3. The charge density will remain zero in interior of conductor static situtation.

4. Conductor is equipotential

5. Electric field at surface of charged conductor is 0

.

6. If conductor has a cavity with no charge inside the cavity then electric field inside cavity is zero, whatever

be the charge on or outside conductor (Electrostatic shielding).

CAPACITANCE

Capacitance of a conductor is measure of ability of conductor to store electric charge and hence electric energyon it.

When charge is given to a conductor its potential increases. It is found that

V Q

or, Q V

Q = CV

where C is the capacitance and its unit is farad (F).

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Capacitance of Isolated Spherical Conductor

C = 40r

Capacitance of Earth Ce = 40Re

= 711 F r

CAPACITOR

It is a device used to store electric energy in the form of electric field.

When an earthed conductor is held near an isolated conductor, a capacitor is obtained.

Working of Capacitor A B

If some charge is given to conductor A its potential increases, and soon

becomes maximum. If some more charge is given to it, it leaks out. Now if

an earthed conductor B is placed near A opposite charges induces on B,

hence more charge can be given to A.

Capacitance of a Parallel Plate Capacitor

1. Electric field in between plates

E = 00

A

Q+

+

+

+

–

–

–

–

d

+ Q – Q

Plate area = A

E

2. Potential difference between the plates = 00

d

A

Qd

3. Capacitance = d

A0

4. Force of attraction between the plates = 222 0

2

0

QEA

A

Q

Parallel Plate Capacitor with Dielectric Slab

(a) Induced charge

KQQi

11 , K is dielectric constant.

(b) Capacitance,

K

ttd

AC

)(

0 .

(c) For conducting slab, K =

Qi = – Q and td

AC

0

(d) The capacitance of a parallel plate capacitor is C. If its plates

are connected by an inclined conducting rod, the new

capacitance is infinity.

C

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Spherical Capacitor

1. Potential difference between plates

ba

abKQV

2. Electric field at any point P between plates

ar

L

b

M

P2r

KQE

3. Potential at any point P between plates

b

KQ

r

KQV

4. Capacitance ab

abC

04

5. Important : If the inner surface is grounded, capacitance ab

bC

2

04

Cylindrical Capacitance

1. Potential difference between plates

a

bnl

l

KQV

2 +

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

– Q

+ Q

a

b

l

2. Field lr

KQE

2

3. Potential at any point between plates

a

rnl

l

KQV

2

4. Capacitance

2 0

a

bnl

lC

System of Two Metal Balls

a b

d

Capacitance

dba

C211

4 0

Dielectric Polarisation

When a dielectre glab is placed between the plates of capacitor it’s polarisation take place. Thus a charge–Q

i, appear on its left face and +Q

i appears on its right face.

+Q – iQ + iQ –Q

0

0

11

A

QE

kQQi

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Effective Capacitance in Some Important Cases

1.

4

4

3

3

2

2

1

1

0

K

t

K

t

K

t

K

t

AC

K1 K2 K3 K4

t1 t2 t3 t4

–+

For two capacitors

If

21

0

21

0

21

11

2

22

2

KKd

A

K

d

K

d

AC

dtt

K1 K2–+

d/2 d/2

21

210

21

21 2A2

KK

KKK

dKK

KKC eq

2.d

AKAKAKC

][ 3322110

–+

K1

K2

K3

A1

A2

A3For two capacitors,

If 221A

AA

d

AK

AK

C

22 210

A/2

A/2 A/2

A/2K1

K222

21021 KKK

d

AKKC eq

COMBINATION OF CAPACITORS

1. Capacitors in series (three capacitors)

11

C

QV ,

22

C

QV and

33

C

QV

V = V1 + V2 + V3

321

111

CCCQV

V

V1 V2 V3

C1 C2 C3

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eqC

QV

321

1111

CCCCeq

2. Two Capacitors in Series

11

C

QV

22

C

QV

21

111

CCCeq

V1 V2

C1 C2

V

21

21

CC

CCCeq

Q = Ceq

V

VCC

CVV

CC

CV

21

12

21

21

Potential dividing rule

3. Capacitors in parallelC1

C2

C3

Q1

Q2

Q3

V

Q1 = C1V, Q2 = C2V, Q3 = C3V

Q = C1V + C2V + C3V

Q = (C1 + C2 + C3)V and Q = Ceq

V

Ceq

= C1 + C

2 + C

3

Energy Stored in a Capacitor

Energy stored in a capacitor of capacitance C, charge Q and potential difference V across it is given by

QVC

QCVU

2

1

22

1 22

Sharing of Charge

Case 1 : Two capacitors charged to potentials V1 and V2 are connected end to end as shown

(a) Final common potential 21

2211

CC

VCVCV

(b) Charge flown through key )( 2121

21 VVCC

CC

(c) Loss of energy = 2

2121

21 )()(2

VVCC

CC

Case 2 : If positive terminal is connected to negative terminal

(a) Final common potential 21

2211

CC

VCVCV

V1

V2(b) Loss of energy = 221

21

21 )()(2

VVCC

CC

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Small Inserting a Dielectric Slab

1 When battery is disconnected (isolated)

Q0 = initial change

C0 = initial capacitance

V0 = initial potential

E0 = initial energy

(a) New capacitance = KC0

(b) New potential difference = K

V

KC

Q 0

0

0

(c) New energy stored = K

E

K

VKC 0

20

0 )(2

1

(d) Note that charge on each plate remains same.

2. When battery is connected

(a) C = KC0

(b) V = V0

(c) Q = KQ0

(d)2

00 )()(2

1VKCE = KE0

Combining Charged Drops

When n droplets of radius r0 having equal charge Q0 colasce to form a bigger drop of radius R.

(a)33

0 3

4

3

4Rrn

03/1 rnR

R(b) C = n 1/3C0

(c) Total charge = nQ0

(d) 03/2

03/1

00 VnCn

nQ

C

nQV

(e) Total energy =0

3/1

20

2

2

)(

2

1

Cn

nQ

C

Q = n5/3 U0

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