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NASSP Self-study Review 0f Electrodynamics

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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!

- Maxwell’s equations:
- Lorentz force:

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

- Mathematical tools:
- Second derivatives
- Product rules

- Potentials

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

- Power (Lorentz)
- Now
- So
- Use Ampere-Maxwell

- Identity
- Use Faraday
- So

- Define
- Mechanical energy density
- Electromagnetic energy density
- Poynting vector

- EM fields carry energy

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

- Maxwell’s equations:
- Curl of Faraday:

- Use Gauss & Ampere-Maxwell; wave equation
- Speed of light
- Monochromatic plane-wave solutions

constant

Transverse

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

Wavelength

Period

- Energy density
- Poynting vector
- Momentum density

- Time average
- Intensity

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.

- Electric field polarizes matter
- Potential in dipole approximation
- Bound charge density

Polarization: dipole moment

per unit volume

- Rewrite Gauss’ law
- Displacement field
- For linear isotropic media

Free charge density

Dielectric constant

- Magnetic field magnetizes matter
- Vector potential

Magnetization: magnetic moment per

unit volume

- Picture
- Dipole approximation
- For arbitrary constant vector
- Therefore

=0

Q.E.D.

- Bound current density

- Induction
- For linear isotropic media

Free current density

- New feature
- Rewrite Ampere-Maxwell

- Maxwell’s equations
- Constitutive relations
- Linear isotropic media

- Boundary conditions

- Energy density
- Poynting vector

- Assume electrical neutrality
- In general there may be mobile charges; use
- Resistivity

Conductivity

- Maxwell’s equations
- Curl of Faraday
- For constant use Ampere-Maxwell

- Wave equation
- In an ideal insulator
- Phase velocity
- Plane wave solution

New

Index of refraction

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

- Take propagation along z-axis
- Complex ‘ansatz’
- Substitution gives
- Solution

- Thus general solution is

Transverse

Phase

Attenuation!

Frequency dependant: dispersion

- Limiting cases
- High frequency
- Low frequency

Good insulator

Good conductor

Note: at very high frequencies conductivity is frequency dependant

- Magnetic field – take for simplicity

Good conductor

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?

- Electric field polarizes matter
- Model

…dynamically

Damping (radiation)

“Restoring force”

Driving force

- Electromagnetic wave
- Rewrite in complex form
- Steady state solution

Natural frequency

- Substitution of steady state solution
- Dipole moment

- Polarization
- Complex permittivity

Number of atoms/molecules per unit volume

- Even for a “good insulator”
- Low density (gases)

Absorption coefficient

Ignore paramagnetism/diamagnetism

- Low density

Frequency dependent: dispersion

Anomalous dispersion

- Electrons free to move; inertia keeps positive ions almost stationary
- Model
- Solution

Electron mass

No restoring force!

- Current density
- Conductivity

Electron number density

Drude model

- Electron collisions rare, so dissipation small
- Recall for conductor

Purely imaginary!!

- As
- 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