2. [25 points] A plane electromagnetic wave propagating through a vacuum is described by the following electric and magnetic fields:

(a) [4 points] In which direction in terms of the (x,y,z) coordinates is the wave traveling?

The wave travels in the direction of the Poynting vector S = (1/m_o)E´B so the direction is y´x = -z , i.e., the wave travels along the –z axis.

[In each of the following give your answers in algebraic from using the above symbols.]
(b) [4 points] Find the wavelength l of the wave.

The wavelength is l = 2p/k .

(c) [4 points] Find the frequency f of the wave.

The frequency is f = w/2p .

(d) [4 points] Find the instantaneous magnitude of the Poynting vector S of the wave.

The Poynting vector is S = (1/m_o)E´B = (1/m_o)[E_oSin(kz - w t)][(E_o/c)Sin(kz - w t)] = (1/ m_oc) E_o² Sin²(kz - w t)

(e) [6 points] An observer now moves at a velocity of v=c/4 toward the oncoming wave. What will he measure for the wavelength l’ and the frequency f’ of the wave?
(Give your answers as fractions of f and l.)

The quantity (v/c) = ¼ . The observed wavelength is l’ = l{[1-(v/c)]/[1+(v/c)]}^½
= l{[1 - ¼]/[1+ ¼]}^½ = l{[3/4]/[5/4]}^½ = l{3/5}^½ = 0.775 l

The observed frequency is f’ = f{[1+(v/c)]/[1-(v/c)]}^½ = f {[1 + ¼]/[1- ¼]}^½
= f {[5/4]/[3/4]}^½ = l{5/3}^½ = 1.29 f

(f) [3 points] What will the observer measure for the velocity of the wave?

The observer will measure the wave velocity to be c (the velocity of light).