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


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


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


q3

rs13rs23

2313

213

03 4

1

ss r

q

r

qqW

πε

q1

q2

rs12

1r

2r

3r


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


Energy of a Continuous Charge Distribution

τρVdW2

1

τε

dVEW

20

s

adEVdVEW

τε

τ20

τε

dEWspaceall 20

2


ELECTROSTATIC ENERGY

in

ii rVqW

12

1

τε

dEWspaceall 20

2

(can be +ve/-ve)

(always +ve)


ELECTROSTATIC ENERGY

•Energy of a point charge is infinite !

•Energy is stored in the field/charge ?

•Doesn’t obey superposition principle !


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


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


CONDUCTORS

Electrostatic Equilibrium:

there is no net motion of charge within the conductor.

Conductor:

charges free to move within the material.


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 ?


A conductor in an electric field:

Electrons move upward in response

to applied field. e-


• Induced charges set up a field E in the

interior.

• Electrons accumulate on top surface.

E0

A conductor in an electric field: (contd.)


Two surfaces of a conductor: sheets of charge

':int 0 EEENeterior

':)( 0 EEEmagnitudes



E0 must move enough electrons to the surface

such that, E = E0

Field of induced charges tends to cancel off the

original field



In the interior of the conductorNET FIELD IS ZERO.

The process is Instantaneous


= 0 inside a conductor.

E 0ερ

00 ρE

same amount of positive and negative charges

NET CHARGE DENSITY IS ZERO.


Any net charge resides on the surface


A conductor is an equipotential.

For any two points, a and b:

0 bVaVE

R r

V

R r


E is to the surface, outside a conductor.

E=0E

Else, the tangential component would cause

charges to move


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


Induced Charges

Conductor+q

Induced charges


+q

A cavity in a conductor

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

Gaussian surface


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


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


Surface charge on a conductor

nEE belowaboveˆ

0ε

σ

Recall electrostatic boundary condition:

=> Field outside a conductor:

nE ˆ0ε

σ


nV 0εσ

The surface charge density :

Knowledge of E or V just outside the conductor

Surface charge on a conductor

nE ˆ0

εσ OR


Force on a conductor


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


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


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


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


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


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


ldEldEVVV

QE QV

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

VQC

CAPACITORS


Capacitance :

• Is a geometrical property

• Units: Farad (= coulomb/volt)

Different possible geometries:• Planer

• Spherical

• Cylindrical


Plates are very large and very close

d

AC 0ε


A Spherical capacitor


Cross section of a spherical capacitor

ab

abC

04πε


A cylindrical capacitor

LL


Capacitance per unit length of a cylindrical capacitor

abC

ln

12 0πε

Cross section of a cylindrical capacitor

Prob 2.39 :


Work done to charge a capacitor

C

qV

C

QW

2

2

1

2

2

1CVW

At any instant,

dqC

qdW

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

Documents