| course |
course_year |
question_number |
tags |
title |
year |
Electromagnetism |
IB |
17 |
|
Paper 3, Section II, 15D |
2021 |
(a) The energy density stored in the electric and magnetic fields $\mathbf{E}$ and $\mathbf{B}$ is given by
$$w=\frac{\epsilon_{0}}{2} \mathbf{E} \cdot \mathbf{E}+\frac{1}{2 \mu_{0}} \mathbf{B} \cdot \mathbf{B}$$
Show that, in regions where no electric current flows,
$$\frac{\partial w}{\partial t}+\boldsymbol{\nabla} \cdot \mathbf{S}=0$$
for some vector field $\mathbf{S}$ that you should determine.
(b) The coordinates $x^{\prime \mu}=\left(c t^{\prime}, \mathbf{x}^{\prime}\right)$ in an inertial frame $\mathcal{S}^{\prime}$ are related to the coordinates $x^{\mu}=(c t, \mathbf{x})$ in an inertial frame $\mathcal{S}$ by a Lorentz transformation $x^{\prime \mu}=\Lambda_{\nu}^{\mu} x^{\nu}$, where
$$\Lambda_{\nu}^{\mu}=\left(\begin{array}{cccc}
\gamma & -\gamma v / c & 0 & 0 \\
-\gamma v / c & \gamma & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right)$$
with $\gamma=\left(1-v^{2} / c^{2}\right)^{-1 / 2}$. Here $v$ is the relative velocity of $\mathcal{S}^{\prime}$ with respect to $\mathcal{S}$ in the x-direction.
In frame $\mathcal{S}^{\prime}$, there is a static electric field $\mathbf{E}^{\prime}\left(\mathbf{x}^{\prime}\right)$ with $\partial \mathbf{E}^{\prime} / \partial t^{\prime}=0$, and no magnetic field. Calculate the electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ in frame $\mathcal{S}$. Show that the energy density in frame $\mathcal{S}$ is given in terms of the components of $\mathbf{E}^{\prime}$ by
$$w=\frac{\epsilon_{0}}{2}\left[E_{x}^{\prime 2}+\left(\frac{c^{2}+v^{2}}{c^{2}-v^{2}}\right)\left(E_{y}^{\prime 2}+E_{z}^{\prime 2}\right)\right]$$
Use the fact that $\partial w / \partial t^{\prime}=0$ to show that
$$\frac{\partial w}{\partial t}+\nabla \cdot\left(w v \mathbf{e}_{x}\right)=0$$
where $\mathbf{e}_{x}$ is the unit vector in the $x$-direction.