2001-54.md

course	course_year	question_number	tags	title	year
Electromagnetism	II	54	II 2001 Electromagnetism	A3.5 B3.3	2001

(i) Develop the theory of electromagnetic waves starting from Maxwell equations in vacuum. You should relate the wave-speed $c$ to $\epsilon_{0}$ and $\mu_{0}$ and establish the existence of plane, plane-polarized waves in which $\mathbf{E}$ takes the form

$$\mathbf{E}=\left(E_{0} \cos (k z-\omega t), 0,0\right) .$$

You should give the form of the magnetic field $\mathbf{B}$ in this case.

(ii) Starting from Maxwell's equation, establish Poynting's theorem.

$$-\mathbf{j} \cdot \mathbf{E}=\frac{\partial W}{\partial t}+\nabla \cdot \mathbf{S},$$

where $W=\frac{\epsilon_{0}}{2} \mathbf{E}^{2}+\frac{1}{2 \mu_{0}} \mathbf{B}^{2}$ and $\mathbf{S}=\frac{1}{\mu_{0}} \mathbf{E} \wedge \mathbf{B}$. Give physical interpretations of $W, S$ and the theorem.

Compute the averages over space and time of $W$ and $\mathbf{S}$ for the plane wave described in (i) and relate them. Comment on the result.