Electric Fields in Matter

1. Electric Fields in Matter • Polarization • Field of a polarized object • Electric displacement • Linear dielectrics Dr. Champak B. Das (BITS, Pilani)

2. Conductors Matter Insulators/Dielectrics All charges are attached to specific atoms/molecules and can only have a restricted motion WITHIN the atom/molecule. Dr. Champak B. Das (BITS, Pilani)

3. A simplified model of a neutral atom electron cloud nucleus • The positively charged nucleus is surrounded by a spherical electron cloud with equal and opposite charge. Dr. Champak B. Das (BITS, Pilani)

4. When the atom is placed in an external electric field (E) E • The electron cloud gets displaced in a direction (w.r.t. the nucleus) opposite to that of the applied electric field. Dr. Champak B. Das (BITS, Pilani)

5. If E is large enough ► the atom gets pulled apart completely => the atom gets IONIZED • For less extreme fields ► an equilibrium is established => the atom gets POLARIZED Dr. Champak B. Das (BITS, Pilani)

6. -e +e ► The net effect is that each atom becomes a small charge dipole which affects the total electric field both inside and outside the material. Dr. Champak B. Das (BITS, Pilani)

7. Induced Dipole Moment: (pointing along E) Atomic Polarizability Dr. Champak B. Das (BITS, Pilani)

8. +q a -q d +q -q E To calculate  : (in a simplified model) The model: an atom consists of a point charge (+q) surrounded by a uniformly charged spherical cloud of charge (-q). At equilibrium, ( produced by the negative charge cloud) Dr. Champak B. Das (BITS, Pilani)

9. At distance d from centre, (where v is the volume of the atom) Dr. Champak B. Das (BITS, Pilani)

10. Prob. 4.4: A point charge q is situated a large distance r from a neutral atom of polarizability . Find the force of attraction between them. Force on q: Dr. Champak B. Das (BITS, Pilani)

11. Alignment of Polar Molecules: Polar molecules: molecules having permanent dipole moment • when put in a uniform external field: Dr. Champak B. Das (BITS, Pilani)

12. Alignment of Polar Molecules: • when put in a non-uniform external field: +q F+ d -q F- Dr. Champak B. Das (BITS, Pilani)

13. +q F+ E+ d -q E- F- Dr. Champak B. Das (BITS, Pilani)

14. For perfect dipole of infinitesimal length, the torque about the centre : the torque about any other point: Dr. Champak B. Das (BITS, Pilani)

15. Prob. 4.9: A dipole p is a distance r from a point charge q, and oriented so that p makes an angle  with the vector r from q to p. (i) What is the force onp? (ii) What is the force onq? Dr. Champak B. Das (BITS, Pilani)

16. Polarization: When a dielectric material is put in an external field: Induced dipoles (for non-polar constituents) Aligned dipoles (for polar constituents) A lot of tiny dipoles pointing along the direction of the field Dr. Champak B. Das (BITS, Pilani)

17. Material becomes POLARIZED A measure of this effect is POLARIZATION defined as: P dipole moment per unit volume Dr. Champak B. Das (BITS, Pilani)

18. The Field of a Polarized Object = sum of the fields produced by infinitesimal dipoles rs p r r Dr. Champak B. Das (BITS, Pilani)

19. p rs r r Total potential : Dr. Champak B. Das (BITS, Pilani)

20. Prove it ! Dr. Champak B. Das (BITS, Pilani)

21. Using Divergence theorem; Dr. Champak B. Das (BITS, Pilani)

22. Defining: Surface Bound Charge Volume Bound Charge Dr. Champak B. Das (BITS, Pilani)

23. Potential due to a surface charge density b & a volume charge density b Dr. Champak B. Das (BITS, Pilani)

24. Field/Potential of a polarized object = Field/Potential produced by a surface bound charge b + Field/Potential produced by a volume bound charge b Dr. Champak B. Das (BITS, Pilani)

25. Physical Interpretation of Bound Charges …… are not only mathematical entities devised for calculation; but represent perfectly genuine accumulations of charge ! Dr. Champak B. Das (BITS, Pilani)

26. BOUND (POLARIZATION) CHARGE DENSITIES Consequence of an external applied field ►Accumulation of b and b Dr. Champak B. Das (BITS, Pilani)

27. P  E  ( n : number of atoms per unit volume ) Dr. Champak B. Das (BITS, Pilani)

28. A A A P  E  Net transfer of charge across A : Dr. Champak B. Das (BITS, Pilani)

29. Net charge transfer per unit area : P is measure of the charge crossing unit area held normal to P when the dielectric gets polarized. Dr. Champak B. Das (BITS, Pilani)

30. When P is uniform : P  N M Q   Q E  … net charge entering the volume is ZERO Dr. Champak B. Das (BITS, Pilani)

31. Volume bound charge P A Net transfer of charge across A : Dr. Champak B. Das (BITS, Pilani)

32. G N M P  E  Surface bound charge Net accumulated charge between M & N : Dr. Champak B. Das (BITS, Pilani)

33. z  P R Field of a uniformly polarized sphere Choose: z-axis || P P is uniform Dr. Champak B. Das (BITS, Pilani)

34. Potential of a uniformly polarized sphere: (Prob. 4.12) Potential of a polarized sphere at a field point ( r ): P is uniform P is constant in each volume element Dr. Champak B. Das (BITS, Pilani)

35. Electric field of a uniformly charged sphere Esphere Dr. Champak B. Das (BITS, Pilani)

36. At a point inside the sphere ( r < R ) Dr. Champak B. Das (BITS, Pilani)

37. ► ► ► ► ► P Field lines inside the sphere : ( Inside the sphere the field is uniform ) Dr. Champak B. Das (BITS, Pilani)

38. At a point outside the sphere ( r > R ) Dr. Champak B. Das (BITS, Pilani)

39. Total dipole moment of the sphere: (potential due to a dipole at the origin) Dr. Champak B. Das (BITS, Pilani)

40. ► ► Field lines outside the sphere : P Dr. Champak B. Das (BITS, Pilani)

41. ► ► ► ► ► ► ► Field lines of a uniformly polarized sphere : Dr. Champak B. Das (BITS, Pilani)

42. Uniformly polarized sphere – A physical analysis Without polarization: Two spheres of opposite charge, superimposed and canceling each other With polarization: The centers get separated, with the positive sphere moving slightly upward and the negative sphere slightly downward Dr. Champak B. Das (BITS, Pilani)

43. + + + + + + + + + + + + + - d - - - - - - - - At the top a cap of LEFTOVER positive charge and at the bottom a cap of negative charge Bound Surface Charge b Dr. Champak B. Das (BITS, Pilani)

44. - _ _ d + + + Recall: Pr. 2.18 Two spheres , each of radius R, overlap partially. Dr. Champak B. Das (BITS, Pilani)

45. + + + + + + + + + + + + + - d - - - - - - - - Electric field in the region of overlap between the two spheres For an outside point: Dr. Champak B. Das (BITS, Pilani)

46. Prob. 4.10: A sphere of radius R carries a polarization where k is a constant and r is the vector from the center. (i) Calculate the bound charges b and b. (ii) Find the field inside and outside the sphere. Dr. Champak B. Das (BITS, Pilani)

47. The Electric Displacement Polarization Accumulation of Bound charges Total field = Field due to bound charges + field due to free charges Dr. Champak B. Das (BITS, Pilani)

48. Gauss’ Law in the presence of dielectrics Within the dielectric the total charge density: free charge bound charge caused by polarization NOT a result of polarization Dr. Champak B. Das (BITS, Pilani)

49. Gauss’ Law Defining Electric Displacement ( D ) : ( Differential form ) ( Integral form ) Dr. Champak B. Das (BITS, Pilani)

50. D & E : … “looks similar” apart from the factor of 0 ( ! ) …….but : Dr. Champak B. Das (BITS, Pilani)