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CONDUCTORS. Conductor: charges free to move within the material. Electrostatic Equilibrium: there is no net motion of charge within the conductor. E = 0 inside a conductor. The existence of electrostatic equilibrium is consistent only with a zero field in the conductor.

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conductors
CONDUCTORS

Conductor:

charges free to move within the material.

Electrostatic Equilibrium:

there is no net motion of charge within the conductor.

Dr. Champak B. Das (BITS, Pilani)

e 0 inside a conductor
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

e-

A conductor in an electric field:

Electrons move upward in response to applied field.

Dr. Champak B. Das (BITS, Pilani)

a conductor in an electric field contd

E0

A conductor in an electric field: (contd.)
  • Electrons accumulate
  • on top surface.
  • Induced charges set up a field E in the interior.

Dr. Champak B. Das (BITS, Pilani)

a conductor in an electric field contd1
A conductor in an electric field: (contd.)

Two surfaces of a conductor:

sheets of charge

Dr. Champak B. Das (BITS, Pilani)

a conductor in an electric field contd2
A conductor in an electric field: (contd.)

Field of induced charges tends to cancel off the original field

 E0must move enough electrons to the surface such that, E = E0

Dr. Champak B. Das (BITS, Pilani)

slide7

In the interior of the conductor

NET FIELD IS ZERO.

The process is Instantaneous

Dr. Champak B. Das (BITS, Pilani)

0 inside a conductor
 = 0 inside a conductor.

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
Any net charge resides on the surface

Dr. Champak B. Das (BITS, Pilani)

slide10

A conductor is an equipotential.

E

V

r

r

R

R

For any two points, a and b:

Dr. Champak B. Das (BITS, Pilani)

slide11

E is  to the surface, outside a conductor.

E

E=0

Else, the tangential component

would cause

charges to move

Dr. Champak B. Das (BITS, Pilani)

slide12

CONDUCTORS

  • E = 0 inside a conductor.
  •  = 0 inside a conductor.
  • Any net charge resides on the surface.
  • A conductor is an equipotential.
  • E is  to the surface, outside a conductor.

Dr. Champak B. Das (BITS, Pilani)

slide13

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

Dr. Champak B. Das (BITS, Pilani)

slide14

Induced Charges

Conductor

+q

Induced charges

Dr. Champak B. Das (BITS, Pilani)

slide15

-

-

-

-

-

-

-

+q

-

-

-

-

-

-

-

A cavity in a conductor

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

Dr. Champak B. Das (BITS, Pilani)

slide16

a

b

R

q

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.

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

Answer:

Dr. Champak B. Das (BITS, Pilani)

slide17

a

b

R

q

Prob. 2.35(b):

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

Answer:

Dr. Champak B. Das (BITS, Pilani)

slide18

Prob. 2.35(c):

a

b

R

q

Now the outer surface is touched to a grounding wire, which lowers its potential to zero (same as infinity). How the answers to part (a) and (b) changes ?

Answer:

Dr. Champak B. Das (BITS, Pilani)

slide19

Surface charge on a conductor

Recall electrostatic boundary condition:

=> Field outside a conductor:

Dr. Champak B. Das (BITS, Pilani)

slide20

The surface charge density :

OR

Knowledge of E or V just outside the conductor

 Surface charge on a conductor

Dr. Champak B. Das (BITS, Pilani)

slide21

Forces on charge distributions

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

On a volume charge distribution :

Dr. Champak B. Das (BITS, Pilani)

slide22

Prob. 2.43:

Find the net force that the southern hemisphere of a uniformly charged sphere exerts on the northern hemisphere.

Z

Ans:

R

r

Y

Q

X

Dr. Champak B. Das (BITS, Pilani)

slide23

Forces on charge distributions

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

On a volume charge distribution :

On a surface charge distribution :

Dr. Champak B. Das (BITS, Pilani)

slide24

Forces on surface charge distributions

“ E is discontinuous across the distribution”

The force per unit area :

Dr. Champak B. Das (BITS, Pilani)

slide25

Force on a conductor

Force (per unit area) on the conductor surface:

Outward Pressure on the conductor surface :

Dr. Champak B. Das (BITS, Pilani)

slide26

Z

R

Y

Q

X

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?

Ans:

Dr. Champak B. Das (BITS, Pilani)

capacitors
CAPACITORS

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

Dr. Champak B. Das (BITS, Pilani)

capacitance
Capacitance :
  • Is a geometrical property
  • Units: Farad (= coulomb/volt)

Different possible geometries:

  • Planer
  • Spherical
  • Cylindrical

Dr. Champak B. Das (BITS, Pilani)

a parallel plate capacitor
A parallel plate capacitor

Dr. Champak B. Das (BITS, Pilani)

plates are very large and very close
Plates are very large and very close

Dr. Champak B. Das (BITS, Pilani)

a spherical capacitor
A Spherical capacitor

Dr. Champak B. Das (BITS, Pilani)

cross section of a spherical capacitor
Cross section of a spherical capacitor

Dr. Champak B. Das (BITS, Pilani)

a cylindrical capacitor
A cylindrical capacitor

Dr. Champak B. Das (BITS, Pilani)

slide35

Work done to charge a capacitor

At any instant,

Dr. Champak B. Das (BITS, Pilani)