Chapter IV

1. Chapter IV Electrostatics in the Presence of Dielectrics

2. Two Types of Dielectrics : 1. Dielectrics with Molecules/atoms carrying permanent dipole moments

4. Or

5. 2. Molecules/atoms acquiring dipole moment by an external electric field :

7. Uniform electron cloud At equilibrium,

8. Non-uniform electron cloud Prob. 4.2 Consider the hydrogen atom in which the electronic charge density is : Calculate the polarizability of hydrogen.

9. Applying Gauss’ law

10. Expanding in a power series in and keeping the lowest power of :

11. The Polarization Vector

12. Potential of a polarized piece of Dielectric

13. Write : Integrand can be written as:

14. Comparing with the potential of surface and volume charges :

15. we conclude : and where, are the surface and volume bound charges Polarized dielectric piece

16. Direction of polarization

17. Prob. 4.14 Prove that the total bound charge is zero However, by Gauss’ theorem,

18. Electric field of a uniformly polarized dielectric sphere : The only non-zero multipole moment of the sphere is the dipole moment.

19. Prob. 4.10 A sphere of radius R carries a polarization : a) Calculate the bound charges b) Find the field inside and outside the sphere

20. a) Total bound charge : b) (Surface charge being uniform, produces zero field inside) (Applying Gauss’ Law)

21. Bound charges Free charge q Dielectric sphere Free and Bound Charges Two types of charges in the presence of dielectrics : 1. Free charge 2. Bound charge The densities of the two types of charges are denoted by respectively

22. The Electric Displacement Vector Denoting by , we get is called the Electric Displacement

23. Gauss’ Law for Electric Displacement Integrating both sides over certain volume,

24. Prob. 4.15 A thick spherical shell of inner radius a and outer radius b is made of a dielectric material with a frozen-in polarization : Find the electric field in two different ways : a) Locate all the bound charge, and find the field produced by it b) Find and hence

25. a)

26. For

27. b) As there is no free charge, in all three regions

28. Surface b.c. can be treated as volume b.c.

29. Linear Dielectrics Linear dielectrics are those in which the polarization vector at every point is proportional to the electric field at that point For homogeneous dielectrics is a constant

30. For a linear dielectric :

31. Example (Prob. 4.32) : Point charge at the centre of a linear dielectric sphere. Find : Inside the sphere :

34. Outside the sphere

35. Is the electric field inside a dielectric, due to given free charges, the same as in free space, only reduced by a a factor ? Dielectric slab

36. Free Space In Dielectric

37. The above result is true only in a homogeneous dielectric filling up all space

38. Energy in Dielectric Systems The above is the work done to assemble all charges, free and bound, bringing all of them from infinity

39. More natural definition : Work done in assembling only the free charges.

40. The first term vanishes as

41. For a linear dielectric :

43. Prob. 4.26 A spherical conductor, of radius a, carries a charge Q. It is surrounded by a linear dielectric material of susceptibility out to a radius b. Find the energy of the configuration