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Maxwell's Equations

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An overview of the 2nd semester of Electricity and Magnetism
Richard Clark
Mind Map by Richard Clark, updated more than 1 year ago
Richard Clark
Created by Richard Clark over 8 years ago
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Maxwell's Equations
  1. Electrodynamics
    1. EMF
      1. Faraday's Law
        1. A changing magnetic field induces an electric field and vice versa
          1. Lenz's Law: the induced emf acts to minimize the change in flux
      2. Potentials
        1. Lienard-Wiechert (for moving point charges)
          1. Fields (moving charge)
              1. General force law (contains all of classical electrodynamics)
      3. Electromagnetic Waves
        1. Linear Medium: no free charge or current, D and H linearly proportional to E and B
          1. Boundary Conditions
            1. Reflection and Transmission
              1. Normal Incidence
                1. Oblique Incidence
                  1. Snell's Law
                    1. Fresnel's Equations
              2. Vacuum
                  1. Transverse, E and B in phase, mutually perpendicular
                2. Conductor
                  1. Wave Guides
                    1. TE or TM but no TEM waves (hollow wave guide)
                      1. Only certain modes propagate
                        1. Cutoff frequency
                      1. Transverse, out of phase, B field lags behind E field, attenuates exponentially
                  2. Relativity
                    1. 4-vectors (e.g. 4-vector potential)
                      1. Einstein Notation
                        1. Classical electrodynamics is already consistent with special relativity
                          1. A relativistic treatment of electricity necessarily leads to magnetism
                            1. Field Transformations
                              1. Field Tensor
                                1. Maxwell's equations expressed by a single 4-vector equation, by applying the d'Alembertian and the Lorenz gauge condition
                          2. Conservation Laws
                            1. Magnets do no work
                              1. Continuity Equation (local conservation of charge)
                                1. 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
                                  1. Poynting Vector: energy flux transported by fields
                                    1. Radiation (carries energy to infinity)
                                      1. Dipole (electric and magnetic)
                                        1. Point Charge
                                    2. Newton's 3rd law doesn't hold for electrodynamics, but momentum is conserved overall when the momentum stored in fields is taken into account.
                                      1. Maxwell Stress Tensor
                                        1. Electromagnetic momentum conservation: Charges and fields exchange momentum, with the total conserved.

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