# Electrodynamics

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Paper 1, Section II, 37C

comment(a) An electromagnetic field is specified by a four-vector potential

$A^{\mu}(\mathbf{x}, t)=(\phi(\mathbf{x}, t) / c, \mathbf{A}(\mathbf{x}, t))$

Define the corresponding field-strength tensor $F^{\mu \nu}$ and state its transformation property under a general Lorentz transformation.

(b) Write down two independent Lorentz scalars that are quadratic in the field strength and express them in terms of the electric and magnetic fields, $\mathbf{E}=-\boldsymbol{\nabla} \phi-\partial \mathbf{A} / \partial t$ and $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$. Show that both these scalars vanish when evaluated on an electromagnetic plane-wave solution of Maxwell's equations of arbitrary wavevector and polarisation.

(c) Find (non-zero) constant, homogeneous background fields $\mathbf{E}(\mathbf{x}, t)=\mathbf{E}_{0}$ and $\mathbf{B}(\mathbf{x}, t)=\mathbf{B}_{0}$ such that both the Lorentz scalars vanish. Show that, for any such background, the field-strength tensor obeys

$F_{\rho}^{\mu} F_{\sigma}^{\rho} F_{\nu}^{\sigma}=0$

(d) Hence find the trajectory of a relativistic particle of mass $m$ and charge $q$ in this background. You should work in an inertial frame where the particle is at rest at the origin at $t=0$ and in which $\mathbf{B}_{0}=\left(0,0, B_{0}\right)$.

Paper 3, Section II, 36C

comment(a) Derive the Larmor formula for the total power $P$ emitted through a large sphere of radius $R$ by a non-relativistic particle of mass $m$ and charge $q$ with trajectory $\mathbf{x}(t)$. You may assume that the electric and magnetic fields describing radiation due to a source localised near the origin with electric dipole moment $\mathbf{p}(t)$ can be approximated as

$\begin{aligned} &\mathbf{B}_{\operatorname{Rad}}(\mathbf{x}, t)=-\frac{\mu_{0}}{4 \pi r c} \widehat{\mathbf{x}} \times \ddot{\mathbf{p}}(t-r / c) \\ &\mathbf{E}_{\mathrm{Rad}}(\mathbf{x}, t)=-c \widehat{\mathbf{x}} \times \mathbf{B}_{\operatorname{Rad}}(\mathbf{x}, t) \end{aligned}$

Here, the radial distance $r=|\mathbf{x}|$ is assumed to be much larger than the wavelength of emitted radiation which, in turn, is large compared to the spatial extent of the source.

(b) A non-relativistic particle of mass $m$, moving at speed $v$ along the $x$-axis in the positive direction, encounters a step potential of width $L$ and height $V_{0}>0$ described by

$V(x)= \begin{cases}0, & x<0 \\ f(x), & 0 \leqslant x \leqslant L \\ V_{0}, & x>L\end{cases}$

where $f(x)$ is a monotonically increasing function with $f(0)=0$ and $f(L)=V_{0}$. The particle carries charge $q$ and loses energy by emitting electromagnetic radiation. Assume that the total energy loss through emission $\Delta E_{\text {Rad }}$ is negligible compared with the particle's initial kinetic energy $E=m v^{2} / 2$. For $E>V_{0}$, show that the total energy lost is

$\Delta E_{\mathrm{Rad}}=\frac{q^{2} \mu_{0}}{6 \pi m^{2} c} \sqrt{\frac{m}{2}} \int_{0}^{L} d x \frac{1}{\sqrt{E-f(x)}}\left(\frac{d f}{d x}\right)^{2}$

Find the total energy lost also for the case $E<V_{0}$.

(c) Take $f(x)=V_{0} x / L$ and explicitly evaluate the particle energy loss $\Delta E_{\text {Rad }}$ in each of the cases $E>V_{0}$ and $E<V_{0}$. What is the maximum value attained by $\Delta E_{\text {Rad }}$ as $E$ is varied?

Paper 4, Section II, 36C

comment(a) Define the electric displacement $\mathbf{D}(\mathbf{x}, t)$ for a medium which exhibits a linear response with polarisation constant $\epsilon$ to an applied electric field $\mathbf{E}(\mathbf{x}, t)$ with polarisation constant $\epsilon$. Write down the effective Maxwell equation obeyed by $\mathbf{D}(\mathbf{x})$ in the timeindependent case and in the absence of any additional mobile charges in the medium. Describe appropriate boundary conditions for the electric field at an interface between two regions with differing values of the polarisation constant. [You should discuss separately the components of the field normal to and tangential to the interface.]

(b) Consider a sphere of radius $a$, centred at the origin, composed of dielectric material with polarisation constant $\epsilon$ placed in a vacuum and subjected to a constant, asymptotically homogeneous, electric field, $\mathbf{E}(\mathbf{x}, t)=\mathbf{E}(\mathbf{x})$ with $\mathbf{E}(\mathbf{x}) \rightarrow \mathbf{E}_{0}$ as $|\mathbf{x}| \rightarrow \infty$. Using the ansatz

$\mathbf{E}(\mathbf{x})= \begin{cases}\alpha \mathbf{E}_{0}, & |\mathbf{x}|<a \\ \mathbf{E}_{0}+\left(\beta\left(\widehat{\mathbf{x}} \cdot \mathbf{E}_{0}\right) \widehat{\mathbf{x}}+\delta \mathbf{E}_{0}\right) /|\mathbf{x}|^{3}, & |\mathbf{x}|>a\end{cases}$

with constants $\alpha, \beta$ and $\delta$ to be determined, find a solution to Maxwell's equations with appropriate boundary conditions at $|\mathbf{x}|=a$.

(c) By comparing your solution with the long-range electric field due to a dipole consisting of electric charges $\pm q$ located at displacements $\pm \mathbf{d} / 2$ find the induced electric dipole moment of the dielectric sphere.

Paper 1, Section II, 37D

commentA relativistic particle of rest mass $m$ and electric charge $q$ follows a worldline $x^{\mu}(\lambda)$ in Minkowski spacetime where $\lambda=\lambda(\tau)$ is an arbitrary parameter which increases monotonically with the proper time $\tau$. We consider the motion of the particle in a background electromagnetic field with four-vector potential $A^{\mu}(x)$ between initial and final values of the proper time denoted $\tau_{i}$ and $\tau_{f}$ respectively.

(i) Write down an action for the particle's motion. Explain what is meant by a gauge transformation of the electromagnetic field. How does the action change under a gauge transformation?

(ii) Derive an equation of motion for the particle by considering the variation of the action with respect to the worldline $x^{\mu}(\lambda)$. Setting $\lambda=\tau$ show that your equation of motion reduces to the Lorentz force law,

$m \frac{d u^{\mu}}{d \tau}=q F^{\mu \nu} u_{\nu}$

where $u^{\mu}=d x^{\mu} / d \tau$ is the particle's four-velocity and $F^{\mu \nu}=\partial^{\mu} A^{\nu}-\partial^{\nu} A^{\mu}$ is the Maxwell field-strength tensor.

(iii) Working in an inertial frame with spacetime coordinates $x^{\mu}=(c t, x, y, z)$, consider the case of a constant, homogeneous magnetic field of magnitude $B$, pointing in the $z$-direction, and vanishing electric field. In a gauge where $A^{\mu}=(0,0, B x, 0)$, show that the equation of motion $(*)$ is solved by circular motion in the $x-y$ plane with proper angular frequency $\omega=q B / m$.

(iv) Let $v$ denote the speed of the particle in this inertial frame with Lorentz factor $\gamma(v)=1 / \sqrt{1-v^{2} / c^{2}}$. Find the radius $R=R(v)$ of the circle as a function of $v$. Setting $\tau_{f}=\tau_{i}+2 \pi / \omega$, evaluate the action $S=S(v)$ for a single period of the particle's motion.

Paper 3, Section II, D

commentThe Maxwell stress tensor $\sigma$ of the electromagnetic fields is a two-index Cartesian tensor with components

$\sigma_{i j}=-\epsilon_{0}\left(E_{i} E_{j}-\frac{1}{2}|\mathbf{E}|^{2} \delta_{i j}\right)-\frac{1}{\mu_{0}}\left(B_{i} B_{j}-\frac{1}{2}|\mathbf{B}|^{2} \delta_{i j}\right)$

where $i, j=1,2,3$, and $E_{i}$ and $B_{i}$ denote the Cartesian components of the electric and magnetic fields $\mathbf{E}(\mathbf{x}, t)$ and $\mathbf{B}(\mathbf{x}, t)$ respectively.

(i) Consider an electromagnetic field sourced by charge and current densities denoted by $\rho(\mathbf{x}, t)$ and $\mathbf{J}(\mathbf{x}, t)$ respectively. Using Maxwell's equations and the Lorentz force law, show that the components of $\sigma$ obey the equation

$\sum_{j=1}^{3} \frac{\partial \sigma_{i j}}{\partial x_{j}}+\frac{\partial g_{i}}{\partial t}=-(\rho \mathbf{E}+\mathbf{J} \times \mathbf{B})_{i}$

where $g_{i}$, for $i=1,2,3$, are the components of a vector field $\mathbf{g}(\mathbf{x}, t)$ which you should give explicitly in terms of $\mathbf{E}$ and $\mathbf{B}$. Explain the physical interpretation of this equation and of the quantities $\sigma$ and $\mathbf{g}$.

(ii) A localised source near the origin, $\mathbf{x}=0$, emits electromagnetic radiation. Far from the source, the resulting electric and magnetic fields can be approximated as

$\mathbf{B}(\mathbf{x}, t) \simeq \mathbf{B}_{0}(\mathbf{x}) \sin (\omega t-\mathbf{k} \cdot \mathbf{x}), \quad \mathbf{E}(\mathbf{x}, t) \simeq \mathbf{E}_{0}(\mathbf{x}) \sin (\omega t-\mathbf{k} \cdot \mathbf{x})$

where $\mathbf{B}_{0}(\mathbf{x})=\frac{\mu_{0} \omega^{2}}{4 \pi r c} \hat{\mathbf{x}} \times \mathbf{p}_{0}$ and $\mathbf{E}_{0}(\mathbf{x})=-c \hat{\mathbf{x}} \times \mathbf{B}_{0}(\mathbf{x})$ with $r=|\mathbf{x}|$ and $\hat{\mathbf{x}}=\mathbf{x} / r$. Here, $\mathbf{k}=(\omega / c) \hat{\mathbf{x}}$ and $\mathbf{p}_{0}$ is a constant vector.

Calculate the pressure exerted by these fields on a spherical shell of very large radius $R$ centred on the origin. [You may assume that $\mathbf{E}$ and $\mathbf{B}$ vanish for $r>R$ and that the shell material is absorbant, i.e. no reflected wave is generated.]

Paper 4 , Section II, 36D

comment(a) A dielectric medium exhibits a linear response if the electric displacement $\mathbf{D}(\mathbf{x}, t)$ and magnetizing field $\mathbf{H}(\mathbf{x}, t)$ are related to the electric and magnetic fields, $\mathbf{E}(\mathbf{x}, t)$ and $\mathbf{B}(\mathbf{x}, t)$, as

$\mathbf{D}=\epsilon \mathbf{E}, \quad \mathbf{B}=\mu \mathbf{H}$

where $\epsilon$ and $\mu$ are constants characterising the electric and magnetic polarisability of the material respectively. Write down the Maxwell equations obeyed by the fields $\mathbf{D}, \mathbf{H}, \mathbf{B}$ and $\mathbf{E}$ in this medium in the absence of free charges or currents.

(b) Two such media with constants $\epsilon_{-}$and $\epsilon_{+}$(but the same $\mu$ ) fill the regions $x<0$ and $x>0$ respectively in three-dimensions with Cartesian coordinates $(x, y, z)$.

(i) Starting from Maxwell's equations, derive the appropriate boundary conditions at $x=0$ for a time-independent electric field $\mathbf{E}(\mathbf{x})$.

(ii) Consider a candidate solution of Maxwell's equations describing the reflection and transmission of an incident electromagnetic wave of wave vector $\mathbf{k}_{I}$ and angular frequency $\omega_{I}$ off the interface at $x=0$. The electric field is given as,

$\mathbf{E}(\mathbf{x}, t)=\left\{\begin{array}{cc} \sum_{X=I, R} \operatorname{Im}\left[\mathbf{E}_{X} \exp \left(i \mathbf{k}_{X} \cdot \mathbf{x}-i \omega_{X} t\right)\right], & x<0 \\ \operatorname{Im}\left[\mathbf{E}_{T} \exp \left(i \mathbf{k}_{T} \cdot \mathbf{x}-i \omega_{T} t\right)\right], & x>0 \end{array}\right.$

where $\mathbf{E}_{I}, \mathbf{E}_{R}$ and $\mathbf{E}_{T}$ are constant real vectors and $\operatorname{Im}[z]$ denotes the imaginary part of a complex number $z$. Give conditions on the parameters $\mathbf{E}_{X}, \mathbf{k}_{X}, \omega_{X}$ for $X=I, R, T$, such that the above expression for the electric field $\mathbf{E}(\mathbf{x}, t)$ solves Maxwell's equations for all $x \neq 0$, together with an appropriate magnetic field $\mathbf{B}(\mathbf{x}, t)$ which you should determine.

(iii) We now parametrize the incident wave vector as $\mathbf{k}_{I}=k_{I}\left(\cos \left(\theta_{I}\right) \hat{\mathbf{i}}_{x}+\sin \left(\theta_{I}\right) \hat{\mathbf{i}}_{z}\right)$, where $\hat{\mathbf{i}}_{x}$ and $\hat{\mathbf{i}}_{z}$ are unit vectors in the $x$ - and $z$-directions respectively, and choose the incident polarisation vector to satisfy $\mathbf{E}_{I} \cdot \hat{\mathbf{i}}_{x}=0$. By imposing appropriate boundary conditions for $\mathbf{E}(\mathbf{x}, t)$ at $x=0$, which you may assume to be the same as those for the time-independent case considered above, determine the Cartesian components of the wavevector $\mathbf{k}_{T}$ as functions of $k_{I}, \theta_{I}, \epsilon_{+}$and $\epsilon_{-}$.

(iv) For $\epsilon_{+}<\epsilon_{-}$find a critical value $\theta_{I}^{\text {cr }}$ of the angle of incidence $\theta_{I}$ above which there is no real solution for the wavevector $\mathbf{k}_{T}$. Write down a solution for $\mathbf{E}(\mathbf{x}, t)$ when $\theta_{I}>\theta_{I}^{\mathrm{cr}}$ and comment on its form.

Paper 1, Section II, E

commentA relativistic particle of charge $q$ and mass $m$ moves in a background electromagnetic field. The four-velocity $u^{\mu}(\tau)$ of the particle at proper time $\tau$ is determined by the equation of motion,

$m \frac{d u^{\mu}}{d \tau}=q F_{\nu}^{\mu} u^{\nu} .$

Here $F_{\nu}^{\mu}=\eta_{\nu \rho} F^{\mu \rho}$, where $F_{\mu \nu}$ is the electromagnetic field strength tensor and Lorentz indices are raised and lowered with the metric tensor $\eta=\operatorname{diag}\{-1,+1,+1,+1\}$. In the case of a constant, homogeneous field, write down the solution of this equation giving $u^{\mu}(\tau)$ in terms of its initial value $u^{\mu}(0)$.

[In the following you may use the relation, given below, between the components of the field strength tensor $F_{\mu \nu}$, for $\mu, \nu=0,1,2,3$, and those of the electric and magnetic fields $\mathbf{E}=\left(E_{1}, E_{2}, E_{3}\right)$ and $\mathbf{B}=\left(B_{1}, B_{2}, B_{3}\right)$,

$F_{i 0}=-F_{0 i}=\frac{1}{c} E_{i}, \quad F_{i j}=\varepsilon_{i j k} B_{k}$

for $i, j=1,2,3 .]$

Suppose that, in some inertial frame with spacetime coordinates $\mathbf{x}=(x, y, z)$ and $t$, the electric and magnetic fields are parallel to the $x$-axis with magnitudes $E$ and $B$ respectively. At time $t=\tau=0$ the 3 -velocity $\mathbf{v}=d \mathbf{x} / d t$ of the particle has initial value $\mathbf{v}(0)=\left(0, v_{0}, 0\right)$. Find the subsequent trajectory of the particle in this frame, giving coordinates $x, y, z$ and $t$ as functions of the proper time $\tau$.

Find the motion in the $x$-direction explicitly, giving $x$ as a function of coordinate time $t$. Comment on the form of the solution at early and late times. Show that, when projected onto the $y-z$ plane, the particle undergoes circular motion which is periodic in proper time. Find the radius $R$ of the circle and proper time period of the motion $\Delta \tau$ in terms of $q, m, E, B$ and $v_{0}$. The resulting trajectory therefore has the form of a helix with varying pitch $P_{n}:=\Delta x_{n} / R$ where $\Delta x_{n}$ is the distance in the $x$-direction travelled by the particle during the $n$ 'th period of its motion in the $y-z$ plane. Show that, for $n \gg 1$,

$P_{n} \sim A \exp \left(\frac{2 \pi E n}{c B}\right),$

where $A$ is a constant which you should determine.

Paper 3, Section II, E

commentA time-dependent charge distribution $\rho(t, \mathbf{x})$ localised in some region of size $a$ near the origin varies periodically in time with characteristic angular frequency $\omega$. Explain briefly the circumstances under which the dipole approximation for the fields sourced by the charge distribution is valid.

Far from the origin, for $r=|\mathbf{x}| \gg a$, the vector potential $\mathbf{A}(t, \mathbf{x})$ sourced by the charge distribution $\rho(t, \mathbf{x})$ is given by the approximate expression

$\mathbf{A}(t, \mathbf{x}) \simeq \frac{\mu_{0}}{4 \pi r} \int d^{3} \mathbf{x}^{\prime} \mathbf{J}\left(t-r / c, \mathbf{x}^{\prime}\right),$

where $\mathbf{J}(t, \mathbf{x})$ is the corresponding current density. Show that, in the dipole approximation, the large-distance behaviour of the magnetic field is given by,

$\mathbf{B}(t, \mathbf{x}) \simeq-\frac{\mu_{0}}{4 \pi r c} \hat{\mathbf{x}} \times \ddot{\mathbf{p}}(t-r / c)$

where $\mathbf{p}(t)$ is the electric dipole moment of the charge distribution. Assuming that, in the same approximation, the corresponding electric field is given as $\mathbf{E}=-c \hat{\mathbf{x}} \times \mathbf{B}$, evaluate the flux of energy through the surface element of a large sphere of radius $R$ centred at the origin. Hence show that the total power $P(t)$ radiated by the charge distribution is given by

$P(t)=\frac{\mu_{0}}{6 \pi c}|\ddot{\mathbf{p}}(t-R / c)|^{2}$

A particle of charge $q$ and mass $m$ undergoes simple harmonic motion in the $x$-direction with time period $T=2 \pi / \omega$ and amplitude $\mathcal{A}$ such that

$\mathbf{x}(t)=\mathcal{A} \sin (\omega t) \mathbf{i}_{x}$

Here $\mathbf{i}_{x}$ is a unit vector in the $x$-direction. Calculate the total power $P(t)$ radiated through a large sphere centred at the origin in the dipole approximation and determine its time averaged value,

$\langle P\rangle=\frac{1}{T} \int_{0}^{T} P(t) d t .$

For what values of the parameters $\mathcal{A}$ and $\omega$ is the dipole approximation valid?

Now suppose that the energy of the particle with trajectory $(\star)$ is given by the usual non-relativistic formula for a harmonic oscillator i.e. $E=m|\dot{\mathbf{x}}|^{2} / 2+m \omega^{2}|\mathbf{x}|^{2} / 2$, and that the particle loses energy due to the emission of radiation at a rate corresponding to the time-averaged power $\langle P\rangle$. Work out the half-life of this system (i.e. the time $t_{1 / 2}$ such that $\left.E\left(t_{1 / 2}\right)=E(0) / 2\right)$. Explain why the non-relativistic approximation for the motion of the particle is reliable as long as the dipole approximation is valid.

Paper 4, Section II, E

commentConsider a medium in which the electric displacement $\mathbf{D}(t, \mathbf{x})$ and magnetising field $\mathbf{H}(t, \mathbf{x})$ are linearly related to the electric and magnetic fields respectively with corresponding polarisation constants $\varepsilon$ and $\mu$;

$\mathbf{D}=\varepsilon \mathbf{E}, \quad \mathbf{B}=\mu \mathbf{H} .$

Write down Maxwell's equations for $\mathbf{E}, \mathbf{B}, \mathbf{D}$ and $\mathbf{H}$ in the absence of free charges and currents.

Consider EM waves of the form,

$\begin{aligned} &\mathbf{E}(t, \mathbf{x})=\mathbf{E}_{0} \sin (\mathbf{k} \cdot \mathbf{x}-\omega t) \\ &\mathbf{B}(t, \mathbf{x})=\mathbf{B}_{0} \sin (\mathbf{k} \cdot \mathbf{x}-\omega t) \end{aligned}$

Find conditions on the electric and magnetic polarisation vectors $\mathbf{E}_{0}$ and $\mathbf{B}_{0}$, wave-vector $\mathbf{k}$ and angular frequency $\omega$ such that these fields satisfy Maxwell's equations for the medium described above. At what speed do the waves propagate?

Consider two media, filling the regions $x<0$ and $x>0$ in three dimensional space, and having two different values $\varepsilon_{-}$and $\varepsilon_{+}$of the electric polarisation constant. Suppose an electromagnetic wave is incident from the region $x<0$ resulting in a transmitted wave in the region $x>0$ and also a reflected wave for $x<0$. The angles of incidence, reflection and transmission are denoted $\theta_{I}, \theta_{R}$ and $\theta_{T}$ respectively. By constructing a corresponding solution of Maxwell's equations, derive the law of reflection $\theta_{I}=\theta_{R}$ and Snell's law of refraction, $n_{-} \sin \theta_{I}=n_{+} \sin \theta_{T}$ where $n_{\pm}=c \sqrt{\varepsilon_{\pm} \mu}$ are the indices of refraction of the two media.

Consider the special case in which the electric polarisation vectors $\mathbf{E}_{I}, \mathbf{E}_{R}$ and $\mathbf{E}_{T}$ of the incident, reflected and transmitted waves are all normal to the plane of incidence (i.e. the plane containing the corresponding wave-vectors). By imposing appropriate boundary conditions for $\mathbf{E}$ and $\mathbf{H}$ at $x=0$, show that,

$\frac{\left|\mathbf{E}_{R}\right|}{\left|\mathbf{E}_{T}\right|}=\frac{1}{2}\left(1-\frac{\tan \theta_{R}}{\tan \theta_{T}}\right)$

Paper 1, Section II, D

commentDefine the field strength tensor $F^{\mu \nu}(x)$ for an electromagnetic field specified by a 4-vector potential $A^{\mu}(x)$. How do the components of $F^{\mu \nu}$ change under a Lorentz transformation? Write down two independent Lorentz-invariant quantities which are quadratic in the field strength tensor.

[Hint: The alternating tensor $\varepsilon^{\mu \nu \rho \sigma}$ takes the values $+1$ and $-1$ when $(\mu, \nu, \rho, \sigma)$ is an even or odd permutation of $(0,1,2,3)$ respectively and vanishes otherwise. You may assume this is an invariant tensor of the Lorentz group. In other words, its components are the same in all inertial frames.]

In an inertial frame with spacetime coordinates $x^{\mu}=(c t, \mathbf{x})$, the 4-vector potential has components $A^{\mu}=(\phi / c, \mathbf{A})$ and the electric and magnetic fields are given as

$\begin{aligned} \mathbf{E} &=-\nabla \phi-\frac{\partial \mathbf{A}}{\partial t} \\ \mathbf{B} &=\nabla \times \mathbf{A} \end{aligned}$

Evaluate the components of $F^{\mu \nu}$ in terms of the components of $\mathbf{E}$ and $\mathbf{B}$. Show that the quantities

$S=|\mathbf{B}|^{2}-\frac{1}{c^{2}}|\mathbf{E}|^{2} \quad \text { and } \quad T=\mathbf{E} \cdot \mathbf{B}$

are the same in all inertial frames.

A relativistic particle of mass $m$, charge $q$ and 4 -velocity $u^{\mu}(\tau)$ moves according to the Lorentz force law,

$\frac{d u^{\mu}}{d \tau}=\frac{q}{m} F_{\nu}^{\mu} u^{\nu}$

Here $\tau$ is the proper time. For the case of a constant, uniform field, write down a solution of $(*)$ giving $u^{\mu}(\tau)$ in terms of its initial value $u^{\mu}(0)$ as an infinite series in powers of the field strength.

Suppose further that the fields are such that both $S$ and $T$ defined above are zero. Work in an inertial frame with coordinates $x^{\mu}=(c t, x, y, z)$ where the particle is at rest at the origin at $t=0$ and the magnetic field points in the positive $z$-direction with magnitude $|\mathbf{B}|=B$. The electric field obeys $\mathbf{E} \cdot \hat{\mathbf{y}}=0$. Show that the particle moves on the curve $y^{2}=A x^{3}$ for some constant $A$ which you should determine.

Paper 3, Section II, D

commentStarting from the covariant form of the Maxwell equations and making a suitable choice of gauge which you should specify, show that the 4-vector potential due to an arbitrary 4-current $J^{\mu}(x)$ obeys the wave equation,

$\left(\nabla^{2}-\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}}\right) A^{\mu}=-\mu_{0} J^{\mu}$

where $x^{\mu}=(c t, \mathbf{x})$.

Use the method of Green's functions to show that, for a localised current distribution, this equation is solved by

$A^{\mu}(t, \mathbf{x})=\frac{\mu_{0}}{4 \pi} \int d^{3} x^{\prime} \frac{J^{\mu}\left(t_{\mathrm{ret}}, \mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}$

for some $t_{\text {ret }}$ that you should specify.

A point particle, of charge $q$, moving along a worldline $y^{\mu}(\tau)$ parameterised by proper time $\tau$, produces a 4 -vector potential

$A^{\mu}(x)=\frac{\mu_{0} q c}{4 \pi} \frac{\dot{y}^{\mu}\left(\tau_{\star}\right)}{\left|R^{\nu}\left(\tau_{\star}\right) \dot{y}_{\nu}\left(\tau_{\star}\right)\right|}$

where $R^{\mu}(\tau)=x^{\mu}-y^{\mu}(\tau)$. Define $\tau_{\star}(x)$ and draw a spacetime diagram to illustrate its physical significance.

Suppose the particle follows a circular trajectory,

$\mathbf{y}(t)=(R \cos (\omega t), R \sin (\omega t), 0)$

(with $y^{0}=c t$ ), in some inertial frame with coordinates $(c t, x, y, z)$. Evaluate the resulting 4 -vector potential at a point on the $z$-axis as a function of $z$ and $t$.

Paper 4 , Section II, D

comment(a) Define the polarisation of a dielectric material and explain what is meant by the term bound charge.

Consider a sample of material with spatially dependent polarisation $\mathbf{P}(\mathbf{x})$ occupying a region $V$ with surface $S$. Show that, in the absence of free charge, the resulting scalar potential $\phi(\mathbf{x})$ can be ascribed to bulk and surface densities of bound charge.

Consider a sphere of radius $R$ consisting of a dielectric material with permittivity $\epsilon$ surrounded by a region of vacuum. A point-like electric charge $q$ is placed at the centre of the sphere. Determine the density of bound charge on the surface of the sphere.

(b) Define the magnetization of a material and explain what is meant by the term bound current.

Consider a sample of material with spatially-dependent magnetization $\mathbf{M}(\mathbf{x})$ occupying a region $V$ with surface $S$. Show that, in the absence of free currents, the resulting vector potential $\mathbf{A}(\mathbf{x})$ can be ascribed to bulk and surface densities of bound current.

Consider an infinite cylinder of radius $r$ consisting of a material with permeability $\mu$ surrounded by a region of vacuum. A thin wire carrying current $I$ is placed along the axis of the cylinder. Determine the direction and magnitude of the resulting bound current density on the surface of the cylinder. What is the magnetization $\mathbf{M}(\mathbf{x})$ on the surface of the cylinder?

Paper 1, Section II, 35D

commentIn some inertial reference frame $S$, there is a uniform electric field $\mathbf{E}$ directed along the positive $y$-direction and a uniform magnetic field $\mathbf{B}$ directed along the positive $z$ direction. The magnitudes of the fields are $E$ and $B$, respectively, with $E<c B$. Show that it is possible to find a reference frame in which the electric field vanishes, and determine the relative speed $\beta c$ of the two frames and the magnitude of the magnetic field in the new frame.

[Hint: You may assume that under a standard Lorentz boost with speed $v=\beta$ c along the $x$-direction, the electric and magnetic field components transform as

$\left(\begin{array}{c} E_{x}^{\prime} \\ E_{y}^{\prime} \\ E_{z}^{\prime} \end{array}\right)=\left(\begin{array}{c} E_{x} \\ \gamma(\beta)\left(E_{y}-v B_{z}\right) \\ \gamma(\beta)\left(E_{z}+v B_{y}\right) \end{array}\right) \quad a n d \quad\left(\begin{array}{c} B_{x}^{\prime} \\ B_{y}^{\prime} \\ B_{z}^{\prime} \end{array}\right)=\left(\begin{array}{c} B_{x} \\ \gamma(\beta)\left(B_{y}+v E_{z} / c^{2}\right) \\ \gamma(\beta)\left(B_{z}-v E_{y} / c^{2}\right) \end{array}\right),$

where the Lorentz factor $\gamma(\beta)=\left(1-\beta^{2}\right)^{-1 / 2}$.]

A point particle of mass $m$ and charge $q$ moves relativistically under the influence of the fields $\mathbf{E}$ and $\mathbf{B}$. The motion is in the plane $z=0$. By considering the motion in the reference frame in which the electric field vanishes, or otherwise, show that, with a suitable choice of origin, the worldline of the particle has components in the frame $S$ of the form

$\begin{aligned} &c t(\tau)=\gamma(u / c) \gamma(\beta)\left[c \tau+\frac{\beta u}{\omega} \sin \omega \tau\right] \\ &x(\tau)=\gamma(u / c) \gamma(\beta)\left[\beta c \tau+\frac{u}{\omega} \sin \omega \tau\right] \\ &y(\tau)=\frac{u \gamma(u / c)}{\omega} \cos \omega \tau \end{aligned}$

Here, $u$ is a constant speed with Lorentz factor $\gamma(u / c), \tau$ is the particle's proper time, and $\omega$ is a frequency that you should determine.

Using dimensionless coordinates,

$\tilde{x}=\frac{\omega}{u \gamma(u / c)} x \quad \text { and } \quad \tilde{y}=\frac{\omega}{u \gamma(u / c)} y$

sketch the trajectory of the particle in the $(\tilde{x}, \tilde{y})$-plane in the limiting cases $2 \pi \beta \ll u / c$ and $2 \pi \beta \gg u / c$.

Paper 3, Section II, D

commentBy considering the force per unit volume $\mathbf{f}=\rho \mathbf{E}+\mathbf{J} \times \mathbf{B}$ on a charge density $\rho$ and current density $\mathbf{J}$ due to an electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$, show that

$\frac{\partial g_{i}}{\partial t}+\frac{\partial \sigma_{i j}}{\partial x_{j}}=-f_{i}$

where $\mathbf{g}=\epsilon_{0} \mathbf{E} \times \mathbf{B}$ and the symmetric tensor $\sigma_{i j}$ should be specified.

Give the physical interpretation of $\mathbf{g}$ and $\sigma_{i j}$ and explain how $\sigma_{i j}$ can be used to calculate the net electromagnetic force exerted on the charges and currents within some region of space in static situations.

The plane $x=0$ carries a uniform charge $\sigma$ per unit area and a current $K$ per unit length along the $z$-direction. The plane $x=d$ carries the opposite charge and current. Show that between these planes

$\sigma_{i j}=\frac{\sigma^{2}}{2 \epsilon_{0}}\left(\begin{array}{ccc} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right)+\frac{\mu_{0} K^{2}}{2}\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{array}\right)$

and $\sigma_{i j}=0$ for $x<0$ and $x>d$.

Use $(*)$ to find the electromagnetic force per unit area exerted on the charges and currents in the $x=0$ plane. Show that your result agrees with direct calculation of the force per unit area based on the Lorentz force law.

If the current $K$ is due to the motion of the charge $\sigma$ with speed $v$, is it possible for the force between the planes to be repulsive?

Paper 4, Section II, D

commentA dielectric material has a real, frequency-independent relative permittivity $\epsilon_{r}$ with $\left|\epsilon_{r}-1\right| \ll 1$. In this case, the macroscopic polarization that develops when the dielectric is placed in an external electric field $\mathbf{E}_{\text {ext }}(t, \mathbf{x})$ is $\mathbf{P}(t, \mathbf{x}) \approx \epsilon_{0}\left(\epsilon_{r}-1\right) \mathbf{E}_{\text {ext }}(t, \mathbf{x})$. Explain briefly why the associated bound current density is

$\mathbf{J}_{\text {bound }}(t, \mathbf{x}) \approx \epsilon_{0}\left(\epsilon_{r}-1\right) \frac{\partial \mathbf{E}_{\text {ext }}(t, \mathbf{x})}{\partial t}$

[You should ignore any magnetic response of the dielectric.]

A sphere of such a dielectric, with radius $a$, is centred on $\mathbf{x}=0$. The sphere scatters an incident plane electromagnetic wave with electric field

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

where $\omega=c|\mathbf{k}|$ and $\mathbf{E}_{0}$ is a constant vector. Working in the Lorenz gauge, show that at large distances $r=|\mathbf{x}|$, for which both $r \gg a$ and $k a^{2} / r \ll 2 \pi$, the magnetic vector potential $\mathbf{A}_{\text {scatt }}(t, \mathbf{x})$ of the scattered radiation is

$\mathbf{A}_{\mathrm{scatt}}(t, \mathbf{x}) \approx-i \omega \mathbf{E}_{0} \frac{e^{i(k r-\omega t)}}{r} \frac{\left(\epsilon_{r}-1\right)}{4 \pi c^{2}} \int_{\left|\mathbf{x}^{\prime}\right| \leqslant a} e^{i \mathbf{q} \cdot \mathbf{x}^{\prime}} d^{3} \mathbf{x}^{\prime}$

where $\mathbf{q}=\mathbf{k}-k \hat{\mathbf{x}}$ with $\hat{\mathbf{x}}=\mathbf{x} / r$.

In the far-field, where $k r \gg 1$, the electric and magnetic fields of the scattered radiation are given by

$\begin{aligned} &\mathbf{E}_{\text {scatt }}(t, \mathbf{x}) \approx-i \omega \hat{\mathbf{x}} \times\left[\hat{\mathbf{x}} \times \mathbf{A}_{\text {scatt }}(t, \mathbf{x})\right] \\ &\mathbf{B}_{\text {scatt }}(t, \mathbf{x}) \approx i k \hat{\mathbf{x}} \times \mathbf{A}_{\text {scatt }}(t, \mathbf{x}) \end{aligned}$

By calculating the Poynting vector of the scattered and incident radiation, show that the ratio of the time-averaged power scattered per unit solid angle to the time-averaged incident power per unit area (i.e. the differential cross-section) is

$\frac{d \sigma}{d \Omega}=\left(\epsilon_{r}-1\right)^{2} k^{4}\left(\frac{\sin (q a)-q a \cos (q a)}{q^{3}}\right)^{2}\left|\hat{\mathbf{x}} \times \hat{\mathbf{E}}_{0}\right|^{2}$

where $\hat{\mathbf{E}}_{0}=\mathbf{E}_{0} /\left|\mathbf{E}_{0}\right|$ and $q=|\mathbf{q}|$.

[You may assume that, in the Lorenz gauge, the retarded potential due to a localised current distribution is

$\mathbf{A}(t, \mathbf{x})=\frac{\mu_{0}}{4 \pi} \int \frac{\mathbf{J}\left(t_{\mathrm{ret}}, \mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} d^{3} \mathbf{x}^{\prime},$

where the retarded time $\left.t_{\text {ret }}=t-\left|\mathbf{x}-\mathbf{x}^{\prime}\right| / c .\right]$

Paper 1, Section II, E

commentA point particle of charge $q$ and mass $m$ moves in an electromagnetic field with 4 -vector potential $A_{\mu}(x)$, where $x^{\mu}$ is position in spacetime. Consider the action

$S=-m c \int\left(-\eta_{\mu \nu} \frac{d x^{\mu}}{d \lambda} \frac{d x^{\nu}}{d \lambda}\right)^{1 / 2} d \lambda+q \int A_{\mu} \frac{d x^{\mu}}{d \lambda} d \lambda$

where $\lambda$ is an arbitrary parameter along the particle's worldline and $\eta_{\mu \nu}=\operatorname{diag}(-1,+1,+1,+1)$ is the Minkowski metric.

(a) By varying the action with respect to $x^{\mu}(\lambda)$, with fixed endpoints, obtain the equation of motion

$m \frac{d u^{\mu}}{d \tau}=q F_{\nu}^{\mu} u^{\nu} \text {, }$

where $\tau$ is the proper time, $u^{\mu}=d x^{\mu} / d \tau$ is the velocity 4-vector, and $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$ is the field strength tensor.

(b) This particle moves in the field generated by a second point charge $Q$ that is held at rest at the origin of some inertial frame. By choosing a suitable expression for $A_{\mu}$ and expressing the first particle's spatial position in spherical polar coordinates $(r, \theta, \phi)$, show from the action $(*)$ that

$\begin{aligned} \mathcal{E} & \equiv \dot{t}-\Gamma / r \\ \ell c & \equiv r^{2} \dot{\phi} \sin ^{2} \theta \end{aligned}$

are constants, where $\Gamma=-q Q /\left(4 \pi \epsilon_{0} m c^{2}\right)$ and overdots denote differentiation with respect to $\tau$.

(c) Show that when the motion is in the plane $\theta=\pi / 2$,

$\mathcal{E}+\frac{\Gamma}{r}=\sqrt{1+\frac{\dot{r}^{2}}{c^{2}}+\frac{\ell^{2}}{r^{2}}}$

Hence show that the particle's orbit is bounded if $\mathcal{E}<1$, and that the particle can reach the origin in finite proper time if $\Gamma>|\ell|$.

Paper 3, Section II, E

commentThe current density in an antenna lying along the $z$-axis takes the form

$\mathbf{J}(t, \mathbf{x})=\left\{\begin{array}{ll} \hat{\mathbf{z}} I_{0} \sin (k d-k|z|) e^{-i \omega t} \delta(x) \delta(y) & |z| \leqslant d \\ \mathbf{0} & |z|>d \end{array},\right.$

where $I_{0}$ is a constant and $\omega=c k$. Show that at distances $r=|\mathbf{x}|$ for which both $r \gg d$ and $r \gg k d^{2} /(2 \pi)$, the retarded vector potential in Lorenz gauge is

$\mathbf{A}(t, \mathbf{x}) \approx \hat{\mathbf{z}} \frac{\mu_{0} I_{0}}{4 \pi r} e^{-i \omega(t-r / c)} \int_{-d}^{d} \sin \left(k d-k\left|z^{\prime}\right|\right) e^{-i k z^{\prime} \cos \theta} d z^{\prime}$

where $\cos \theta=\hat{\mathbf{r}} \cdot \hat{\mathbf{z}}$ and $\hat{\mathbf{r}}=\mathbf{x} /|\mathbf{x}|$. Evaluate the integral to show that

$\mathbf{A}(t, \mathbf{x}) \approx \hat{\mathbf{z}} \frac{\mu_{0} I_{0}}{2 \pi k r}\left(\frac{\cos (k d \cos \theta)-\cos (k d)}{\sin ^{2} \theta}\right) e^{-i \omega(t-r / c)}$

In the far-field, where $k r \gg 1$, the electric and magnetic fields are given by

$\begin{aligned} &\mathbf{E}(t, \mathbf{x}) \approx-i \omega \hat{\mathbf{r}} \times[\hat{\mathbf{r}} \times \mathbf{A}(t, \mathbf{x})] \\ &\mathbf{B}(t, \mathbf{x}) \approx i k \hat{\mathbf{r}} \times \mathbf{A}(t, \mathbf{x}) \end{aligned}$

By calculating the Poynting vector, show that the time-averaged power radiated per unit solid angle is

$\frac{d \mathcal{P}}{d \Omega}=\frac{c \mu_{0} I_{0}^{2}}{8 \pi^{2}}\left(\frac{\cos (k d \cos \theta)-\cos (k d)}{\sin \theta}\right)^{2}$

[You may assume that in Lorenz gauge, the retarded potential due to a localised current distribution is

$\mathbf{A}(t, \mathbf{x})=\frac{\mu_{0}}{4 \pi} \int \frac{\mathbf{J}\left(t_{\mathrm{ret}}, \mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} d^{3} \mathbf{x}^{\prime}$

where the retarded time $\left.t_{\text {ret }}=t-\left|\mathbf{x}-\mathbf{x}^{\prime}\right| / c .\right]$

Paper 4, Section II, E

comment(a) A uniform, isotropic dielectric medium occupies the half-space $z>0$. The region $z<0$ is in vacuum. State the boundary conditions that should be imposed on $\mathbf{E}, \mathbf{D}, \mathbf{B}$ and $\mathbf{H}$ at $z=0$.

(b) A linearly polarized electromagnetic plane wave, with magnetic field in the $(x, y)$-plane, is incident on the dielectric from $z<0$. The wavevector $\mathbf{k}$ makes an acute angle $\theta_{I}$ to the normal $\hat{\mathbf{z}}$. If the dielectric has frequency-independent relative permittivity $\epsilon_{r}$, show that the fraction of the incident power that is reflected is

$\mathcal{R}=\left(\frac{n \cos \theta_{I}-\cos \theta_{T}}{n \cos \theta_{I}+\cos \theta_{T}}\right)^{2}$

where $n=\sqrt{\epsilon_{r}}$, and the angle $\theta_{T}$ should be specified. [You should ignore any magnetic response of the dielectric.]

(c) Now suppose that the dielectric moves at speed $\beta c$ along the $x$-axis, the incident angle $\theta_{I}=0$, and the magnetic field of the incident radiation is along the $y$-direction. Show that the reflected radiation propagates normal to the surface $z=0$, has the same frequency as the incident radiation, and has magnetic field also along the $y$-direction. [Hint: You may assume that under a standard Lorentz boost with speed $v=\beta c$ along the $x$-direction, the electric and magnetic field components transform as

$\left(\begin{array}{c} E_{x}^{\prime} \\ E_{y}^{\prime} \\ E_{z}^{\prime} \end{array}\right)=\left(\begin{array}{c} E_{x} \\ \gamma\left(E_{y}-v B_{z}\right) \\ \gamma\left(E_{z}+v B_{y}\right) \end{array}\right) \quad \text { and } \quad\left(\begin{array}{c} B_{x}^{\prime} \\ B_{y}^{\prime} \\ B_{z}^{\prime} \end{array}\right)=\left(\begin{array}{c} B_{x} \\ \gamma\left(B_{y}+v E_{z} / c^{2}\right) \\ \gamma\left(B_{z}-v E_{y} / c^{2}\right) \end{array}\right)$

where $\gamma=\left(1-\beta^{2}\right)^{-1 / 2}$.]

(d) Show that the fraction of the incident power reflected from the moving dielectric

$\mathcal{R}_{\beta}=\left(\frac{n / \gamma-\sqrt{1-\beta^{2} / n^{2}}}{n / \gamma+\sqrt{1-\beta^{2} / n^{2}}}\right)^{2}$

Paper 1, Section II, A

commentBriefly explain how to interpret the components of the relativistic stress-energy tensor in terms of the density and flux of energy and momentum in some inertial frame.

(i) The stress-energy tensor of the electromagnetic field is

$T_{\mathrm{em}}^{\mu \nu}=\frac{1}{\mu_{0}}\left(F^{\mu \alpha} F_{\alpha}^{\nu}-\frac{1}{4} \eta^{\mu \nu} F^{\alpha \beta} F_{\alpha \beta}\right)$

where $F_{\mu \nu}$ is the field strength, $\eta_{\mu \nu}$ is the Minkowski metric, and $\mu_{0}$ is the permeability of free space. Show that $\partial_{\mu} T_{\mathrm{em}}^{\mu \nu}=-F_{\mu}^{\nu} J^{\mu}$, where $J^{\mu}$ is the current 4-vector.

[ Maxwell's equations are $\partial_{\mu} F^{\mu \nu}=-\mu_{0} J^{\nu}$ and $\partial_{\rho} F_{\mu \nu}+\partial_{\nu} F_{\rho \mu}+\partial_{\mu} F_{\nu \rho}=0 .$ ]

(ii) A fluid consists of point particles of rest mass $m$ and charge $q$. The fluid can be regarded as a continuum, with 4 -velocity $u^{\mu}(x)$ depending on the position $x$ in spacetime. For each $x$ there is an inertial frame $S_{x}$ in which the fluid particles at $x$ are at rest. By considering components in $S_{x}$, show that the fluid has a current 4-vector field

$J^{\mu}=q n_{0} u^{\mu}$

and a stress-energy tensor

$T_{\text {fluid }}^{\mu \nu}=m n_{0} u^{\mu} u^{\nu},$

where $n_{0}(x)$ is the proper number density of particles (the number of particles per unit spatial volume in $S_{x}$ in a small region around $x$ ). Write down the Lorentz 4-force on a fluid particle at $x$. By considering the resulting 4 -acceleration of the fluid, show that the total stress-energy tensor is conserved, i.e.

$\partial_{\mu}\left(T_{\mathrm{em}}^{\mu \nu}+T_{\text {fluid }}^{\mu \nu}\right)=0 .$

Paper 3, Section II, 34A

comment(i) Consider the action

$S=-\frac{1}{4 \mu_{0} c} \int\left(F_{\mu \nu} F^{\mu \nu}+2 \lambda^{2} A_{\mu} A^{\mu}\right) d^{4} x+\frac{1}{c} \int A_{\mu} J^{\mu} d^{4} x$

where $A_{\mu}(x)$ is a 4-vector potential, $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$ is the field strength tensor, $J^{\mu}(x)$ is a conserved current, and $\lambda \geqslant 0$ is a constant. Derive the field equation

$\partial_{\mu} F^{\mu \nu}-\lambda^{2} A^{\nu}=-\mu_{0} J^{\nu} .$

For $\lambda=0$ the action $S$ describes standard electromagnetism. Show that in this case the theory is invariant under gauge transformations of the field $A_{\mu}(x)$, which you should define. Is the theory invariant under these same gauge transformations when $\lambda>0$ ?

Show that when $\lambda>0$ the field equation above implies

$\partial_{\mu} \partial^{\mu} A^{\nu}-\lambda^{2} A^{\nu}=-\mu_{0} J^{\nu}$

Under what circumstances does $(*)$ hold in the case $\lambda=0$ ?

(ii) Now suppose that $A_{\mu}(x)$ and $J_{\mu}(x)$ obeying $(*)$ reduce to static 3 -vectors $\mathbf{A}(\mathbf{x})$ and $\mathbf{J}(\mathbf{x})$ in some inertial frame. Show that there is a solution

$\mathbf{A}(\mathbf{x})=-\mu_{0} \int G\left(\left|\mathbf{x}-\mathbf{x}^{\prime}\right|\right) \mathbf{J}\left(\mathbf{x}^{\prime}\right) d^{3} \mathbf{x}^{\prime}$

for a suitable Green's function $G(R)$ with $G(R) \rightarrow 0$ as $R \rightarrow \infty$. Determine $G(R)$ for any $\lambda \geqslant 0$. [Hint: You may find it helpful to consider first the case $\lambda=0$ and then the case $\lambda>0$, using the result $\nabla^{2}\left(\frac{1}{R} f(R)\right)=\nabla^{2}\left(\frac{1}{R}\right) f(R)+\frac{1}{R} f^{\prime \prime}(R)$, where $\left.R=\left|\mathbf{x}-\mathbf{x}^{\prime}\right| .\right]$

If $\mathbf{J}(\mathbf{x})$ is zero outside some bounded region, comment on the effect of the value of $\lambda$ on the behaviour of $\mathbf{A}(\mathbf{x})$ when $|\mathbf{x}|$ is large. [No further detailed calculations are required.]

Paper 4, Section II, A

commentA point particle of charge $q$ has trajectory $y^{\mu}(\tau)$ in Minkowski space, where $\tau$ is its proper time. The resulting electromagnetic field is given by the Liénard-Wiechert 4-potential

$A^{\mu}(x)=-\frac{q \mu_{0} c}{4 \pi} \frac{u^{\mu}\left(\tau_{*}\right)}{R^{\nu}\left(\tau_{*}\right) u_{\nu}\left(\tau_{*}\right)}, \quad \text { where } \quad R^{\nu}=x^{\nu}-y^{\nu}(\tau) \quad \text { and } \quad u^{\mu}=d y^{\mu} / d \tau$

Write down the condition that determines the point $y^{\mu}\left(\tau_{*}\right)$ on the trajectory of the particle for a given value of $x^{\mu}$. Express this condition in terms of components, setting $x^{\mu}=(c t, \mathbf{x})$ and $y^{\mu}=\left(c t^{\prime}, \mathbf{y}\right)$, and define the retarded time $t_{r}$.

Suppose that the 3 -velocity of the particle $\mathbf{v}\left(t^{\prime}\right)=\dot{\mathbf{y}}\left(t^{\prime}\right)=d \mathbf{y} / d t^{\prime}$ is small in size compared to $c$, and suppose also that $r=|\mathbf{x}| \gg|\mathbf{y}|$. Working to leading order in $1 / r$ and to first order in $\mathbf{v}$, show that

$\phi(x)=\frac{q \mu_{0} c}{4 \pi r}\left(c+\hat{\mathbf{r}} \cdot \mathbf{v}\left(t_{r}\right)\right), \quad \mathbf{A}(x)=\frac{q \mu_{0}}{4 \pi r} \mathbf{v}\left(t_{r}\right), \quad \text { where } \quad \hat{\mathbf{r}}=\mathbf{x} / r$

Now assume that $t_{r}$ can be replaced by $t_{-}=t-(r / c)$ in the expressions for $\phi$ and $\mathbf{A}$ above. Calculate the electric and magnetic fields to leading order in $1 / r$ and hence show that the Poynting vector is (in this approximation)

$\mathbf{N}(x)=\frac{q^{2} \mu_{0}}{(4 \pi)^{2} c} \frac{\hat{\mathbf{r}}}{r^{2}}\left|\hat{\mathbf{r}} \times \dot{\mathbf{v}}\left(t_{-}\right)\right|^{2}$

If the charge $q$ is performing simple harmonic motion $\mathbf{y}\left(t^{\prime}\right)=A \mathbf{n} \cos \omega t^{\prime}$, where $\mathbf{n}$ is a unit vector and $A \omega \ll c$, find the total energy radiated during one period of oscillation.

Paper 1, Section II, 36C

(i) Starting from the field-strength tensor $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$