1. NASSP Self-study Review 0f Electrodynamics
Created by Dr G B Tupper

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 EM fields carry energy
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 …dynamically
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

51. Electromagnetic waves in Plasma
Electron collisions rare, so dissipation small
Recall for conductor
Purely imaginary!!

52. Electromagnetic waves in Plasma
Above the plasma frequency: waves propagate with negligible loss
Below the plasma frequency: no propagation, only exponential damping
Dispersion relation
Plasma frequency
F&F 2013 L46

53. Plasma - Ionosphere