NASSP Self-study Review 0f Electrodynamics Created by Dr G B Tupper gary.tupper@uct.ac.za

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!

Basics Maxwell’s equations: Lorentz force:

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

Basics Mathematical tools: Second derivatives Product rules Potentials

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

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

Conservation of energy Identity Use Faraday So

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

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

Waves in vacuum Maxwell’s equations: Curl of Faraday:

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

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

Monochromatic plane-wave

Monochromatic plane-wave Energy density Poynting vector Momentum density

Monochromatic plane-wave Time average Intensity

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.

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

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

Dielectric constant

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

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

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

Electrodynamics in matter New feature Rewrite Ampere-Maxwell

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

Electrodynamics in matter Boundary conditions

Electrodynamics in matter Energy density Poynting vector

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

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

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

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

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

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

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

Electromagnetic waves in matter Magnetic field – take for simplicity

Electromagnetic waves in matter Good conductor

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?  

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

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

Frequency dependence Substitution of steady state solution Dipole moment

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

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

Frequency dependence Low density Frequency dependent: dispersion

Frequency dependence Anomalous dispersion

Questions

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

Electromagnetic waves in Plasma Current density Conductivity Electron number density Drude model

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

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

Plasma - Ionosphere