| course |
course_year |
question_number |
tags |
title |
year |
Electromagnetism |
IB |
17 |
|
Paper 3, Section II, C |
2018 |
Use Maxwell's equations to show that
$$\frac{d}{d t} \int_{\Omega}\left(\frac{\epsilon_{0}}{2} \mathbf{E} \cdot \mathbf{E}+\frac{1}{2 \mu_{0}} \mathbf{B} \cdot \mathbf{B}\right) d V+\int_{\Omega} \mathbf{J} \cdot \mathbf{E} d V=-\frac{1}{\mu_{0}} \int_{\partial \Omega}(\mathbf{E} \times \mathbf{B}) \cdot \mathbf{n} d S$$
where $\Omega \subset \mathbb{R}^{3}$ is a bounded region, $\partial \Omega$ its boundary and $\mathbf{n}$ its outward-pointing normal. Give an interpretation for each of the terms in this equation.
A certain capacitor consists of two conducting, circular discs, each of large area $A$, separated by a small distance along their common axis. Initially, the plates carry charges $q_{0}$ and $-q_{0}$. At time $t=0$ the plates are connected by a resistive wire, causing the charge on the plates to decay slowly as $q(t)=q_{0} \mathrm{e}^{-\lambda t}$ for some constant $\lambda$. Construct the Poynting vector and show that energy flows radially out of the capacitor as it discharges.