dr. champak b. das (bits, pilani) work and energy work done by an external agency to move a charge :...

Dr. Champak B. Das (BITS, Pilani)

Work and Energy

Work done by an external agency to move a charge :

aVbVQW

Work done is path independent

Electrostatic force is conservative

Dr. Champak B. Das (BITS, Pilani)

Work done to bring a charge from infinity to :

rQVW

Potential is the work required to create the system

(potential energy) per unit charge

r

Dr. Champak B. Das (BITS, Pilani)

Energy of a Point Charge

DistributionEx: Case of assembling three point charges

01 Wrs12

12

12

02 4

1

sr

qqW

πε

q1

1r

q2

2r

Dr. Champak B. Das (BITS, Pilani)

q3

rs13rs23

2313

213

03 4

1

ss r

q

r

qqW

πε

q1

q2

rs12

1r

2r

3r

Dr. Champak B. Das (BITS, Pilani)

Energy of a Point Charge

DistributionWork necessary to assemble n number of point charges

n

i

n

ijj s

ji

ijr

qqW

1 ,108

1

πε

in

ii rVqW

12

1

Dr. Champak B. Das (BITS, Pilani)

Energy of a Continuous Charge Distribution

τρVdW2

1

τε

dVEW

20

s

adEVdVEW

τε

τ20

τε

dEWspaceall 20

2

Dr. Champak B. Das (BITS, Pilani)

ELECTROSTATIC ENERGY

in

ii rVqW

12

1

τε

dEWspaceall 20

2

(can be +ve/-ve)

(always +ve)

Dr. Champak B. Das (BITS, Pilani)

ELECTROSTATIC ENERGY

•Energy of a point charge is infinite !

•Energy is stored in the field/charge ?

•Doesn’t obey superposition principle !

Dr. Champak B. Das (BITS, Pilani)

Prob. 2.32 (a) :Find the energy stored in a uniformly charged solid sphere of radius R and charge q using:

τρVdW2

1

2

2

0

32

1

4 R

r

R

qrVWhere

πε

Ans (a):

R

qW

2

0 5

3

4

1

πε

Dr. Champak B. Das (BITS, Pilani)

Prob. 2.32 (b) :Find the energy stored in a uniformly charged solid sphere of radius R and charge q using:

τε

dEWspaceall 20

2

RrforR

qr

Rrforr

qEwhere

30

20

4

1

4

1

πε

πε

Ans (b):

R

qW

2

0 5

3

4

1

πε

Dr. Champak B. Das (BITS, Pilani)

CONDUCTORS

Electrostatic Equilibrium:

there is no net motion of charge within the conductor.

Conductor:

charges free to move within the material.

Dr. Champak B. Das (BITS, Pilani)

E = 0 inside a conductor.

The existence of electrostatic equilibrium is consistent only with a zero field in the conductor.

When an external field is applied ?

Dr. Champak B. Das (BITS, Pilani)

A conductor in an electric field:

Electrons move upward in response

to applied field. e-

Dr. Champak B. Das (BITS, Pilani)

• Induced charges set up a field E in the

interior.

• Electrons accumulate on top surface.

E0

A conductor in an electric field: (contd.)

Dr. Champak B. Das (BITS, Pilani)

Two surfaces of a conductor: sheets of charge

':int 0 EEENeterior

':)( 0 EEEmagnitudes

A conductor in an electric field: (contd.)

Dr. Champak B. Das (BITS, Pilani)

E0 must move enough electrons to the surface

such that, E = E0

Field of induced charges tends to cancel off the

original field

A conductor in an electric field: (contd.)

Dr. Champak B. Das (BITS, Pilani)

In the interior of the conductorNET FIELD IS ZERO.

The process is Instantaneous

Dr. Champak B. Das (BITS, Pilani)

= 0 inside a conductor.

E 0ερ

00 ρE

same amount of positive and negative charges

NET CHARGE DENSITY IS ZERO.

Dr. Champak B. Das (BITS, Pilani)

Any net charge resides on the surface

Dr. Champak B. Das (BITS, Pilani)

A conductor is an equipotential.

For any two points, a and b:

0 bVaVE

R r

V

R r

Dr. Champak B. Das (BITS, Pilani)

E is to the surface, outside a conductor.

E=0E

Else, the tangential component would cause

charges to move

Dr. Champak B. Das (BITS, Pilani)

A justification for surface distribution of charges in a conductor :

go for a configuration to minimize the potential energy

Example : Solid sphere carrying charge q

R

qW

0

2

8πεσ

R

qW

0

2

85

6

περ

ρσ WW

Dr. Champak B. Das (BITS, Pilani)

Induced Charges

Conductor+q

Induced charges

Dr. Champak B. Das (BITS, Pilani)

+q

A cavity in a conductor

If +q is placed in the cavity, -q is induced on the surface of the cavity.

Gaussian surface

Dr. Champak B. Das (BITS, Pilani)

Prob. 2.35:A metal sphere of radius R, carrying charge q is

surrounded by a thick concentric metal shell. The shell carries no net charge.

24 R

qR πσ

24 a

qa πσ

24 b

qb πσ

(a) Find the surface charge density at R, a and b

Answer:

a

bR

q

Dr. Champak B. Das (BITS, Pilani)

Prob. 2.35(b):

a

q

R

q

b

qV

04

10

πε

Find the potential at the centre, using infinity as the reference point.

Answer:a

bR

q

Dr. Champak B. Das (BITS, Pilani)

Surface charge on a conductor

nEE belowaboveˆ

0ε

σ

Recall electrostatic boundary condition:

=> Field outside a conductor:

nE ˆ0ε

σ

Dr. Champak B. Das (BITS, Pilani)

nV 0εσ

The surface charge density :

Knowledge of E or V just outside the conductor

Surface charge on a conductor

nE ˆ0

εσ OR

Dr. Champak B. Das (BITS, Pilani)

Force on a conductor

Dr. Champak B. Das (BITS, Pilani)

Forces on charge distributions

Force on a charge element dq placed in an external field E(e) :

eEdqF

Edq

On a volume charge distribution :

τ

τρ dEF

Dr. Champak B. Das (BITS, Pilani)

Prob. 2.43:Find the net force that the southern hemisphere

of a uniformly charged sphere exerts on the northern hemisphere.

Z

X

YrR

Q

Ans:

kR

QF ˆ

16

3

4

12

2

0πε

Dr. Champak B. Das (BITS, Pilani)

Forces on charge distributionsForce on a charge element dq placed in an external field E(e) :

eEdqF

Edq

On a volume charge distribution :

τ

τρ dEF

On a surface charge distribution :

s

daEF

σ

Dr. Champak B. Das (BITS, Pilani)

Forces on surface charge distributions “ E is discontinuous across the

distribution ”

n̂

odqbb EEE

odqaa EEE

above

below

da

The force per unit area : belowabove EEf

σ2

1

Dr. Champak B. Das (BITS, Pilani)

nf ˆ2

1 2

0

σε

Force on a conductor

Outward Pressure on the conductor surface :

202

1EP ε

Force (per unit area) on the conductor surface:

The direction of the force is “outward” or “into the

field”….. whether is positive or negative

Dr. Champak B. Das (BITS, Pilani)

Prob. 2.38:A metal sphere of radius R carries a total charge Q. What is the force of repulsion between the northern

hemisphere and the southern hemisphere?Z

X

YR

Q

Ans:

kR

QF ˆ

84

12

2

0πε

Dr. Champak B. Das (BITS, Pilani)

ldEldEVVV

QE QV

Potential difference between two conductors carrying +Q and –Q charge:

VQC

CAPACITORS

Dr. Champak B. Das (BITS, Pilani)

Capacitance :

• Is a geometrical property

• Units: Farad (= coulomb/volt)

Different possible geometries:• Planer

• Spherical

• Cylindrical

Dr. Champak B. Das (BITS, Pilani)

Plates are very large and very close

d

AC 0ε

Dr. Champak B. Das (BITS, Pilani)

A Spherical capacitor

Dr. Champak B. Das (BITS, Pilani)

Cross section of a spherical capacitor

ab

abC

04πε

Dr. Champak B. Das (BITS, Pilani)

A cylindrical capacitor

LL

Dr. Champak B. Das (BITS, Pilani)

Capacitance per unit length of a cylindrical capacitor

abC

ln

12 0πε

Cross section of a cylindrical capacitor

Prob 2.39 :

Dr. Champak B. Das (BITS, Pilani)

Work done to charge a capacitor

C

qV

C

QW

2

2

1

2

2

1CVW

At any instant,

dqC

qdW