# Electromagnetism

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Paper 1, Section II, 15D

comment(a) Show that the magnetic flux passing through a simple, closed curve $C$ can be written as

$\Phi=\oint_{C} \mathbf{A} \cdot \mathbf{d} \mathbf{x},$

where $\mathbf{A}$ is the magnetic vector potential. Explain why this integral is independent of the choice of gauge.

(b) Show that the magnetic vector potential due to a static electric current density $\mathbf{J}$, in the Coulomb gauge, satisfies Poisson's equation

$-\nabla^{2} \mathbf{A}=\mu_{0} \mathbf{J}$

Hence obtain an expression for the magnetic vector potential due to a static, thin wire, in the form of a simple, closed curve $C$, that carries an electric current $I$. [You may assume that the electric current density of the wire can be written as

$\mathbf{J}(\mathbf{x})=I \int_{C} \delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) \mathbf{d} \mathbf{x}^{\prime}$

where $\delta^{(3)}$ is the three-dimensional Dirac delta function.]

(c) Consider two thin wires, in the form of simple, closed curves $C_{1}$ and $C_{2}$, that carry electric currents $I_{1}$ and $I_{2}$, respectively. Let $\Phi_{i j}$ (where $i, j \in\{1,2\}$ ) be the magnetic flux passing through the curve $C_{i}$ due to the current $I_{j}$ flowing around $C_{j}$. The inductances are defined by $L_{i j}=\Phi_{i j} / I_{j}$. By combining the results of parts (a) and (b), or otherwise, derive Neumann's formula for the mutual inductance,

$L_{12}=L_{21}=\frac{\mu_{0}}{4 \pi} \oint_{C_{1}} \oint_{C_{2}} \frac{\mathbf{d} \mathbf{x}_{1} \cdot \mathbf{d} \mathbf{x}_{2}}{\left|\mathbf{x}_{1}-\mathbf{x}_{2}\right|} .$

Suppose that $C_{1}$ is a circular loop of radius $a$, centred at $(0,0,0)$ and lying in the plane $z=0$, and that $C_{2}$ is a different circular loop of radius $b$, centred at $(0,0, c)$ and lying in the plane $z=c$. Show that the mutual inductance of the two loops is

$\frac{\mu_{0}}{4} \sqrt{a^{2}+b^{2}+c^{2}} f(q)$

where

$q=\frac{2 a b}{a^{2}+b^{2}+c^{2}}$

and the function $f(q)$ is defined, for $0<q<1$, by the integral

$f(q)=\int_{0}^{2 \pi} \frac{q \cos \theta d \theta}{\sqrt{1-q \cos \theta}}$

Paper 2, Section I, $4 \mathrm{D}$

commentState Gauss's Law in the context of electrostatics.

A simple coaxial cable consists of an inner conductor in the form of a perfectly conducting, solid cylinder of radius $a$, surrounded by an outer conductor in the form of a perfectly conducting, cylindrical shell of inner radius $b>a$ and outer radius $c>b$. The cylinders are coaxial and the gap between them is filled with a perfectly insulating material. The cable may be assumed to be straight and arbitrarily long.

In a steady state, the inner conductor carries an electric charge $+Q$ per unit length, and the outer conductor carries an electric charge $-Q$ per unit length. The charges are distributed in a cylindrically symmetric way and no current flows through the cable.

Determine the electrostatic potential and the electric field as functions of the cylindrical radius $r$, for $0<r<\infty$. Calculate the capacitance $C$ of the cable per unit length and the electrostatic energy $U$ per unit length, and verify that these are related by

$U=\frac{Q^{2}}{2 C}$

Paper 2, Section II, $16 \mathrm{D}$

comment(a) Show that, for $|\mathbf{x}| \gg|\mathbf{y}|$,

$\frac{1}{|\mathbf{x}-\mathbf{y}|}=\frac{1}{|\mathbf{x}|}\left[1+\frac{\mathbf{x} \cdot \mathbf{y}}{|\mathbf{x}|^{2}}+\frac{3(\mathbf{x} \cdot \mathbf{y})^{2}-|\mathbf{x}|^{2}|\mathbf{y}|^{2}}{2|\mathbf{x}|^{4}}+O\left(\frac{|\mathbf{y}|^{3}}{|\mathbf{x}|^{3}}\right)\right]$

(b) A particle with electric charge $q>0$ has position vector $(a, 0,0)$, where $a>0$. An earthed conductor (held at zero potential) occupies the plane $x=0$. Explain why the boundary conditions can be satisfied by introducing a fictitious 'image' particle of appropriate charge and position. Hence determine the electrostatic potential and the electric field in the region $x>0$. Find the leading-order approximation to the potential for $|\mathbf{x}| \gg a$ and compare with that of an electric dipole. Directly calculate the total flux of the electric field through the plane $x=0$ and comment on the result. Find the induced charge distribution on the surface of the conductor, and the total induced surface charge. Sketch the electric field lines in the plane $z=0$.

(c) Now consider instead a particle with charge $q$ at position $(a, b, 0)$, where $a>0$ and $b>0$, with earthed conductors occupying the planes $x=0$ and $y=0$. Find the leading-order approximation to the potential in the region $x, y>0$ for $|\mathbf{x}| \gg a, b$ and state what type of multipole potential this is.

Paper 3, Section II, 15D

comment(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.

Paper 4, Section I, $5 \mathrm{D}$

commentWrite down Maxwell's equations in a vacuum. Show that they admit wave solutions with

$\mathbf{B}(\mathbf{x}, t)=\operatorname{Re}\left[\mathbf{B}_{0} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right]$

where $\mathbf{B}_{0}, \mathbf{k}$ and $\omega$ must obey certain conditions that you should determine. Find the corresponding electric field $\mathbf{E}(\mathbf{x}, t)$.

A light wave, travelling in the $x$-direction and linearly polarised so that the magnetic field points in the $z$-direction, is incident upon a conductor that occupies the half-space $x>0$. The electric and magnetic fields obey the boundary conditions $\mathbf{E} \times \mathbf{n}=\mathbf{0}$ and $\mathbf{B} \cdot \mathbf{n}=0$ on the surface of the conductor, where $\mathbf{n}$ is the unit normal vector. Determine the contributions to the magnetic field from the incident and reflected waves in the region $x \leqslant 0$. Compute the magnetic field tangential to the surface of the conductor.

Paper 1, Section II, D

commentWrite down the electric potential due to a point charge $Q$ at the origin.

A dipole consists of a charge $Q$ at the origin, and a charge $-Q$ at position $-\mathbf{d}$. Show that, at large distances, the electric potential due to such a dipole is given by

$\Phi(\mathbf{x})=\frac{1}{4 \pi \epsilon_{0}} \frac{\mathbf{p} \cdot \mathbf{x}}{|\mathbf{x}|^{3}}$

where $\mathbf{p}=Q \mathbf{d}$ is the dipole moment. Hence show that the potential energy between two dipoles $\mathbf{p}_{1}$ and $\mathbf{p}_{2}$, with separation $\mathbf{r}$, where $|\mathbf{r}| \gg|\mathbf{d}|$, is

$U=\frac{1}{8 \pi \epsilon_{0}}\left(\frac{\mathbf{p}_{1} \cdot \mathbf{p}_{2}}{r^{3}}-\frac{3\left(\mathbf{p}_{1} \cdot \mathbf{r}\right)\left(\mathbf{p}_{2} \cdot \mathbf{r}\right)}{r^{5}}\right)$

Dipoles are arranged on an infinite chessboard so that they make an angle $\theta$ with the horizontal in an alternating pattern as shown in the figure. Compute the energy between a given dipole and its four nearest neighbours, and show that this is independent of $\theta$.

Paper 2, Section I, D

commentTwo concentric spherical shells with radii $R$ and $2 R$ carry fixed, uniformly distributed charges $Q_{1}$ and $Q_{2}$ respectively. Find the electric field and electric potential at all points in space. Calculate the total energy of the electric field.

Paper 2, Section II, D

comment(a) A surface current $\mathbf{K}=K \mathbf{e}_{x}$, with $K$ a constant and $\mathbf{e}_{x}$ the unit vector in the $x$-direction, lies in the plane $z=0$. Use Ampère's law to determine the magnetic field above and below the plane. Confirm that the magnetic field is discontinuous across the surface, with the discontinuity given by

$\lim _{z \rightarrow 0^{+}} \mathbf{e}_{z} \times \mathbf{B}-\lim _{z \rightarrow 0^{-}} \mathbf{e}_{z} \times \mathbf{B}=\mu_{0} \mathbf{K}$

where $\mathbf{e}_{z}$ is the unit vector in the $z$-direction.

(b) A surface current $\mathbf{K}$ flows radially in the $z=0$ plane, resulting in a pile-up of charge $Q$ at the origin, with $d Q / d t=I$, where $I$ is a constant.

Write down the electric field $\mathbf{E}$ due to the charge at the origin, and hence the displacement current $\epsilon_{0} \partial \mathbf{E} / \partial t$.

Confirm that, away from the plane and for $\theta<\pi / 2$, the magnetic field due to the displacement current is given by

$\mathbf{B}(r, \theta)=\frac{\mu_{0} I}{4 \pi r} \tan \left(\frac{\theta}{2}\right) \mathbf{e}_{\phi}$

where $(r, \theta, \phi)$ are the usual spherical polar coordinates. [Hint: Use Stokes' theorem applied to a spherical cap that subtends an angle $\theta$.]

Paper 1, Section II, A

commentLet $\mathbf{E}(\mathbf{x})$ be the electric field and $\varphi(\mathbf{x})$ the scalar potential due to a static charge density $\rho(\mathbf{x})$, with all quantities vanishing as $r=|\mathbf{x}|$ becomes large. The electrostatic energy of the configuration is given by

$U=\frac{\varepsilon_{0}}{2} \int|\mathbf{E}|^{2} d V=\frac{1}{2} \int \rho \varphi d V$

with the integrals taken over all space. Verify that these integral expressions agree.

Suppose that a total charge $Q$ is distributed uniformly in the region $a \leqslant r \leqslant b$ and that $\rho=0$ otherwise. Use the integral form of Gauss's Law to determine $\mathbf{E}(\mathbf{x})$ at all points in space and, without further calculation, sketch graphs to indicate how $|\mathbf{E}|$ and $\varphi$ depend on position.

Consider the limit $b \rightarrow a$ with $Q$ fixed. Comment on the continuity of $\mathbf{E}$ and $\varphi$. Verify directly from each of the integrals in $(*)$ that $U=Q \varphi(a) / 2$ in this limit.

Now consider a small change $\delta Q$ in the total charge $Q$. Show that the first-order change in the energy is $\delta U=\delta Q \varphi(a)$ and interpret this result.

Paper 2, Section I, A

commentWrite 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: $\left.\frac{\partial}{\partial x_{j}}\left(x_{i} J_{j}\right)=J_{i}+x_{i} \frac{\partial J_{j}}{\partial x_{j}} .\right]$

Paper 2, Section II, A

commentConsider a conductor in the shape of a closed curve $C$ moving in the presence of a magnetic field B. State Faraday's Law of Induction, defining any quantities that you introduce.

Suppose $C$ is a square horizontal loop that is allowed to move only vertically. The location of the loop is specified by a coordinate $z$, measured vertically upwards, and the edges of the loop are defined by $x=\pm a,-a \leqslant y \leqslant a$ and $y=\pm a,-a \leqslant x \leqslant a$. If the magnetic field is

$\mathbf{B}=b(x, y,-2 z),$

where $b$ is a constant, find the induced current $I$, given that the total resistance of the loop is $R$.

Calculate the resulting electromagnetic force on the edge of the loop $x=a$, and show that this force acts at an angle $\tan ^{-1}(2 z / a)$ to the vertical. Find the total electromagnetic force on the loop and comment on its direction.

Now suppose that the loop has mass $m$ and that gravity is the only other force acting on it. Show that it is possible for the loop to fall with a constant downward velocity $R m g /\left(8 b a^{2}\right)^{2}$.

Paper 3, Section II, A

commentThe electric and magnetic fields $\mathbf{E}, \mathbf{B}$ in an inertial frame $\mathcal{S}$ are related to the fields $\mathbf{E}^{\prime}, \mathbf{B}^{\prime}$ in a frame $\mathcal{S}^{\prime}$ by a Lorentz transformation. Given that $\mathcal{S}^{\prime}$ moves in the $x$-direction with speed $v$ relative to $\mathcal{S}$, and that

$E_{y}^{\prime}=\gamma\left(E_{y}-v B_{z}\right), \quad B_{z}^{\prime}=\gamma\left(B_{z}-\left(v / c^{2}\right) E_{y}\right),$

write down equations relating the remaining field components and define $\gamma$. Use your answers to show directly that $\mathbf{E}^{\prime} \cdot \mathbf{B}^{\prime}=\mathbf{E} \cdot \mathbf{B}$.

Give an expression for an additional, independent, Lorentz-invariant function of the fields, and check that it is invariant for the special case when $E_{y}=E$ and $B_{y}=B$ are the only non-zero components in the frame $\mathcal{S}$.

Now suppose in addition that $c B=\lambda E$ with $\lambda$ a non-zero constant. Show that the angle $\theta$ between the electric and magnetic fields in $\mathcal{S}^{\prime}$ is given by

$\cos \theta=f(\beta)=\frac{\lambda\left(1-\beta^{2}\right)}{\left\{\left(1+\lambda^{2} \beta^{2}\right)\left(\lambda^{2}+\beta^{2}\right)\right\}^{1 / 2}}$

where $\beta=v / c$. By considering the behaviour of $f(\beta)$ as $\beta$ approaches its limiting values, show that the relative velocity of the frames can be chosen so that the angle takes any value in one of the ranges $0 \leqslant \theta<\pi / 2$ or $\pi / 2<\theta \leqslant \pi$, depending on the sign of $\lambda$.

Paper 4, Section I, A

commentWrite down Maxwell's Equations for electric and magnetic fields $\mathbf{E}(\mathbf{x}, t)$ and $\mathbf{B}(\mathbf{x}, t)$ in the absence of charges and currents. Show that there are solutions of the form

$\mathbf{E}(\mathbf{x}, t)=\operatorname{Re}\left\{\mathbf{E}_{0} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right\}, \quad \mathbf{B}(\mathbf{x}, t)=\operatorname{Re}\left\{\mathbf{B}_{0} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right\}$

if $\mathbf{E}_{0}$ and $\mathbf{k}$ satisfy a constraint and if $\mathbf{B}_{0}$ and $\omega$ are then chosen appropriately.

Find the solution with $\mathbf{E}_{0}=E(1, i, 0)$, where $E$ is real, and $\mathbf{k}=k(0,0,1)$. Compute the Poynting vector and state its physical significance.

Paper 1, Section II, C

commentStarting from the Lorentz force law acting on a current distribution $\mathbf{J}$ obeying $\boldsymbol{\nabla} \cdot \mathbf{J}=0$, show that the energy of a magnetic dipole $\mathbf{m}$ in the presence of a time independent magnetic field $\mathbf{B}$ is

$U=-\mathbf{m} \cdot \mathbf{B}$

State clearly any approximations you make.

[You may use without proof the fact that

$\int(\mathbf{a} \cdot \mathbf{r}) \mathbf{J}(\mathbf{r}) \mathrm{d} V=-\frac{1}{2} \mathbf{a} \times \int(\mathbf{r} \times \mathbf{J}(\mathbf{r})) \mathrm{d} V$

for any constant vector $\mathbf{a}$, and the identity

$(\mathbf{b} \times \boldsymbol{\nabla}) \times \mathbf{c}=\boldsymbol{\nabla}(\mathbf{b} \cdot \mathbf{c})-\mathbf{b}(\boldsymbol{\nabla} \cdot \mathbf{c})$

which holds when $\mathbf{b}$ is constant.]

A beam of slowly moving, randomly oriented magnetic dipoles enters a region where the magnetic field is

$\mathbf{B}=\hat{\mathbf{z}} B_{0}+(y \hat{\mathbf{x}}+x \hat{\mathbf{y}}) B_{1},$

with $B_{0}$ and $B_{1}$ constants. By considering their energy, briefly describe what happens to those dipoles that are parallel to, and those that are anti-parallel to the direction of $\mathbf{B}$.

Paper 2, Section I, $\mathbf{6 C}$

commentDerive the Biot-Savart law

$\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \int_{V} \frac{\mathbf{j}\left(\mathbf{r}^{\prime}\right) \times\left(\mathbf{r}-\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{3}} \mathrm{~d} V$

from Maxwell's equations, where the time-independent current $\mathbf{j}(\mathbf{r})$ vanishes outside $V$. [You may assume that the vector potential can be chosen to be divergence-free.]

Paper 2, Section II, C

commentA plane with unit normal $\mathbf{n}$ supports a charge density and a current density that are each time-independent. Show that the tangential components of the electric field and the normal component of the magnetic field are continuous across the plane.

Albert moves with constant velocity $\mathbf{v}=v \mathbf{n}$ relative to the plane. Find the boundary conditions at the plane on the normal component of the magnetic field and the tangential components of the electric field as seen in Albert's frame.

Paper 3, Section II, C

commentUse 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.

Paper 4, Section I, $7 \mathrm{C}$

commentShow that Maxwell's equations imply the conservation of charge.

A conducting medium has $\mathbf{J}=\sigma \mathbf{E}$ where $\sigma$ is a constant. Show that any charge density decays exponentially in time, at a rate to be determined.

Paper 1, Section II, C

commentWrite down Maxwell's equations for the electric field $\mathbf{E}(\mathbf{x}, t)$ and the magnetic field $\mathbf{B}(\mathbf{x}, t)$ in a vacuum. Deduce that both $\mathbf{E}$ and $\mathbf{B}$ satisfy a wave equation, and relate the wave speed $c$ to the physical constants $\epsilon_{0}$ and $\mu_{0}$.

Verify that there exist plane-wave solutions of the form

$\begin{aligned} &\mathbf{E}(\mathbf{x}, t)=\operatorname{Re}\left[\mathbf{e} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right] \\ &\mathbf{B}(\mathbf{x}, t)=\operatorname{Re}\left[\mathbf{b} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right] \end{aligned}$

where $\mathbf{e}$ and $\mathbf{b}$ are constant complex vectors, $\mathbf{k}$ is a constant real vector and $\omega$ is a real constant. Derive the dispersion relation that relates the angular frequency $\omega$ of the wave to the wavevector $\mathbf{k}$, and give the algebraic relations between the vectors $\mathbf{e}, \mathbf{b}$ and $\mathbf{k}$ implied by Maxwell's equations.

Let $\mathbf{n}$ be a constant real unit vector. Suppose that a perfect conductor occupies the region $\mathbf{n} \cdot \mathbf{x}<0$ with a plane boundary $\mathbf{n} \cdot \mathbf{x}=0$. In the vacuum region $\mathbf{n} \cdot \mathbf{x}>0$, a plane electromagnetic wave of the above form, with $\mathbf{k} \cdot \mathbf{n}<0$, is incident on the plane boundary. Write down the boundary conditions on $\mathbf{E}$ and $\mathbf{B}$ at the surface of the conductor. Show that Maxwell's equations and the boundary conditions are satisfied if the solution in the vacuum region is the sum of the incident wave given above and a reflected wave of the form

$\begin{aligned} &\mathbf{E}^{\prime}(\mathbf{x}, t)=\operatorname{Re}\left[\mathbf{e}^{\prime} e^{i\left(\mathbf{k}^{\prime} \cdot \mathbf{x}-\omega t\right)}\right] \\ &\mathbf{B}^{\prime}(\mathbf{x}, t)=\operatorname{Re}\left[\mathbf{b}^{\prime} e^{i\left(\mathbf{k}^{\prime} \cdot \mathbf{x}-\omega t\right)}\right] \end{aligned}$

where

$\begin{aligned} &\mathbf{e}^{\prime}=-\mathbf{e}+2(\mathbf{n} \cdot \mathbf{e}) \mathbf{n} \\ &\mathbf{b}^{\prime}=\mathbf{b}-2(\mathbf{n} \cdot \mathbf{b}) \mathbf{n} \\ &\mathbf{k}^{\prime}=\mathbf{k}-2(\mathbf{n} \cdot \mathbf{k}) \mathbf{n} \end{aligned}$

Paper 2, Section I, $\mathbf{6 C}$

commentState Gauss's Law in the context of electrostatics.

A spherically symmetric capacitor consists of two conductors in the form of concentric spherical shells of radii $a$ and $b$, with $b>a$. The inner sphere carries a charge $Q$ and the outer sphere carries a charge $-Q$. Determine the electric field $\mathbf{E}$ and the electrostatic potential $\phi$ in the regions $r<a, a<r<b$ and $r>b$. Show that the capacitance is

$C=\frac{4 \pi \epsilon_{0} a b}{b-a}$

and calculate the electrostatic energy of the system in terms of $Q$ and $C$.

Paper 2, Section II, C

commentIn special relativity, the electromagnetic fields can be derived from a 4-vector potential $A^{\mu}=(\phi / c, \mathbf{A})$. Using the Minkowski metric tensor $\eta_{\mu \nu}$ and its inverse $\eta^{\mu \nu}$, state how the electromagnetic tensor $F_{\mu \nu}$ is related to the 4-potential, and write out explicitly the components of both $F_{\mu \nu}$ and $F^{\mu \nu}$ in terms of those of $\mathbf{E}$ and $\mathbf{B}$.

If $x^{\prime \mu}=\Lambda_{\nu}^{\mu} x^{\nu}$ is a Lorentz transformation of the spacetime coordinates from one inertial frame $\mathcal{S}$ to another inertial frame $\mathcal{S}^{\prime}$, state how $F^{\prime \mu \nu}$ is related to $F^{\mu \nu}$.

Write down the Lorentz transformation matrix for a boost in standard configuration, such that frame $\mathcal{S}^{\prime}$ moves relative to frame $\mathcal{S}$ with speed $v$ in the $+x$ direction. Deduce the transformation laws

$\begin{aligned} E_{x}^{\prime} &=E_{x} \\ E_{y}^{\prime} &=\gamma\left(E_{y}-v B_{z}\right) \\ E_{z}^{\prime} &=\gamma\left(E_{z}+v B_{y}\right) \\ B_{x}^{\prime} &=B_{x} \\ B_{y}^{\prime} &=\gamma\left(B_{y}+\frac{v}{c^{2}} E_{z}\right) \\ B_{z}^{\prime} &=\gamma\left(B_{z}-\frac{v}{c^{2}} E_{y}\right) \end{aligned}$

where $\gamma=\left(1-\frac{v^{2}}{c^{2}}\right)^{-1 / 2}$

In frame $\mathcal{S}$, an infinitely long wire of negligible thickness lies along the $x$ axis. The wire carries $n$ positive charges $+q$ per unit length, which travel at speed $u$ in the $+x$ direction, and $n$ negative charges $-q$ per unit length, which travel at speed $u$ in the $-x$ direction. There are no other sources of the electromagnetic field. Write down the electric and magnetic fields in $\mathcal{S}$ in terms of Cartesian coordinates. Calculate the electric field in frame $\mathcal{S}^{\prime}$, which is related to $\mathcal{S}$ by a boost by speed $v$ as described above. Give an explanation of the physical origin of your expression.

Paper 3, Section II, C

comment(i) Two point charges, of opposite sign and unequal magnitude, are placed at two different locations. Show that the combined electrostatic potential vanishes on a sphere that encloses only the charge of smaller magnitude.

(ii) A grounded, conducting sphere of radius $a$ is centred at the origin. A point charge $q$ is located outside the sphere at position vector $\mathbf{p}$. Formulate the differential equation and boundary conditions for the electrostatic potential outside the sphere. Using the result of part (i) or otherwise, show that the electric field outside the sphere is identical to that generated (in the absence of any conductors) by the point charge $q$ and an image charge $q^{\prime}$ located inside the sphere at position vector $\mathbf{p}^{\prime}$, provided that $\mathbf{p}^{\prime}$ and $q^{\prime}$ are chosen correctly.

Calculate the magnitude and direction of the force experienced by the charge $q$.

Paper 4 , Section I, $7 \mathrm{C}$

commentA thin wire, in the form of a closed curve $C$, carries a constant current $I$. Using either the Biot-Savart law or the magnetic vector potential, show that the magnetic field far from the loop is of the approximate form

$\mathbf{B}(\mathbf{r}) \approx \frac{\mu_{0}}{4 \pi}\left[\frac{3(\mathbf{m} \cdot \mathbf{r}) \mathbf{r}-\mathbf{m}|\mathbf{r}|^{2}}{|\mathbf{r}|^{5}}\right]$

where $\mathbf{m}$ is the magnetic dipole moment of the loop. Derive an expression for $\mathbf{m}$ in terms of $I$ and the vector area spanned by the curve $C$.

Paper 1, Section II, D

comment(a) From the differential form of Maxwell's equations with $\mathbf{J}=\mathbf{0}, \mathbf{B}=\mathbf{0}$ and a time-independent electric field, derive the integral form of Gauss's law.

(b) Derive an expression for the electric field $\mathbf{E}$ around an infinitely long line charge lying along the $z$-axis with charge per unit length $\mu$. Find the electrostatic potential $\phi$ up to an arbitrary constant.

(c) Now consider the line charge with an ideal earthed conductor filling the region $x>d$. State the boundary conditions satisfied by $\phi$ and $\mathbf{E}$ on the surface of the conductor.

(d) Show that the same boundary conditions at $x=d$ are satisfied if the conductor is replaced by a second line charge at $x=2 d, y=0$ with charge per unit length $-\mu$.

(e) Hence or otherwise, returning to the setup in (c), calculate the force per unit length acting on the line charge.

(f) What is the charge per unit area $\sigma(y, z)$ on the surface of the conductor?

Paper 2, Section I, $6 \mathrm{D}$

comment(a) Derive the integral form of Ampère's law from the differential form of Maxwell's equations with a time-independent magnetic field, $\rho=0$ and $\mathbf{E}=\mathbf{0}$.

(b) Consider two perfectly-conducting concentric thin cylindrical shells of infinite length with axes along the $z$-axis and radii $a$ and $b(a<b)$. Current $I$ flows in the positive $z$-direction in each shell. Use Ampère's law to calculate the magnetic field in the three regions: (i) $r<a$, (ii) $a<r<b$ and (iii) $r>b$, where $r=\sqrt{x^{2}+y^{2}}$.

(c) If current $I$ now flows in the positive $z$-direction in the inner shell and in the negative $z$-direction in the outer shell, calculate the magnetic field in the same three regions.

Paper 2, Section II, D

comment(a) State the covariant form of Maxwell's equations and define all the quantities that appear in these expressions.

(b) Show that $\mathbf{E} \cdot \mathbf{B}$ is a Lorentz scalar (invariant under Lorentz transformations) and find another Lorentz scalar involving $\mathbf{E}$ and $\mathbf{B}$.

(c) In some inertial frame $S$ the electric and magnetic fields are respectively $\mathbf{E}=\left(0, E_{y}, E_{z}\right)$ and $\mathbf{B}=\left(0, B_{y}, B_{z}\right)$. Find the electric and magnetic fields, $\mathbf{E}^{\prime}=\left(0, E_{y}^{\prime}, E_{z}^{\prime}\right)$ and $\mathbf{B}^{\prime}=\left(0, B_{y}^{\prime}, B_{z}^{\prime}\right)$, in another inertial frame $S^{\prime}$ that is related to $S$ by the Lorentz transformation,

$\Lambda_{\nu}^{\mu}=\left(\begin{array}{cccc} \gamma & -\gamma v / c & 0 & 0 \\ -\gamma v / c & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

where $v$ is the velocity of $S^{\prime}$ in $S$ and $\gamma=\left(1-v^{2} / c^{2}\right)^{-1 / 2}$.

(d) Suppose that $\mathbf{E}=E_{0}(0,1,0)$ and $\mathbf{B}=\frac{E_{0}}{c}(0, \cos \theta, \sin \theta)$ where $0 \leqslant \theta \leqslant \pi / 2$, and $E_{0}$ is a real constant. An observer is moving in $S$ with velocity $v$ parallel to the $x$-axis. What must $v$ be for the electric and magnetic fields to appear to the observer to be parallel? Comment on the case $\theta=\pi / 2$.

Paper 3, Section II, D

comment(a) State Faraday's law of induction for a moving circuit in a time-dependent magnetic field and define all the terms that appear.

(b) Consider a rectangular circuit DEFG in the $z=0$ plane as shown in the diagram below. There are two rails parallel to the $x$-axis for $x>0$ starting at $\mathrm{D}$ at $(x, y)=(0,0)$ and $G$ at $(0, L)$. A battery provides an electromotive force $\mathcal{E}_{0}$ between $D$ and $G$ driving current in a positive sense around DEFG. The circuit is completed with a bar resistor of resistance $R$, length $L$ and mass $m$ that slides without friction on the rails; it connects $E$ at $(s(t), 0)$ and $F$ at $(s(t), L)$. The rest of the circuit has no resistance. The circuit is in a constant uniform magnetic field $B_{0}$ parallel to the $z$-axis.

[In parts (i)-(iv) you can neglect any magnetic field due to current flow.]

(i) Calculate the current in the bar and indicate its direction on a diagram of the circuit.

(ii) Find the force acting on the bar.

(iii) If the initial velocity and position of the bar are respectively $\dot{s}(0)=v_{0}>0$ and $s(0)=s_{0}>0$, calculate $\dot{s}(t)$ and $s(t)$ for $t>0$.

(iv) If $\mathcal{E}_{0}=0$, find the total energy dissipated in the circuit after $t=0$ and verify that total energy is conserved.

(v) Describe qualitatively the effect of the magnetic field caused by the induced current flowing in the circuit when $\mathcal{E}_{0}=0$.

Paper 4, Section I, D

comment(a) Starting from Maxwell's equations, show that in a vacuum,

$\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{E}}{\partial t^{2}}-\nabla^{2} \mathbf{E}=\mathbf{0} \quad \text { and } \quad \boldsymbol{\nabla} \cdot \mathbf{E}=0 \quad \text { where } \quad c=\sqrt{\frac{1}{\epsilon_{0} \mu_{0}}} .$

(b) Suppose that $\mathbf{E}=\frac{E_{0}}{\sqrt{2}}(1,1,0) \cos (k z-\omega t)$ where $E_{0}, k$ and $\omega$ are real constants.

(i) What are the wavevector and the polarisation? How is $\omega$ related to $k$ ?

(ii) Find the magnetic field $\mathbf{B}$.

(iii) Compute and interpret the time-averaged value of the Poynting vector, $\mathbf{S}=\frac{1}{\mu_{0}} \mathbf{E} \times \mathbf{B}$.

Paper 1, Section II, A

comment(i) Write down the Lorentz force law for $d \mathbf{p} / d t$ due to an electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ acting on a particle of charge $q$ moving with velocity $\dot{\mathbf{x}}$.

(ii) Write down Maxwell's equations in terms of $c$ (the speed of light in a vacuum), in the absence of charges and currents.

(iii) Show that they can be manipulated into a wave equation for each component of $\mathbf{E}$.

(iv) Show that Maxwell's equations admit solutions of the form

$\mathbf{E}(\mathbf{x}, t)=\operatorname{Re}\left(\mathbf{E}_{0} e^{i(\omega t-\mathbf{k} \cdot \mathbf{x})}\right)$

where $\mathbf{E}_{\mathbf{0}}$ and $\mathbf{k}$ are constant vectors and $\omega$ is a constant (all real). Derive a condition on $\mathbf{k} \cdot \mathbf{E}_{\mathbf{0}}$ and relate $\omega$ and $\mathbf{k}$.

(v) Suppose that a perfect conductor occupies the region $z<0$ and that a plane wave with $\mathbf{k}=(0,0,-k), \mathbf{E}_{0}=\left(E_{0}, 0,0\right)$ is incident from the vacuum region $z>0$. Write down boundary conditions for the $\mathbf{E}$ and $\mathbf{B}$ fields. Show that they can be satisfied if a suitable reflected wave is present, and determine the total $\mathbf{E}$ and $\mathbf{B}$ fields in real form.

(vi) At time $t=\pi /(4 \omega)$, a particle of charge $q$ and mass $m$ is at $(0,0, \pi /(4 k))$ moving with velocity $(c / 2,0,0)$. You may assume that the particle is far enough away from the conductor so that we can ignore its effect upon the conductor and that $q E_{0}>0$. Give a unit vector for the direction of the Lorentz force on the particle at time $t=\pi /(4 \omega)$.

(vii) Ignoring relativistic effects, find the magnitude of the particle's rate of change of velocity in terms of $E_{0}, q$ and $m$ at time $t=\pi /(4 \omega)$. Why is this answer inaccurate?

Paper 2, Section I, A

commentIn a constant electric field $\mathbf{E}=(E, 0,0)$ a particle of rest mass $m$ and charge $q>0$ has position $\mathbf{x}$ and velocity $\dot{\mathbf{x}}$. At time $t=0$, the particle is at rest at the origin. Including relativistic effects, calculate $\dot{\mathbf{x}}(t)$.

Sketch a graph of $|\dot{\mathbf{x}}(t)|$ versus $t$, commenting on the $t \rightarrow \infty$ limit.

Calculate $|\mathbf{x}(t)|$ as an explicit function of $t$ and find the non-relativistic limit at small times $t$.

Paper 2, Section II, A

commentConsider the magnetic field

$\mathbf{B}=b[\mathbf{r}+(k \hat{\mathbf{z}}+l \hat{\mathbf{y}}) \hat{\mathbf{z}} \cdot \mathbf{r}+p \hat{\mathbf{x}}(\hat{\mathbf{y}} \cdot \mathbf{r})+n \hat{\mathbf{z}}(\hat{\mathbf{x}} \cdot \mathbf{r})],$

where $b \neq 0, \mathbf{r}=(x, y, z)$ and $\hat{\mathbf{x}}, \hat{\mathbf{y}}, \hat{\mathbf{z}}$ are unit vectors in the $x, y$ and $z$ directions, respectively. Imposing that this satisfies the expected equations for a static magnetic field in a vacuum, find $k, l, n$ and $p$.

A circular wire loop of radius $a$, mass $m$ and resistance $R$ lies in the $(x, y)$ plane with its centre on the $z$-axis at $z$ and a magnetic field as given above. Calculate the magnetic flux through the loop arising from this magnetic field and also the force acting on the loop when a current $I$ is flowing around the loop in a clockwise direction about the $z$-axis.

At $t=0$, the centre of the loop is at the origin, travelling with velocity $(0,0, v(t=0))$, where $v(0)>0$. Ignoring gravity and relativistic effects, and assuming that $I$ is only the induced current, find the time taken for the speed to halve in terms of $a, b, R$ and $m$. By what factor does the rate of heat generation change in this time?

Where is the loop as $t \rightarrow \infty$ as a function of $a, b, R, v(0) ?$

Paper 3, Section II, A

commentA charge density $\rho=\lambda / r$ fills the region of 3-dimensional space $a<r<b$, where $r$ is the radial distance from the origin and $\lambda$ is a constant. Compute the electric field in all regions of space in terms of $Q$, the total charge of the region. Sketch a graph of the magnitude of the electric field versus $r$ (assuming that $Q>0$ ).

Now let $\Delta=b-a \rightarrow 0$. Derive the surface charge density $\sigma$ in terms of $\Delta, a$ and $\lambda$ and explain how a finite surface charge density may be obtained in this limit. Sketch the magnitude of the electric field versus $r$ in this limit. Comment on any discontinuities, checking a standard result involving $\sigma$ for this particular case.

A second shell of equal and opposite total charge is centred on the origin and has a radius $c<a$. Sketch the electric potential of this system, assuming that it tends to 0 as $r \rightarrow \infty$.

Paper 4, Section I, A

commentFrom Maxwell's equations, derive the Biot-Savart law

$\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \int_{V} \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right) \times\left(\mathbf{r}-\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{3}} d^{3} \mathbf{r}^{\prime}$

giving the magnetic field $\mathbf{B}(\mathbf{r})$ produced by a steady current density $\mathbf{J}(\mathbf{r})$ that vanishes outside a bounded region $V$.

[You may assume that you can choose a gauge such that the divergence of the magnetic vector potential is zero.]

Paper 1, Section II, A

commentThe region $z<0$ is occupied by an ideal earthed conductor and a point charge $q$ with mass $m$ is held above it at $(0,0, d)$.

(i) What are the boundary conditions satisfied by the electric field $\mathbf{E}$ on the surface of the conductor?

(ii) Consider now a system without the conductor mentioned above. A point charge $q$ with mass $m$ is held at $(0,0, d)$, and one of charge $-q$ is held at $(0,0,-d)$. Show that the boundary condition on $\mathbf{E}$ at $z=0$ is identical to the answer to (i). Explain why this represents the electric field due to the charge at $(0,0, d)$ under the influence of the conducting boundary.

(iii) The original point charge in (i) is released with zero initial velocity. Find the time taken for the point charge to reach the plane (ignoring gravity).

[You may assume that the force on the point charge is equal to $m d^{2} \mathbf{x} / d t^{2}$, where $\mathbf{x}$ is the position vector of the charge, and $t$ is time.]

Paper 2, Section I, A

commentStarting from Maxwell's equations, deduce that

$\frac{d \Phi}{d t}=-\mathcal{E}$

for a moving circuit $C$, where $\Phi$ is the flux of $\mathbf{B}$ through the circuit and where the electromotive force $\mathcal{E}$ is defined to be

$\mathcal{E}=\oint_{\mathcal{C}}(\mathbf{E}+\mathbf{v} \times \mathbf{B}) \cdot \mathbf{d} \mathbf{r}$

where $\mathbf{v}=\mathbf{v}(\mathbf{r})$ denotes the velocity of a point $\mathbf{r}$ on $C$.

[Hint: Consider the closed surface consisting of the surface $S(t)$ bounded by $C(t)$, the surface $S(t+\delta t)$ bounded by $C(t+\delta t)$ and the surface $S^{\prime}$ stretching from $C(t)$ to $C(t+\delta t)$. Show that the flux of $\mathbf{B}$ through $S^{\prime}$ is $-\delta t \oint_{C} \mathbf{B} \cdot(\mathbf{v} \times \mathbf{d r})$.]

Paper 2, Section II, A

commentWhat is the relationship between the electric field $\mathbf{E}$ and the charge per unit area $\sigma$ on the surface of a perfect conductor?

Consider a charge distribution $\rho(\mathbf{r})$ distributed with potential $\phi(\mathbf{r})$ over a finite volume $V$ within which there is a set of perfect conductors with charges $Q_{i}$, each at a potential $\phi_{i}$ (normalised such that the potential at infinity is zero). Using Maxwell's equations and the divergence theorem, derive a relationship between the electrostatic energy $W$ and a volume integral of an explicit function of the electric field $\mathbf{E}$, where

$W=\frac{1}{2} \int_{V} \rho \phi d \tau+\frac{1}{2} \sum_{i} Q_{i} \phi_{i}$

Consider $N$ concentric perfectly conducting spherical shells. Shell $n$ has radius $r_{n}$ (where $r_{n}>r_{n-1}$ ) and charge $q$ for $n=1$, and charge $2(-1)^{(n+1)} q$ for $n>1$. Show that

$W \propto \frac{1}{r_{1}},$

and determine the constant of proportionality.

Paper 3, Section II, A

comment(i) Consider charges $-q$ at $\pm \mathbf{d}$ and $2 q$ at $(0,0,0)$. Write down the electric potential.

(ii) Take $\mathbf{d}=(0,0, d)$. A quadrupole is defined in the limit that $q \rightarrow \infty, d \rightarrow 0$ such that $q d^{2}$ tends to a constant $p$. Find the quadrupole's potential, showing that it is of the form

$\phi(\mathbf{r})=A \frac{\left(r^{2}+C z^{D}\right)}{r^{B}}$

where $r=|\mathbf{r}|$. Determine the constants $A, B, C$ and $D$.

(iii) The quadrupole is fixed at the origin. At time $t=0$ a particle of charge $-Q(Q$ has the same sign as $q)$ and mass $m$ is at $(1,0,0)$ travelling with velocity $d \mathbf{r} / d t=(-\kappa, 0,0)$, where

$\kappa=\sqrt{\frac{Q p}{2 \pi \epsilon_{0} m}} .$

Neglecting gravity, find the time taken for the particle to reach the quadrupole in terms of $\kappa$, given that the force on the particle is equal to $m d^{2} \mathbf{r} / d t^{2}$.

Paper 4, Section I, A

commentA continuous wire of resistance $R$ is wound around a very long right circular cylinder of radius $a$, and length $l$ (long enough so that end effects can be ignored). There are $N \gg 1$ turns of wire per unit length, wound in a spiral of very small pitch. Initially, the magnetic field $\mathbf{B}$ is $\mathbf{0}$.

Both ends of the coil are attached to a battery of electromotance $\mathcal{E}_{0}$ at $t=0$, which induces a current $I(t)$. Use Ampère's law to derive $\mathbf{B}$ inside and outside the cylinder when the displacement current may be neglected. Write the self-inductance of the coil $L$ in terms of the quantities given above. Using Ohm's law and Faraday's law of induction, find $I(t)$ explicitly in terms of $\mathcal{E}_{0}, R, L$ and $t$.

Paper 1, Section II, $16 \mathrm{D}$

commentBriefly explain the main assumptions leading to Drude's theory of conductivity. Show that these assumptions lead to the following equation for the average drift velocity $\langle\mathbf{v}(t)\rangle$ of the conducting electrons:

$\frac{d\langle\mathbf{v}\rangle}{d t}=-\tau^{-1}\langle\mathbf{v}\rangle+(e / m) \mathbf{E}$

where $m$ and $e$ are the mass and charge of each conducting electron, $\tau^{-1}$ is the probability that a given electron collides with an ion in unit time, and $\mathbf{E}$ is the applied electric field.

Given that $\langle\mathbf{v}\rangle=\mathbf{v}_{0} e^{-i \omega t}$ and $\mathbf{E}=\mathbf{E}_{0} e^{-i \omega t}$, where $\mathbf{v}_{0}$ and $\mathbf{E}_{0}$ are independent of $t$, show that

$\mathbf{J}=\sigma \mathbf{E}$

Here, $\sigma=\sigma_{s} /(1-i \omega \tau), \sigma_{s}=n e^{2} \tau / m$ and $n$ is the number of conducting electrons per unit volume.

Now let $\mathbf{v}_{0}=\widetilde{\mathbf{v}}_{0} e^{i \mathbf{k} \cdot \mathbf{x}}$ and $\mathbf{E}_{0}=\widetilde{\mathbf{E}}_{0} e^{i \mathbf{k} \cdot \mathbf{x}}$, where $\widetilde{\mathbf{v}}_{0}$ and $\widetilde{\mathbf{E}}_{0}$ are constant. Assuming that $(*)$ remains valid, use Maxwell's equations (taking the charge density to be everywhere zero but allowing for a non-zero current density) to show that

$k^{2}=\frac{\omega^{2}}{c^{2}} \epsilon_{r}$

where the relative permittivity $\epsilon_{r}=1+i \sigma /\left(\omega \epsilon_{0}\right)$ and $k=|\mathbf{k}|$.

In the case $\omega \tau \gg 1$ and $\omega<\omega_{p}$, where $\omega_{p}^{2}=\sigma_{s} / \tau \epsilon_{0}$, show that the wave decays exponentially with distance inside the conductor.

Paper 2, Section I, D

commentUse Maxwell's equations to obtain the equation of continuity

$\frac{\partial \rho}{\partial t}+\nabla \cdot \mathbf{J}=0$

Show that, for a body made from material of uniform conductivity $\sigma$, the charge density at any fixed internal point decays exponentially in time. If the body is finite and isolated, explain how this result can be consistent with overall charge conservation.

Paper 2, Section II, D

commentStarting with the expression

$\mathbf{A}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \int \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right) d V^{\prime}}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}$

for the magnetic vector potential at the point $r$ due to a current distribution of density $\mathbf{J}(\mathbf{r})$, obtain the Biot-Savart law for the magnetic field due to a current $I$ flowing in a simple loop $C$ :

$\mathbf{B}(\mathbf{r})=-\frac{\mu_{0} I}{4 \pi} \oint_{C} \frac{d \mathbf{r}^{\prime} \times\left(\mathbf{r}^{\prime}-\mathbf{r}\right)}{\left|\mathbf{r}^{\prime}-\mathbf{r}\right|^{3}} \quad(\mathbf{r} \notin C) .$

Verify by direct differentiation that this satisfies $\boldsymbol{\nabla} \times \mathbf{B}=\mathbf{0}$. You may use without proof the identity $\boldsymbol{\nabla} \times(\mathbf{a} \times \mathbf{v})=\mathbf{a}(\boldsymbol{\nabla} \cdot \mathbf{v})-(\mathbf{a} \cdot \boldsymbol{\nabla}) \mathbf{v}$, where $\mathbf{a}$ is a constant vector and $\mathbf{v}$ is a vector field.

Given that $C$ is planar, and is described in cylindrical polar coordinates by $z=0$, $r=f(\theta)$, show that the magnetic field at the origin is

$\widehat{\mathbf{z}} \frac{\mu_{0} I}{4 \pi} \oint \frac{d \theta}{f(\theta)}$

If $C$ is the ellipse $r(1-e \cos \theta)=\ell$, find the magnetic field at the focus due to a current $I$.

Paper 3, Section II, D

commentThree sides of a closed rectangular circuit $C$ are fixed and one is moving. The circuit lies in the plane $z=0$ and the sides are $x=0, y=0, x=a(t), y=b$, where $a(t)$ is a given function of time. A magnetic field $\mathbf{B}=\left(0,0, \frac{\partial f}{\partial x}\right)$ is applied, where $f(x, t)$ is a given function of $x$ and $t$ only. Find the magnetic flux $\Phi$ of $\mathbf{B}$ through the surface $S$ bounded by $C$.

Find an electric field $\mathbf{E}_{\mathbf{0}}$ that satisfies the Maxwell equation

$\boldsymbol{\nabla} \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$

and then write down the most general solution $\mathbf{E}$ in terms of $\mathbf{E}_{0}$ and an undetermined scalar function independent of $f$.

Verify that

$\oint_{C}(\mathbf{E}+\mathbf{v} \times \mathbf{B}) \cdot d \mathbf{r}=-\frac{d \Phi}{d t},$

where $\mathbf{v}$ is the velocity of the relevant side of $C$. Interpret the left hand side of this equation.

If a unit current flows round $C$, what is the rate of work required to maintain the motion of the moving side of the rectangle? You should ignore any electromagnetic fields produced by the current.

Paper 4, Section I, D

commentThe infinite plane $z=0$ is earthed and the infinite plane $z=d$ carries a charge of $\sigma$ per unit area. Find the electrostatic potential between the planes.

Show that the electrostatic energy per unit area (of the planes $z=$ constant) between the planes can be written as either $\frac{1}{2} \sigma^{2} d / \epsilon_{0}$ or $\frac{1}{2} \epsilon_{0} V^{2} / d$, where $V$ is the potential at $z=d$.

The distance between the planes is now increased by $\alpha d$, where $\alpha$ is small. Show that the change in the energy per unit area is $\frac{1}{2} \sigma V \alpha$ if the upper plane $(z=d)$ is electrically isolated, and is approximately $-\frac{1}{2} \sigma V \alpha$ if instead the potential on the upper plane is maintained at $V$. Explain briefly how this difference can be accounted for.

Paper 1, Section II, B

commentA sphere of radius a carries an electric charge $Q$ uniformly distributed over its surface. Calculate the electric field outside and inside the sphere. Also calculate the electrostatic potential outside and inside the sphere, assuming it vanishes at infinity. State the integral formula for the energy $U$ of the electric field and use it to evaluate $U$ as a function of $Q$

Relate $\frac{d U}{d Q}$ to the potential on the surface of the sphere and explain briefly the physical interpretation of the relation.

Paper 2, Section I, B

commentWrite down the expressions for a general, time-dependent electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ in terms of a vector potential $\mathbf{A}$ and scalar potential $\phi$. What is meant by a gauge transformation of $\mathbf{A}$ and $\phi$ ? Show that $\mathbf{E}$ and $\mathbf{B}$ are unchanged under a gauge transformation.

A plane electromagnetic wave has vector and scalar potentials

$\begin{aligned} \mathbf{A}(\mathbf{x}, t) &=\mathbf{A}_{0} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)} \\ \phi(\mathbf{x}, t) &=\phi_{0} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)} \end{aligned}$

where $\mathbf{A}_{0}$ and $\phi_{0}$ are constants. Show that $\left(\mathbf{A}_{0}, \phi_{0}\right)$ can be modified to $\left(\mathbf{A}_{0}+\mu \mathbf{k}, \phi_{0}+\mu \omega\right)$ by a gauge transformation. What choice of $\mu$ leads to the modified $\mathbf{A}(\mathbf{x}, t)$ satisfying the Coulomb gauge condition $\boldsymbol{\nabla} \cdot \mathbf{A}=0$ ?

Paper 2, Section II, B

commentA straight wire has $n$ mobile, charged particles per unit length, each of charge $q$. Assuming the charges all move with velocity $v$ along the wire, show that the current is $I=n q v$.

Using the Lorentz force law, show that if such a current-carrying wire is placed in a uniform magnetic field of strength $B$ perpendicular to the wire, then the force on the wire, per unit length, is $B I$.

Consider two infinite parallel wires, with separation $L$, carrying (in the same sense of direction) positive currents $I_{1}$ and $I_{2}$, respectively. Find the force per unit length on each wire, determining both its magnitude and direction.

Paper 3, Section II, B

commentUsing the Maxwell equations

$\begin{gathered} \nabla \cdot \mathbf{E}=\frac{\rho}{\epsilon_{0}}, \quad \nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \cdot \mathbf{B}=0, \quad \boldsymbol{\nabla} \times \mathbf{B}-\epsilon_{0} \mu_{0} \frac{\partial \mathbf{E}}{\partial t}=\mu_{0} \mathbf{j} \end{gathered}$

show that in vacuum, E satisfies the wave equation

$\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{E}}{\partial t^{2}}-\nabla^{2} \mathbf{E}=0$

where $c^{2}=\left(\epsilon_{0} \mu_{0}\right)^{-1}$, as well as $\nabla \cdot \mathbf{E}=0$. Also show that at a planar boundary between two media, $\mathbf{E}_{t}$ (the tangential component of $\mathbf{E}$ ) is continuous. Deduce that if one medium is of negligible resistance, $\mathbf{E}_{t}=0$.

Consider an empty cubic box with walls of negligible resistance on the planes $x=0$, $x=a, y=0, y=a, z=0, z=a$, where $a>0$. Show that an electric field in the interior of the form

$\begin{aligned} &E_{x}=f(x) \sin \left(\frac{m \pi y}{a}\right) \sin \left(\frac{n \pi z}{a}\right) e^{-i \omega t} \\ &E_{y}=g(y) \sin \left(\frac{l \pi x}{a}\right) \sin \left(\frac{n \pi z}{a}\right) e^{-i \omega t} \\ &E_{z}=h(z) \sin \left(\frac{l \pi x}{a}\right) \sin \left(\frac{m \pi y}{a}\right) e^{-i \omega t} \end{aligned}$

with $l, m$ and $n$ positive integers, satisfies the boundary conditions on all six walls. Now suppose that

$f(x)=f_{0} \cos \left(\frac{l \pi x}{a}\right), \quad g(y)=g_{0} \cos \left(\frac{m \pi y}{a}\right), \quad h(z)=h_{0} \cos \left(\frac{n \pi z}{a}\right)$

where $f_{0}, g_{0}$ and $h_{0}$ are constants. Show that the wave equation $(*)$ is satisfied, and determine the frequency $\omega$. Find the further constraint on $f_{0}, g_{0}$ and $h_{0}$ ?

Paper 4, Section I, B

commentDefine the notions of magnetic flux, electromotive force and resistance, in the context of a single closed loop of wire. Use the Maxwell equation

$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$

to derive Faraday's law of induction for the loop, assuming the loop is at rest.

Suppose now that the magnetic field is $\mathbf{B}=(0,0, B \tanh t)$ where $B$ is a constant, and that the loop of wire, with resistance $R$, is a circle of radius a lying in the $(x, y)$ plane. Calculate the current in the wire as a function of time.

Explain briefly why, even in a time-independent magnetic field, an electromotive force may be produced in a loop of wire that moves through the field, and state the law of induction in this situation.

Paper 1, Section II, D

commentStarting from the relevant Maxwell equation, derive Gauss's law in integral form.

Use Gauss's law to obtain the potential at a distance $r$ from an infinite straight wire with charge $\lambda$ per unit length.

Write down the potential due to two infinite wires parallel to the $z$-axis, one at $x=y=0$ with charge $\lambda$ per unit length and the other at $x=0, y=d$ with charge $-\lambda$ per unit length.

Find the potential and the electric field in the limit $d \rightarrow 0$ with $\lambda d=p$ where $p$ is fixed. Sketch the equipotentials and the electric field lines.

Paper 2, Section I, $\mathbf{6 C}$

Maxwell's equations are

$\begin{gathered} \boldsymbol{\nabla} \cdot \mathbf{E}=\frac{\rho}{\epsilon_{0}}, \quad \boldsymbol{\nabla} \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \cdot \mathbf{B}=0, \quad \nabla \times \mathbf{B}=\mu_{0} \mathbf{J}+\epsilon_{0} \mu_{0} \frac{\partial \mathbf{E}}{\partial t} \end{gathered}$