THE UNIVERSITY OF BRITISH COLUMBIA
 
Physics 401 Assignment # 6:
 
ELECTROMAGNETIC WAVES
 
Wed. 8 Feb. 2006 - finish by Wed. 22 Feb.

1. CMBR: Most of the electromagnetic energy in the universe is in the cosmic microwave background radiation (CMBR), sometimes referred to as the $3^\circ$ Kelvin background. Penzias and Wilson discovered the CMBR in 1965 using a radio telescope, and subsequently received the Nobel Prize for this discovery. This background radiation has wavelength $\lambda \sim 1.1$ mm. The energy density of the CMBR is about 4.0 x 10^-14 J/m³. What is the rms electric field strength of the CMBR?

2. STANDING WAVES: Consider standing electromagnetic waves:
   
   $$\Vec{E} = E_0 \left( \sin k z \; \sin \omega t \right) \Hat . . . . . . } = B_0 \left( \cos k z \; \cos \omega t \right) \Hat{y} \; .$$
   
   
   1. Show that these satisfy the wave equation (9.2).
   2. Show that we must also have $c = \omega / k$ and E₀ = c B₀.
   3. Show that the time-averaged power flow across any area will be zero.
   4. Show that the Poynting vector will also be zero, i.e. there is no net energy flow.

3. (p. 386, Problem 9.14) - REFLECTED & TRANSMITTED POLARIZATION: In Eqs. (9.76) and (9.77) it was tacitly assumed that the reflected and transmitted waves have the same polarization as the incident wave, namely along the $\Hat{x}$ direction. Prove that this must be so. [Hint: Let the polarization vectors of the reflected and transmitted waves be
   
   $$\Hat{n}_T = \cos \theta_T \Hat{x} + \sin \theta_T \Hat{y} . . . . . . d \Hat{n}_R = \cos \theta_R \Hat{x} + \sin \theta_R \Hat{y}$$
   
   
   and prove from the boundary conditions that $\theta_T = \theta_R = 0$.]

4. (p. 392, Problem 9.15) - COMPLEX ALGEBRA EXERCISE: Suppose that we have six nonzero constants A,B,C,a,b,c such than Ae^iax + Be^ibx = Ce^icx for all x. Prove that a=b=c and A+B=C.

5. (p. 392, Problem 9.17) - DIAMOND: The index of refraction of diamond is 2.42. Construct the graph analogous to Figure 9.16 for the air/diamond interface. (Assume $\mu_1 = \mu_2 = \muz$.) In particular, calculate
   1. the amplitudes at normal incidence;
   2. Brewster's angle; and
   3. the "crossover" angle at which the reflected and transmitted amplitudes are equal.

6. PLANE WAVE STRESS TENSOR: Find all the elements of the Maxwell stress tensor of a monochromatic plane wave traveling in the z-direction, polarized in the x-direction:
   
   $$\begin{eqnarray*} \Vec{E}(z,t) &=& E_0 \cos(kz - \omega t + \delta) \Hat{x} \cr . . . . . . }(z,t) &=& {E_0\over c} \cos(kz - \omega t + \delta) \Hat{y} \cr \end{eqnarray*}$$
   
   In what direction does this EM wave transport momentum?  Does this agree with the form of the Maxwell stress tensor you just deduced? 

7. (p. 412, Problem 9.33) - SPHERICAL WAVES: Suppose
   
   $$\Vec{E}(r,\theta,\phi,t) = A {\sin \theta \over r} \left[ \ . . . . . . hi} \qquad \hbox{\rm with} \qquad c = {\omega \over k} \; ,$$
   
   
   as usual.  [This is, incidentally, the simplest possible spherical wave.  For notational convenience, let $(k r - \omega t) \equiv u$ in your calculations.]
   1. Show that $\Vec{E}$ obeys all four of Maxwell's equations, in vacuum, and find the associated magnetic field.
   2. Calculate the Poynting vector. Average $\Vec{S}$ over a full cycle to get the intensity vector $\Vec{I}$.  Does $\Vec{I}$ point in the expected direction?  Does it fall off like r^-2, as it should?
   3. Integrate $\Vec{I}\cdot d\Vec{a}$ over a spherical surface to determine the total power radiated.  [You should get $P = 4 \pi A^2 / 3\muz c$.]