2004-82.md

course	course_year	question_number	tags	title	year
Electromagnetism	IB	82	IB 2004 Electromagnetism	3.II.19B	2004

Starting from Maxwell's equations, derive the law of energy conservation in the form

$$\frac{\partial W}{\partial t}+\nabla \cdot \mathbf{S}+\mathbf{J} \cdot \mathbf{E}=0$$

where $W=\frac{\epsilon_{0}}{2} E^{2}+\frac{1}{2 \mu_{0}} B^{2}$ and $\mathbf{S}=\frac{1}{\mu_{0}} \mathbf{E} \times \mathbf{B}$.

Evaluate $W$ and $\mathbf{S}$ for the plane electromagnetic wave in vacuum

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

where the relationships between $E_{0}, B_{0}, \omega$ and $k$ should be determined. Show that the electromagnetic energy propagates at speed $c^{2}=1 /\left(\epsilon_{0} \mu_{0}\right)$, i.e. show that $S=W c$.