1. Work and Energy Electrostatic force is conservative
Work done by an external agency to move a charge :
 Work done is path independent
Electrostatic force is conservative
Dr. Champak B. Das (BITS, Pilani)

2. Dr. Champak B. Das (BITS, Pilani)
Work done to bring a charge from infinity to :
Potential is the work required to create the system (potential energy) per unit charge
Dr. Champak B. Das (BITS, Pilani)

3. Energy of a Point Charge Distribution
Ex: Case of assembling three point charges q2
rs12 q1
Dr. Champak B. Das (BITS, Pilani)

4. Dr. Champak B. Das (BITS, Pilani)q3
rs23
rs13 q1 q2
rs12
Dr. Champak B. Das (BITS, Pilani)

5. Energy of a Point Charge Distribution
Work necessary to assemble n number of point charges
Dr. Champak B. Das (BITS, Pilani)

6. Energy of a Continuous Charge Distribution
Dr. Champak B. Das (BITS, Pilani)

7. Dr. Champak B. Das (BITS, Pilani)
ELECTROSTATIC ENERGY
(can be +ve/-ve)
(always +ve)
Dr. Champak B. Das (BITS, Pilani)

8. 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)

9. Dr. Champak B. Das (BITS, Pilani)
Prob (a) :
Find the energy stored in a uniformly charged solid sphere of radius R and charge q using:
Ans (a):
Dr. Champak B. Das (BITS, Pilani)

10. Dr. Champak B. Das (BITS, Pilani)
Prob (b) :
Find the energy stored in a uniformly charged solid sphere of radius R and charge q using:
Ans (b):
Dr. Champak B. Das (BITS, Pilani)

11. 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)

12. When an external field is applied ?
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)

13. A conductor in an electric field:
Electrons move upward in response to applied field.
Dr. Champak B. Das (BITS, Pilani)

14. 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)

15. A conductor in an electric field: (contd.)
Two surfaces of a conductor:
sheets of charge
Dr. Champak B. Das (BITS, Pilani)

16. A conductor in an electric field: (contd.)
Field of induced charges tends to cancel off the original field
 E0 must move enough electrons to the surface such that, E = E0
Dr. Champak B. Das (BITS, Pilani)

17. In the interior of the conductor NET FIELD IS ZERO.
The process is Instantaneous
Dr. Champak B. Das (BITS, Pilani)

18. same amount of positive and negative charges
 = 0 inside a conductor.
same amount of positive and negative charges
NET CHARGE DENSITY IS ZERO.
Dr. Champak B. Das (BITS, Pilani)

19. Any net charge resides on the surface
Dr. Champak B. Das (BITS, Pilani)

20. A conductor is an equipotential.
For any two points, a and b: E R r V R r
Dr. Champak B. Das (BITS, Pilani)

21. E is  to the surface, outside a conductor.
Else, the tangential component
would cause
charges to move
Dr. Champak B. Das (BITS, Pilani)

22. 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)

23. Dr. Champak B. Das (BITS, Pilani)
Induced Charges
Conductor +q
Induced charges
Dr. Champak B. Das (BITS, Pilani)

24. Dr. Champak B. Das (BITS, Pilani)
A cavity in a conductor +q
Gaussian surface
If +q is placed in the cavity, -q is induced on the surface of the cavity.
Dr. Champak B. Das (BITS, Pilani)

25. 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: a b R q
Dr. Champak B. Das (BITS, Pilani)

26. Dr. Champak B. Das (BITS, Pilani)
Prob. 2.35(b):
Find the potential at the centre, using infinity as the reference point. a b R q
Answer:
Dr. Champak B. Das (BITS, Pilani)

27. Surface charge on a conductor
Recall electrostatic boundary condition:
=> Field outside a conductor:
Dr. Champak B. Das (BITS, Pilani)

28. 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)

29. Dr. Champak B. Das (BITS, Pilani)
Force on a conductor
Dr. Champak B. Das (BITS, Pilani)

30. 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)

31. 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
Ans: R  r Y Q X
Dr. Champak B. Das (BITS, Pilani)

32. 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)

33. Forces on surface charge distributions
“ E is discontinuous across the distribution ”
below
above
The force per unit area :
Dr. Champak B. Das (BITS, Pilani)

34. Dr. Champak B. Das (BITS, Pilani)
Force on a conductor
Force (per unit area) on the conductor surface:
Outward Pressure 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)

35. 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 Y R Q
Ans: 
Dr. Champak B. Das (BITS, Pilani)

36. Dr. Champak B. Das (BITS, Pilani)
CAPACITORS
Potential difference between two conductors carrying +Q and –Q charge:
Dr. Champak B. Das (BITS, Pilani)

37. Different possible geometries:
Capacitance :
Is a geometrical property
Units: Farad (= coulomb/volt)
Different possible geometries:
Planer
Spherical
Cylindrical
Dr. Champak B. Das (BITS, Pilani)

38. Plates are very large and very close
Dr. Champak B. Das (BITS, Pilani)

39. Dr. Champak B. Das (BITS, Pilani)
A Spherical capacitor
Dr. Champak B. Das (BITS, Pilani)

40. Cross section of a spherical capacitor
Dr. Champak B. Das (BITS, Pilani)

41. A cylindrical capacitor
Dr. Champak B. Das (BITS, Pilani)

42. Capacitance per unit length of a cylindrical capacitor
Cross section of a cylindrical capacitor
Prob 2.39 :
Capacitance per unit length of a cylindrical capacitor
Dr. Champak B. Das (BITS, Pilani)

43. Work done to charge a capacitor
At any instant,
Dr. Champak B. Das (BITS, Pilani)