NASSP Self-study Review 0f Electrodynamics

1. NASSP Self-studyReview 0f Electrodynamics • Created by Dr G B Tupper • gary.tupper@uct.ac.za

2. The following is intended to provide a review of classical electrodynamics at the 2nd and 3rd year physics level, i.e. up to chapter 9 of Griffiths book, preparatory to Honours. • You will notice break points with questions. Try your best to answer them before proceeding on – it is an important part of the process!

3. Basics • Maxwell’s equations: • Lorentz force:

4. Basics • Mathematical tools: • Gauss’ Theorem • Stokes’ Theorem • Gradient Theorem • Green’s function

5. Basics • Mathematical tools: • Second derivatives • Product rules • Potentials

6. Questions • Where is “charge conservation”? • Where is Coulomb’s “law”? • Where is Biot-Savart “law”? • What about Ohm’s “law”?

7. Work done on charge • Power (Lorentz) • Now • So • Use Ampere-Maxwell

8. Conservation of energy • Identity • Use Faraday • So

9. Poynting’s Theorem • Define • Mechanical energy density • Electromagnetic energy density • Poynting vector • EM fields carry energy

10. Questions • Problem: an infinite line charge along z-axis moves with velocity : Determine

11. Waves in vacuum • Maxwell’s equations: • Curl of Faraday:

12. Waves in vacuum • Use Gauss & Ampere-Maxwell; wave equation • Speed of light • Monochromatic plane-wave solutions constant Transverse

13. Questions • What is the meaning of the wave-number ? • What is the meaning of angular frequency ? • What is the associated magnetic field? Wavelength Period

14. Monochromatic plane-wave

17. Monochromatic plane-wave • Energy density • Poynting vector • Momentum density

18. Monochromatic plane-wave • Time average • Intensity

19. Questions A monochromatic plane-polarized wave propagating in the z-direction has Cartesian components in phase: . In contrast, a circularly-polarized wave propagating in the z-direction has Cartesian components out of phase: Describe in words what such a circularly-polarized wave looks like. One of the two casess “left-handed”, and the other is “right handed” – which is which? i Determine the corresponding magnetic field. Determine the instantaneous energy-density and Poynting vector.

20. Electrostatics in matter • Electric field polarizes matter • Potential in dipole approximation • Bound charge density Polarization: dipole moment per unit volume

21. Electrostatics in matter • Rewrite Gauss’ law • Displacement field • For linear isotropic media Free charge density

22. Dielectric constant

23. Magnetostatics in matter • Magnetic field magnetizes matter • Vector potential Magnetization: magnetic moment per unit volume

24. Magnetostatics in matter;Dipole moment proof • Picture • Dipole approximation • For arbitrary constant vector • Therefore =0 Q.E.D.

25. Magnetostatics in matter • Bound current density • Rewrite Ampere’s law • Induction • For linear isotropic media Free current density

27. Electrodynamics in matter • New feature • Rewrite Ampere-Maxwell

28. Electrodynamics in matter • Maxwell’s equations • Constitutive relations • Linear isotropic media

29. Electrodynamics in matter • Boundary conditions

30. Electrodynamics in matter • Energy density • Poynting vector

31. Electromagnetic waves in matter • Assume electrical neutrality • In general there may be mobile charges; use • Resistivity Conductivity

32. Electromagnetic waves in matter • Maxwell’s equations • Curl of Faraday • For constant use Ampere-Maxwell

33. Electromagnetic waves in matter • Wave equation • In an ideal insulator • Phase velocity • Plane wave solution New Index of refraction

34. Questions • What do you expect happens in real matter where the conductivity doesn’t vanish? • Which is more basic: wavelength or frequency?

35. Electromagnetic waves in matter • Take propagation along z-axis • Complex ‘ansatz’ • Substitution gives • Solution

36. Electromagnetic waves in matter • Thus general solution is Transverse Phase Attenuation! Frequency dependant: dispersion

37. Electromagnetic waves in matter • Limiting cases • High frequency • Low frequency Good insulator Good conductor Note: at very high frequencies conductivity is frequency dependant

38. Electromagnetic waves in matter • Magnetic field – take for simplicity

39. Electromagnetic waves in matter Good conductor

40. Questions What one calls a “good conductor” or “good insulator” is actually frequency dependant; i.e. is or ? Find the value of for pure water and for copper metal. Where does it lie in the electromagnetic spectrum in each case? For each determine the high-frequency skin depth. For each determine the skin depth of infrared radiation ( ). In the case of copper, what is the phase velocity of infrared radiation? In the case of copper, what is the ratio for infrared radiation?

41. Frequency dependence • Electric field polarizes matter • Model …dynamically Damping (radiation) “Restoring force” Driving force

42. Frequency dependence • Electromagnetic wave • Rewrite in complex form • Steady state solution Natural frequency

43. Frequency dependence • Substitution of steady state solution • Dipole moment

44. Frequency dependence • Polarization • Complex permittivity Number of atoms/molecules per unit volume

45. Frequency dependence • Even for a “good insulator” • Low density (gases) Absorption coefficient Ignore paramagnetism/diamagnetism

46. Frequency dependence • Low density Frequency dependent: dispersion

47. Frequency dependence Anomalous dispersion

48. Questions

49. Electromagnetic waves in Plasma • Electrons free to move; inertia keeps positive ions almost stationary • Model • Solution Electron mass No restoring force!

50. Electromagnetic waves in Plasma • Current density • Conductivity Electron number density Drude model