course |
course_year |
question_number |
tags |
title |
year |
Electromagnetism |
IB |
14 |
|
Paper 2, Section I, A |
2019 |
Write down the solution for the scalar potential $\varphi(\mathbf{x})$ that satisfies
$$\nabla^{2} \varphi=-\frac{1}{\varepsilon_{0}} \rho,$$
with $\varphi(\mathbf{x}) \rightarrow 0$ as $r=|\mathbf{x}| \rightarrow \infty$. You may assume that the charge distribution $\rho(\mathbf{x})$ vanishes for $r>R$, for some constant $R$. In an expansion of $\varphi(\mathbf{x})$ for $r \gg R$, show that the terms of order $1 / r$ and $1 / r^{2}$ can be expressed in terms of the total charge $Q$ and the electric dipole moment $\mathbf{p}$, which you should define.
Write down the analogous solution for the vector potential $\mathbf{A}(\mathbf{x})$ that satisfies
$$\nabla^{2} \mathbf{A}=-\mu_{0} \mathbf{J}$$
with $\mathbf{A}(\mathbf{x}) \rightarrow \mathbf{0}$ as $r \rightarrow \infty$. You may assume that the current $\mathbf{J}(\mathbf{x})$ vanishes for $r>R$ and that it obeys $\nabla \cdot \mathbf{J}=0$ everywhere. In an expansion of $\mathbf{A}(\mathbf{x})$ for $r \gg R$, show that the term of order $1 / r$ vanishes.
$\left[\right.$ Hint: $\left.\frac{\partial}{\partial x_{j}}\left(x_{i} J_{j}\right)=J_{i}+x_{i} \frac{\partial J_{j}}{\partial x_{j}} .\right]$