Maxwell's Equations

Maxwell's Equations

1. Electrodynamics

   • EMF
     • Faraday's Law
       • A changing magnetic field induces an electric field and vice versa
         • Lenz's Law: the induced emf acts to minimize the change in flux
   • Potentials
     • Lienard-Wiechert (for moving point charges)
       • Fields (moving charge)
         • • General force law (contains all of classical electrodynamics)

2. Electromagnetic Waves

   • Linear Medium: no free charge or current, D and H linearly proportional to E and B
     • Boundary Conditions
       • Reflection and Transmission
         • Normal Incidence
         • Oblique Incidence
           • Snell's Law
           • Fresnel's Equations
   • Vacuum
     • • Transverse, E and B in phase, mutually perpendicular
   • Conductor
     • Wave Guides
       • TE or TM but no TEM waves (hollow wave guide)
         • Only certain modes propagate
           • Cutoff frequency
     • • Transverse, out of phase, B field lags behind E field, attenuates exponentially

3. Relativity

   • 4-vectors (e.g. 4-vector potential)
   • Einstein Notation
   • Classical electrodynamics is already consistent with special relativity
     • A relativistic treatment of electricity necessarily leads to magnetism
     • Field Transformations
       • Field Tensor
         • Maxwell's equations expressed by a single 4-vector equation, by applying the d'Alembertian and the Lorenz gauge condition

4. Conservation Laws

   • Magnets do no work
   • Continuity Equation (local conservation of charge)
   • Poynting's Theorem: the work done on a charge is the decrease in energy in the fields minus the energy that flowed out of the system
     • Poynting Vector: energy flux transported by fields
       • Radiation (carries energy to infinity)
         • Dipole (electric and magnetic)
         • Point Charge
   • Newton's 3rd law doesn't hold for electrodynamics, but momentum is conserved overall when the momentum stored in fields is taken into account.
   • Maxwell Stress Tensor
     • Electromagnetic momentum conservation: Charges and fields exchange momentum, with the total conserved.

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