| course |
course_year |
question_number |
tags |
title |
year |
Electromagnetism |
IB |
52 |
|
3.II.17B |
2008 |
(i) From Maxwell's equations in vacuum,
$$\begin{array}{ll}
\nabla \cdot \mathbf{E}=0 & \nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\
\nabla \cdot \mathbf{B}=0 & \nabla \times \mathbf{B}=\mu_{0} \epsilon_{0} \frac{\partial \mathbf{E}}{\partial t}
\end{array}$$
obtain the wave equation for the electric field E. [You may find the following identity useful: $\left.\nabla \times(\nabla \times \mathbf{A})=\nabla(\nabla \cdot \mathbf{A})-\nabla^{2} \mathbf{A} .\right]$
(ii) If the electric and magnetic fields of a monochromatic plane wave in vacuum are
$$\mathbf{E}(z, t)=\mathbf{E}{0} \mathrm{e}^{i(k z-\omega t)} \text { and } \mathbf{B}(z, t)=\mathbf{B}{0} \mathrm{e}^{i(k z-\omega t)}$$
show that the corresponding electromagnetic waves are transverse (that is, both fields have no component in the direction of propagation).
(iii) Use Faraday's law for these fields to show that
$$\mathbf{B}{0}=\frac{k}{\omega}\left(\hat{\mathbf{e}}{z} \times \mathbf{E}_{0}\right)$$
(iv) Explain with symmetry arguments how these results generalise to
$$\mathbf{E}(\mathbf{r}, t)=E_{0} \mathrm{e}^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)} \hat{\mathbf{n}} \quad \text { and } \quad \mathbf{B}(\mathbf{r}, t)=\frac{1}{c} E_{0} \mathrm{e}^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)}(\hat{\mathbf{k}} \times \hat{\mathbf{n}})$$
where $\hat{\mathbf{n}}$ is the polarisation vector, i.e., the unit vector perpendicular to the direction of motion and along the direction of the electric field, and $\hat{\mathbf{k}}$ is the unit vector in the direction of propagation of the wave.
(v) Using Maxwell's equations in vacuum prove that:
$$\oint_{\mathcal{A}}\left(1 / \mu_{0}\right)(\mathbf{E} \times \mathbf{B}) \cdot d \mathcal{A}=-\frac{\partial}{\partial t} \int_{\mathcal{V}}\left(\frac{\epsilon_{0} E^{2}}{2}+\frac{B^{2}}{2 \mu_{0}}\right) d V$$
where $\mathcal{V}$ is the closed volume and $\mathcal{A}$ is the bounding surface. Comment on the differing time dependencies of the left-hand-side of (1) for the case of (a) linearly-polarized and (b) circularly-polarized monochromatic plane waves.