Electric Fields in Matter and Polarization

1. Electric Fields in Matter • Polarization • Field of a polarized object • Electric displacement • Linear dielectrics

2. Conductors Matter Insulators/Dielectrics All charges are attached to specific atoms/molecules and can only have a restricted motion WITHIN the atom/molecule.

3. When a neutral atom is placed in an external electric field (E): … positively charged core (nucleus) is pushed along E; … centre of the negatively charged cloud is pushed in the opposite direction of E; • If E is large enough ► the atom gets pulled apart completely => the atom gets IONIZED

4. For less extreme fields ► an equilibrium is established ……. the attraction between the nucleus and the electrons AND ……. the repulsion between them caused by E => the atom gets POLARIZED

5. Induced Dipole Moment: (pointing along E) Atomic Polarizability

6. a +q +q -q -q d 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)

7. At distance d from centre, (where v is the volume of the atom)

8. 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:

9. Alignment of Polar Molecules: Polar molecules: molecules having permanent dipole moment • when put in a uniform external field:

10. Alignment of Polar Molecules: • when put in a non-uniform external field: +q F+ d -q F-

11. +q F+ E+ d -q E- F-

12. For perfect dipole of infinitesimal length, the torque about the centre : the torque about any other point:

13. 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?

14. 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

15. Material becomes POLARIZED A measure of this effect is POLARIZATION defined as: P dipole moment per unit volume

16. rs p The Field of a Polarized Object = sum of the fields produced by infinitesimal dipoles

17. Dividing the whole object into small elements, the dipole moment in each volume element d’ : Total potential :

18. Prove it ! Use a product rule :

20. Using Divergence theorem;

21. Defining: Surface Bound Charge Volume Bound Charge

22. surface charge density b volume charge density b

23. Field/Potential of a polarized object = Field/Potential produced by a surface bound charge b + Field/Potential produced by a volume bound charge b

24. Physical Interpretation of Bound Charges …… are not only mathematical entities devised for calculation; but represent perfectly genuine accumulations of charge !

25. Surface Bound Charge d P A dielectric tube Dipole momentof the small piece: = -q +q A Surface charge density:

26.  P A If the cut is not to P : A’ In general:

27. Volume Bound Charge + + + _ _ _ _ _ + + _ _ _ _ + + A non-uniform polarization accumulation of bound charge within the volume diverging P pile-up of negative charge +

28. = Net accumulated charge with a volume Opposite to the amount of charge pushed out of the volume through the surface

29. z  P R Field of a uniformly polarized sphere Choose: z-axis || P P is uniform

30. 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

31. Electric field of a uniformly charged sphere

32. At a point inside the sphere ( r < R )

33. Inside the sphere the field is uniform

34. At a point outside the sphere ( r > R )

35. Total dipole moment of the sphere: (potential due to a dipole at the origin)

36. 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

37. + + + + + + + + + + + + + - d - - - - - - - - At the top a cap of LEFTOVER positive charge and at the bottom a cap of negative charge Bound Surface Charge b

38. - _ _ d + + + Recall: Pr. 2.18 Two spheres , each of radius R, overlap partially.

39. + + + + + + + + + + + + + - d - - - - - - - - Electric field in the region of overlap between the two spheres For an outside point:

40. 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.

41. The Electric Displacement Polarization Accumulation of Bound charges Total field = Field due to bound charges + field due to free charges

42. 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

43. Gauss’ Law : Electric Displacement ( D ) :

44. Gauss’ Law

45. D & E :

46. Boundary Conditions: On normal components: On tangential components:

47. Linear Dielectrics Recall: Cause of polarization is an Electric field For some material (if E is not TOO strong) Electric susceptibility of the medium Total field due to (bound + free) charges

48. In a dielectric material, if e is independent of : Location ► Homogeneous ► Linear Magnitude of E ► Isotropic Direction of E

49. In linear (& isotropic) dielectrics; Permittivity of the material The dimensionless quantity: Relative permittivity or Dielectric constant of the material

50. Electric Constitutive Relations and / or Represent the behavior of materials