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3.3.2.     Solution in a Vacuum – EM Waves   <CJ chap 24.4-24.7  >

3.3.2.1. Discussion

3.3.2.1.1.     Maxwell solved his equations in a vacuum – meaning no charges or currents on the RHS

3.3.2.1.2.     He found solutions:

3.3.2.1.2.1. With oscillating E & B perpendicular fields at any frequency, and any amplitude such that E = cB

3.3.2.1.2.2. The oscillations move at exactly the speed of light, c = (e₀ m₀)^-1\2 with E & B perpendicular to c

3.3.2.1.3.     The waves carry both energy and momenta and are transverse with the E direction giving polarization

3.3.2.1.3.1. Polarization can also be circular (left or right handed) corresponding to the spin of the photon

3.3.2.1.4.     The Doppler effect applies to EM waves (like to sound) and raises frequencies of oncoming waves 

3.3.2.2. Mathematical

3.3.2.2.1.     The wave is given by E(x,t) = E₀ cos (wt + ky + d) where d  is the phase in radians and

3.3.2.2.1.1. The angular frequency w is the angular velocity & related to the period T (=1/f) by wT = 2p

3.3.2.2.1.2. The wave number k is related to the wave length of a full wave by k l =  2p  

3.3.2.2.1.3. And  E₀ is the amplitude of the wave restricted to  E₀ = c B₀

3.3.2.2.1.4. Likewise, B(x,t) = B₀ cos (wt + ky + d) with the same values and such that  lf = w/k = c

3.3.2.2.2.     Energy and momenta densities of the wave:

3.3.2.2.2.1. The energy density is given generally by u = (½)e₀ E²  + (1/(2m₀ )) B²

3.3.2.2.2.1.1.     but one must insert the root mean square value for the oscillating fields as E_rms = E₀/(2)^1/2

3.3.2.2.2.1.2.     and likewise for the B field 

3.3.2.2.2.1.3.     The energy and momenta are equally distributed in the E and B fields.

3.3.2.2.2.2. The intensity of the EM wave is the power/m² = S = c u where u is the energy density above

3.3.2.2.3.     Doppler effect is given when V_rel << c by   f_o = f_s (1  v_rel/c) where refers to approach or recede  

3.3.2.3. Advanced