Electrodynamics

2021

• Paper 1, Section II, 37C

  Electrodynamics | Part II, 2021

  (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 $\mathbf{B}_{0}=\left(0,0, B_{0}\right)$.

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• Paper 3, Section II, 36C

  Electrodynamics | Part II, 2021

  (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?

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• Paper 4, Section II, 36C

  Electrodynamics | Part II, 2021

  (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.

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2020

• Paper 1, Section II, 37D

  Electrodynamics | Part II, 2020

  A 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.

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• Paper 3, Section II, D

  Electrodynamics | Part II, 2020

  The 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.]

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• Paper 4 , Section II, 36D

  Electrodynamics | Part II, 2020

  (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.

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2019

• Paper 1, Section II, E

  Electrodynamics | Part II, 2019

  A 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.

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• Paper 3, Section II, E

  Electrodynamics | Part II, 2019

  A 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.

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• Paper 4, Section II, E

  Electrodynamics | Part II, 2019

  Consider 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,

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

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2018

• Paper 1, Section II, D

  Electrodynamics | Part II, 2018

  Define 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.

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• Paper 3, Section II, D

  Electrodynamics | Part II, 2018

  Starting 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 $t$.

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• Paper 4 , Section II, D

  Electrodynamics | Part II, 2018

  (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?

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2017

• Paper 1, Section II, 35D

  Electrodynamics | Part II, 2017

  In 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 $2 \pi \beta \gg u / c$.

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• Paper 3, Section II, D

  Electrodynamics | Part II, 2017

  By 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?

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• Paper 4, Section II, D

  Electrodynamics | Part II, 2017

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

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2016

• Paper 1, Section II, E

  Electrodynamics | Part II, 2016

  A 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 $\Gamma>|\ell|$.

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• Paper 3, Section II, E

  Electrodynamics | Part II, 2016

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

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• Paper 4, Section II, E

  Electrodynamics | Part II, 2016

  (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

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

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2015

• Paper 1, Section II, A

  Electrodynamics | Part II, 2015

  Briefly 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.

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

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• Paper 3, Section II, 34A

  Electrodynamics | Part II, 2015

  (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.]

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• Paper 4, Section II, A

  Electrodynamics | Part II, 2015

  A 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.

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2014

• Paper 1, Section II, 36C

  Electrodynamics | Part II, 2014

  (i) Starting from the field-strength tensor $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$, where $A^{\mu}=(\phi / c, \mathbf{A})$ is the 4-vector potential with components such that

  $$\mathbf{E}=-\frac{\partial \mathbf{A}}{\partial t}-\boldsymbol{\nabla} \phi \quad \text { and } \quad \mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$$

  derive the transformation laws for the components of the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ under the standard Lorentz boost $x^{\prime \mu}=\Lambda_{\nu}^{\mu} x^{\nu}$ with

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

  (ii) Two point charges, each with electric charge $q$, are at rest and separated by a distance $d$ in some inertial frame $S$. By transforming the fields from the rest frame $S$, calculate the magnitude and direction of the force between the two charges in an inertial frame in which the charges are moving with speed $\beta c$ in a direction perpendicular to their separation.

  (iii) The 4-force for a particle with 4-momentum $p^{\mu}$ is $F^{\mu}=d p^{\mu} / d \tau$, where $\tau$ is proper time. Show that the components of $F^{\mu}$ in an inertial frame in which the particle has 3 -velocity $\mathbf{v}$ are

  $$F^{\mu}=\gamma(\mathbf{F} \cdot \mathbf{v} / c, \mathbf{F})$$

  where $\gamma=\left(1-\mathbf{v} \cdot \mathbf{v} / c^{2}\right)^{-1 / 2}$ and $\mathbf{F}$ is the 3-force acting on the particle. Hence verify that your result in (ii) above is consistent with Lorentz transforming the electromagnetic 3 -force from the rest frame $S$.

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• Paper 3, Section II, C

  Electrodynamics | Part II, 2014

  The 4-vector potential $A^{\mu}(t, \mathbf{x})$ (in the Lorenz gauge $\partial_{\mu} A^{\mu}=0$ ) due to a localised source with conserved 4-vector current $J^{\mu}$ is

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

  where $t_{\text {ret }}=t-\left|\mathbf{x}-\mathbf{x}^{\prime}\right| / c$. For a source that varies slowly in time, show that the spatial components of $A^{\mu}$ at a distance $r=|\mathbf{x}|$ that is large compared to the spatial extent of the source are

  $$\left.\mathbf{A}(t, \mathbf{x}) \approx \frac{\mu_{0}}{4 \pi r} \frac{d \mathbf{P}}{d t}\right|_{t-r / c}$$

  where $\mathbf{P}$ is the electric dipole moment of the source, which you should define. Explain what is meant by the far-field region, and calculate the leading-order part of the magnetic field there.

  A point charge $q$ moves non-relativistically in a circle of radius $a$ in the $(x, y)$ plane with angular frequency $\omega$ (such that $a \omega \ll c$ ). Show that the magnetic field in the far-field at the point $\mathbf{x}$ with spherical polar coordinates $r, \theta$ and $\phi$ has components along the $\theta$ and $\phi$ directions given by

  $$\begin{aligned} &B_{\theta} \approx-\frac{\mu_{0} \omega^{2} q a}{4 \pi r c} \sin [\omega(t-r / c)-\phi] \\ &B_{\phi} \approx \frac{\mu_{0} \omega^{2} q a}{4 \pi r c} \cos [\omega(t-r / c)-\phi] \cos \theta \end{aligned}$$

  Calculate the total power radiated by the charge.

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• Paper 4, Section II, 35C

  Electrodynamics | Part II, 2014

  (i) The action $S$ for a point particle of rest mass $m$ and charge $q$ moving along a trajectory $x^{\mu}(\lambda)$ in the presence of an electromagnetic 4 -vector potential $A^{\mu}$ is

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

  where $\lambda$ is an arbitrary parametrization of the path and $\eta_{\mu \nu}$ is the Minkowski metric. By varying the action with respect to $x^{\mu}(\lambda)$, derive the equation of motion $m \ddot{x}^{\mu}=q F^{\mu} \dot{x}^{\nu}$, where $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$ and overdots denote differentiation with respect to proper time for the particle.

  (ii) The particle moves in constant electric and magnetic fields with non-zero Cartesian components $E_{z}=E$ and $B_{y}=B$, with $B>E / c>0$ in some inertial frame. Verify that a suitable 4-vector potential has components

  $$A^{\mu}=(0,0,0,-B x-E t)$$

  in that frame.

  Find the equations of motion for $x, y, z$ and $t$ in terms of proper time $\tau$. For the case of a particle that starts at rest at the spacetime origin at $\tau=0$, show that

  $$\ddot{z}+\frac{q^{2}}{m^{2}}\left(B^{2}-\frac{E^{2}}{c^{2}}\right) z=\frac{q E}{m} .$$

  Find the trajectory $x^{\mu}(\tau)$ and sketch its projection onto the $(x, z)$ plane.

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2013

• Paper 1, Section II, 36B

  Electrodynamics | Part II, 2013

  (i) Starting from

  $$F^{\mu \nu}=\left(\begin{array}{cccc} 0 & E_{1} / c & E_{2} / c & E_{3} / c \\ -E_{1} / c & 0 & B_{3} & -B_{2} \\ -E_{2} / c & -B_{3} & 0 & B_{1} \\ -E_{3} / c & B_{2} & -B_{1} & 0 \end{array}\right)$$

  and performing a Lorentz transformation with $\gamma=1 / \sqrt{1-u^{2} / c^{2}}$, using

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

  show how $\mathbf{E}$ and $\mathbf{B}$ transform under a Lorentz transformation.

  (ii) By taking the limit $c \rightarrow \infty$, obtain the behaviour of $\mathbf{E}$ and $\mathbf{B}$ under a Galilei transfomation and verify the invariance under Galilei transformations of the nonrelativistic equation

  $$m \frac{d \mathbf{v}}{d t}=q(\mathbf{E}+\mathbf{v} \times \mathbf{B})$$

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

  $$\mathbf{E}=\mathbf{E}_{0} f(t-\mathbf{n} \cdot \mathbf{x} / c), \quad \mathbf{B}=\mathbf{B}_{0} f(t-\mathbf{n} \cdot \mathbf{x} / c)$$

  where $f$ is an arbitrary function, $\mathbf{n}$ is a unit vector, and the constant vectors $\mathbf{E}_{0}$ and $\mathbf{B}_{0}$ are subject to restrictions which should be stated.

  (iv) Perform a Galilei transformation of a solution $(\star)$, with $\mathbf{n}=(1,0,0)$. Show that, by a particular choice of $u$, the solution may brought to the form

  $$\tilde{\mathbf{E}}=\tilde{\mathbf{E}}_{0} g(\tilde{x}), \quad \tilde{\mathbf{B}}=\tilde{\mathbf{B}}_{0} g(\tilde{x}),$$

  where $g$ is an arbitrary function and $\tilde{x}$ is a spatial coordinate in the rest frame. By showing that $(t)$ is not a solution of Maxwell's equations in the boosted frame, conclude that Maxwell's equations are not invariant under Galilei transformations.

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• Paper 3, Section II, 36B

  Electrodynamics | Part II, 2013

  (i) Obtain Maxwell's equations in empty space from the action functional

  $$S\left[A_{\mu}\right]=-\frac{1}{\mu_{0}} \int d^{4} x \frac{1}{4} F_{\mu \nu} F^{\mu \nu}$$

  where $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$.

  (ii) A modification of Maxwell's equations has the action functional

  $$\tilde{S}\left[A_{\mu}\right]=-\frac{1}{\mu_{0}} \int d^{4} x\left\{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}+\frac{1}{2 \lambda^{2}} A_{\mu} A^{\mu}\right\},$$

  where again $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$ and $\lambda$ is a constant. Obtain the equations of motion of this theory and show that they imply $\partial_{\mu} A^{\mu}=0$.

  (iii) Show that the equations of motion derived from $\tilde{S}$ admit solutions of the form

  $$A^{\mu}=A_{0}^{\mu} e^{i k_{\nu} x^{\nu}}$$

  where $A_{0}^{\mu}$ is a constant 4-vector, and the 4 -vector $k_{\mu}$ satisfies $A_{0}^{\mu} k_{\mu}=0$ and $k_{\mu} k^{\mu}=-1 / \lambda^{2}$.

  (iv) Show further that the tensor

  $$T_{\mu \nu}=\frac{1}{\mu_{0}}\left\{F_{\mu \sigma} F_{\nu}^{\sigma}-\frac{1}{4} \eta_{\mu \nu} F_{\alpha \beta} F^{\alpha \beta}-\frac{1}{2 \lambda^{2}}\left(\eta_{\mu \nu} A_{\alpha} A^{\alpha}-2 A_{\mu} A_{\nu}\right)\right\}$$

  is conserved, that is $\partial^{\mu} T_{\mu \nu}=0$.

  comment

• Paper 4, Section II, 35B

  Electrodynamics | Part II, 2013

  (i) For a time-dependent source, confined within a domain $D$, show that the time derivative $\dot{d}$ of the dipole moment $\mathbf{d}$ satisfies

  $$\dot{\mathbf{d}}=\int_{D} d^{3} y \mathbf{J}(\mathbf{y})$$

  where $\mathbf{J}$ is the current density.

  (ii) The vector potential $\mathbf{A}(\mathbf{x}, t)$ due to a time-dependent source is given by

  $$\mathbf{A}=\frac{1}{r} f(t-r / c) \mathbf{k}$$

  where $r=|\mathbf{x}| \neq 0$, and $\mathbf{k}$ is the unit vector in the $z$ direction. Calculate the resulting magnetic field $\mathbf{B}(\mathbf{x}, t)$. By considering the magnetic field for small $r$ show that the dipole moment of the effective source satisfies

  $$\frac{\mu_{0}}{4 \pi} \dot{\mathbf{d}}=f(t) \mathbf{k}$$

  Calculate the asymptotic form of the magnetic field $\mathbf{B}$ at very large $r$.

  (iii) Using the equation

  $$\frac{\partial \mathbf{E}}{\partial t}=c^{2} \nabla \times \mathbf{B}$$

  calculate $\mathbf{E}$ at very large $r$. Show that $\mathbf{E}, \mathbf{B}$ and $\hat{\mathbf{r}}=\mathbf{x} /|\mathbf{x}|$ form a right-handed triad, and moreover $|\mathbf{E}|=c|\mathbf{B}|$. How do $|\mathbf{E}|$ and $|\mathbf{B}|$ depend on $r ?$ What is the significance of this?

  (iv) Calculate the power $P(\theta, \phi)$ emitted per unit solid angle and sketch its dependence on $\theta$. Show that the emitted radiation is polarised and describe how the plane of polarisation (that is, the plane in which $\mathbf{E}$ and $\hat{\mathbf{r}}$ lie) depends on the direction of the dipole. Suppose the dipole moment has constant amplitude and constant frequency and so the radiation is monochromatic with wavelength $\lambda$. How does the emitted power depend on $\lambda$ ?

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2012

• Paper 1, Section II, B

  Electrodynamics | Part II, 2012

  A particle of mass $m$ and charge $q$ moves relativistically under the influence of a constant electric field $E$ in the positive $z$-direction, and a constant magnetic field $B$ also in the positive $z$-direction.

  In some inertial observer's coordinate system, the particle starts at

  $$x=R, \quad y=0, \quad z=0, \quad t=0$$

  with velocity given by

  $$\dot{x}=0, \quad \dot{y}=u, \quad \dot{z}=0,$$

  where the dot indicates differentiation with respect to the proper time of the particle. Show that the subsequent motion of the particle, as seen by the inertial observer, is a helix.

  a) What is the radius of the helix as seen by the inertial observer?

  b) What are the $x$ and $y$ coordinates of the axis of the helix?

  c) What is the $z$ coordinate of the particle after a proper time $\tau$ has elapsed, as measured by the particle?

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• Paper 3, Section II, B

  Electrodynamics | Part II, 2012

  The non-relativistic Larmor formula for the power, $P$, radiated by a particle of charge $q$ and mass $m$ that is being accelerated with an acceleration a is

  $$P=\frac{\mu_{0}}{6 \pi} q^{2}|\mathbf{a}|^{2} .$$

  Starting from the Liénard-Wiechert potentials, sketch a derivation of this result. Explain briefly why the relativistic generalization of this formula is

  $$P=\frac{\mu_{0}}{6 \pi} \frac{q^{2}}{m^{2}}\left(\frac{d p^{\mu}}{d \tau} \frac{d p^{\nu}}{d \tau} \eta_{\mu \nu}\right)$$

  where $p^{\mu}$ is the relativistic momentum of the particle and $\tau$ is the proper time along the worldline of the particle.

  A particle of mass $m$ and charge $q$ moves in a plane perpendicular to a constant magnetic field $B$. At time $t=0$ as seen by an observer $\mathbf{O}$ at rest, the particle has energy $E=\gamma m$. At what rate is electromagnetic energy radiated by this particle?

  At time $t$ according to the observer $\mathbf{O}$, the particle has energy $E^{\prime}=\gamma^{\prime} m$. Find an expression for $\gamma^{\prime}$ in terms of $\gamma$ and $t$.

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• Paper 4, Section II, B

  Electrodynamics | Part II, 2012

  The charge and current densities are given by $\rho(t, \mathbf{x}) \neq 0$ and $\mathbf{j}(t, \mathbf{x})$ respectively. The electromagnetic scalar and vector potentials are given by $\phi(t, \mathbf{x})$ and $\mathbf{A}(t, \mathbf{x})$ respectively. Explain how one can regard $j^{\mu}=(\rho, \mathbf{j})$ as a four-vector that obeys the current conservation rule $\partial_{\mu} j^{\mu}=0$.

  In the Lorenz gauge $\partial_{\mu} A^{\mu}=0$, derive the wave equation that relates $A^{\mu}=(\phi, \mathbf{A})$ to $j^{\mu}$ and hence show that it is consistent to treat $A^{\mu}$ as a four-vector.

  In the Lorenz gauge, with $j^{\mu}=0$, a plane wave solution for $A^{\mu}$ is given by

  $$A^{\mu}=\epsilon^{\mu} \exp \left(i k_{\nu} x^{\nu}\right)$$

  where $\epsilon^{\mu}, k^{\mu}$ and $x^{\mu}$ are four-vectors with

  $$\epsilon^{\mu}=\left(\epsilon^{0}, \boldsymbol{\epsilon}\right), \quad k^{\mu}=\left(k^{0}, \mathbf{k}\right), \quad x^{\mu}=(t, \mathbf{x}) .$$

  Show that $k_{\mu} k^{\mu}=k_{\mu} \epsilon^{\mu}=0$.

  Interpret the components of $k^{\mu}$ in terms of the frequency and wavelength of the wave.

  Find what residual gauge freedom there is and use it to show that it is possible to set $\epsilon^{0}=0$. What then is the physical meaning of the components of $\boldsymbol{\epsilon}$ ?

  An observer at rest in a frame $S$ measures the angular frequency of a plane wave travelling parallel to the $z$-axis to be $\omega$. A second observer travelling at velocity $v$ in $S$ parallel to the $z$-axis measures the radiation to have frequency $\omega^{\prime}$. Express $\omega^{\prime}$ in terms of $\omega$.

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2011

• Paper 1, Section II, C

  Electrodynamics | Part II, 2011

  In the Landau-Ginzburg model of superconductivity, the energy of the system is given, for constants $\alpha$ and $\beta$, by

  $$E=\int\left\{\frac{1}{2 \mu_{0}} \mathbf{B}^{2}+\frac{1}{2 m}\left[(i \hbar \nabla-q \mathbf{A}) \psi^{*} \cdot(-i \hbar \nabla-q \mathbf{A}) \psi\right]+\alpha \psi^{*} \psi+\beta\left(\psi^{*} \psi\right)^{2}\right\} d^{3} \mathbf{x},$$

  where $\mathbf{B}$ is the time-independent magnetic field derived from the vector potential $\mathbf{A}$, and $\psi$ is the wavefunction of the charge carriers, which have mass $m$ and charge $q$.

  Describe the physical meaning of each of the terms in the integral.

  Explain why in a superconductor one must choose $\alpha<0$ and $\beta>0$. Find an expression for the number density $n$ of the charge carriers in terms of $\alpha$ and $\beta$.

  Show that the energy is invariant under the gauge transformations

  $$\mathbf{A} \rightarrow \mathbf{A}+\nabla \Lambda, \quad \psi \rightarrow \psi e^{i q \Lambda / \hbar}$$

  Assuming that the number density $n$ is uniform, show that, if $E$ is a minimum under variations of $\mathbf{A}$, then

  $$\operatorname{curl} \mathbf{B}=-\frac{\mu_{0} q^{2} n}{m}\left(\mathbf{A}-\frac{\hbar}{q} \nabla \phi\right)$$

  where $\phi=\arg \psi$.

  Find a formula for $\nabla^{2} \mathbf{B}$ and use it to explain why there cannot be a magnetic field inside the bulk of a superconductor.

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• Paper 3, Section II, C

  Electrodynamics | Part II, 2011

  Explain how time-dependent distributions of electric charge $\rho(\mathbf{x}, t)$ and current $\mathbf{j}(\mathbf{x}, t)$ can be combined into a four-vector $j^{a}(x)$ that obeys $\partial_{a} j^{a}=0$.

  This current generates a four-vector potential $A^{a}(x)$. Explain how to find $A^{a}$ in the gauge $\partial_{a} A^{a}=0$.

  A small circular loop of wire of radius $r$ is centred at the origin. The unit vector normal to the plane of the loop is $\mathbf{n}$. A current $I_{o} \sin \omega t$ flows in the loop. Find the three-vector potential $\mathbf{A}(\mathbf{x}, t)$ to leading order in $r /|\mathbf{x}|$.

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• Paper 4, Section II, C

  Electrodynamics | Part II, 2011

  Suppose that there is a distribution of electric charge given by the charge density $\rho(\mathbf{x})$. Develop the multipole expansion, up to quadrupole terms, for the electrostatic potential $\phi$ and define the dipole and quadrupole moments of the charge distribution.

  A tetrahedron has a vertex at $(1,1,1)$ where there is a point charge of strength $3 q$. At each of the other vertices located at $(1,-1,-1),(-1,1,-1)$ and $(-1,-1,1)$ there is a point charge of strength $-q$.

  What is the dipole moment of this charge distribution?

  What is the quadrupole moment?

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2010

• Paper 1, Section II, B

  Electrodynamics | Part II, 2010

  The vector potential $A^{\mu}$ is determined by a current density distribution $j^{\mu}$ in the gauge $\partial_{\mu} A^{\mu}=0$ by

  $$\square A^{\mu}=-\mu_{0} j^{\mu}, \quad \square=-\frac{\partial^{2}}{\partial t^{2}}+\nabla^{2},$$

  in units where $c=1$.

  Describe how to justify the result

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

  A plane square loop of thin wire, edge lengths $l$, has its centre at the origin and lies in the $(x, y)$ plane. For $t<0$, no current is flowing in the loop, but at $t=0$ a constant current $I$ is turned on.

  Find the vector potential at the point $(0,0, z)$ as a function of time due to a single edge of the loop.

  What is the electric field due to the entire loop at $(0,0, z)$ as a function of time? Give a careful justification of your answer.

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• Paper 3, Section II, B

  Electrodynamics | Part II, 2010

  A particle of rest-mass $m$, electric charge $q$, is moving relativistically along the path $x^{\mu}(s)$ where $s$ parametrises the path.

  Write down an action for which the extremum determines the particle's equation of motion in an electromagnetic field given by the potential $A^{\mu}(x)$.

  Use your action to derive the particle's equation of motion in a form where $s$ is the proper time.

  Suppose that the electric and magnetic fields are given by

  $$\begin{aligned} \mathbf{E} &=(0,0, E) \\ \mathbf{B} &=(0, B, 0) \end{aligned}$$

  where $E$ and $B$ are constants and $B>E>0$.

  Find $x^{\mu}(s)$ given that the particle starts at rest at the origin when $s=0$.

  Describe qualitatively the motion of the particle.

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• Paper 4, Section II, B

  Electrodynamics | Part II, 2010

  In a superconductor the number density of charge carriers of charge $q$ is $n_{s}$. Suppose that there is a time-independent magnetic field described by the three-vector potential $\mathbf{A} .$

  Derive an expression for the superconducting current.

  Explain how your answer is gauge invariant.

  Suppose that for $z<0$ there is a constant magnetic field $\mathbf{B}_{0}$ in a vacuum and, for $z>0$, there is a uniform superconductor. Derive the magnetic field for $z>0$.

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2009

• Paper 1, Section II, C

  Electrodynamics | Part II, 2009

  The action for a modified version of electrodynamics is given by

  $$I=\int d^{4} x\left(-\frac{1}{4} F_{a b} F^{a b}-\frac{1}{2} m^{2} A_{a} A^{a}+\mu_{0} J^{a} A_{a}\right)$$

  where $m$ is an arbitrary constant, $F_{a b}=\partial_{a} A_{b}-\partial_{b} A_{a}$ and $J^{a}$ is a conserved current.

  (i) By varying $A_{a}$, derive the field equations analogous to Maxwell's equations by demanding that $\delta I=0$ for an arbitrary variation $\delta A_{a}$.

  (ii) Show that $\partial_{a} A^{a}=0$.

  (iii) Suppose that the current $J^{a}(x)$ is a function of position only. Show that

  $$A^{a}(\mathbf{x})=\frac{\mu_{0}}{4 \pi} \int d^{3} x^{\prime} \frac{J^{a}\left(\mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} e^{-m\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}$$

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• Paper 3, Section II, C

  Electrodynamics | Part II, 2009

  A particle of charge of $q$ moves along a trajectory $y^{a}(s)$ in spacetime where $s$ is the proper time on the particle's world-line.

  Explain briefly why, in the gauge $\partial_{a} A^{a}=0$, the potential at the spacetime point $x$ is given by

  $$A^{a}(x)=\frac{\mu_{0} q}{2 \pi} \int d s \frac{d y^{a}}{d s} \theta\left(x^{0}-y^{0}(s)\right) \delta\left(\left(x^{c}-y^{c}(s)\right)\left(x^{d}-y^{d}(s)\right) \eta_{c d}\right)$$

  Evaluate this integral for a point charge moving relativistically along the $z$-axis, $x=y=0$, at constant velocity $v$ so that $z=v t .$

  Check your result by starting from the potential of a point charge at rest

  $$\begin{aligned} \mathbf{A} &=0 \\ \phi &=\frac{\mu_{0} q}{4 \pi\left(x^{2}+y^{2}+z^{2}\right)^{1 / 2}} \end{aligned}$$

  and making an appropriate Lorentz transformation.

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• Paper 4, Section II, C

  Electrodynamics | Part II, 2009

  In a superconductor, the charge carriers have a charge $q$, mass $m$ and number density $n$. Describe how to construct the superconducting current in terms of the vector potential A and the wavefunction of the charge carriers.

  Show that the current is gauge invariant.

  Derive the Helmholtz equation

  $$\nabla^{2} \mathbf{B}=\mathbf{B} / \ell^{2}$$

  for a time-independent magnetic field and evaluate the length scale $\ell$ in terms of $n, q$ and $m$.

  Why does this imply that magnetic flux cannot exist in a superconductor?

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2008

• $3 . \mathrm{II} . 35 \mathrm{D} \quad$

  Electrodynamics | Part II, 2008

  The retarded scalar potential $\varphi(t, \mathbf{x})$ produced by a charge distribution $\rho(t, \mathbf{x})$ is given by

  $$\varphi(t, \mathbf{x})=\frac{1}{4 \pi \epsilon_{0}} \int_{\Omega} d^{3} x^{\prime} \frac{\rho\left(t-\left|\mathbf{x}-\mathbf{x}^{\prime}\right|, \mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}$$

  where $\Omega$ denotes all 3 -space. Describe briefly and qualitatively the physics underlying this formula.

  Write the integrand in the formula above as a 1-dimensional integral over a new time coordinate $\tau$. Next consider a special source, a point charge $q$ moving along a trajectory $\mathbf{x}=\mathbf{x}_{0}(t)$ so that

  $$\rho(t, \mathbf{x})=q \delta^{(3)}\left(\mathbf{x}-\mathbf{x}_{0}(t)\right),$$

  where $\delta^{(3)}(\mathbf{x})$ denotes the 3 -dimensional delta function. By reversing the order of integration, or otherwise, obtain the Liénard-Wiechert potential

  $$\varphi(t, \mathbf{x})=\frac{1}{4 \pi \epsilon_{0}} \frac{q}{R-\mathbf{v} \cdot \mathbf{R}},$$

  where $\mathbf{v}$ and $\mathbf{R}$ are to be determined.

  Write down the corresponding formula for the vector potential $\mathbf{A}(t, \mathbf{x})$.

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

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• 1.II $. 34 \mathrm{D} \quad$

  Electrodynamics | Part II, 2008

  Frame $\mathcal{S}^{\prime}$ is moving with uniform speed $v$ in the $x$-direction relative to a laboratory frame $\mathcal{S}$. The components of the electric and magnetic fields $\mathbf{E}$ and $\mathbf{B}$ in the two frames are related by the Lorentz transformation

  where $\gamma=1 / \sqrt{1-v^{2}}$ and units are chosen so that $c=1$. How do the components of the spatial vector $\mathbf{F}=\mathbf{E}+i \mathbf{B}$ (where $i=\sqrt{-1}$ ) transform?

  Show that $\mathbf{F}^{\prime}$ is obtained from $\mathbf{F}$ by a rotation through $\theta$ about a spatial axis $\mathbf{n}$, where $\mathbf{n}$ and $\theta$ should be determined. Hence, or otherwise, show that there are precisely two independent scalars associated with $\mathbf{F}$ which are preserved by the Lorentz transformation, and obtain them.

  [Hint: since $|v|<1$ there exists a unique real $\psi$ such that $v=\tanh \psi$.]

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• 4.II $. 35 \mathrm{D} \quad$

  Electrodynamics | Part II, 2008

  The Maxwell field tensor is given by

  $$F^{a b}=\left(\begin{array}{cccc} 0 & -E_{x} & -E_{y} & -E_{z} \\ E_{x} & 0 & -B_{z} & B_{y} \\ E_{y} & B_{z} & 0 & -B_{x} \\ E_{z} & -B_{y} & B_{x} & 0 \end{array}\right)$$

  A general 4-velocity is written as $U^{a}=\gamma(1, \mathbf{v})$, where $\gamma=\left(1-|\mathbf{v}|^{2}\right)^{-1 / 2}$, and $c=1$. A general 4-current density is written as $J^{a}=(\rho, \mathbf{j})$, where $\rho$ is the charge density and $\mathbf{j}$ is the 3 -current density. Show that

  $$F^{a b} U_{b}=\gamma(\mathbf{E} \cdot \mathbf{v}, \mathbf{E}+\mathbf{v} \times \mathbf{B})$$

  In the rest frame of a conducting medium, Ohm's law states that $\mathbf{j}=\sigma \mathbf{E}$ where $\sigma$ is the conductivity. Show that the relativistic generalization to a frame in which the medium moves with uniform velocity $\mathbf{v}$ is

  $$J^{a}-\left(J^{b} U_{b}\right) U^{a}=\sigma F^{a b} U_{b}$$

  Show that this implies

  $$\mathbf{j}=\rho \mathbf{v}+\sigma \gamma(\mathbf{E}+\mathbf{v} \times \mathbf{B}-(\mathbf{v} \cdot \mathbf{E}) \mathbf{v})$$

  Simplify this formula, given that the charge density vanishes in the rest frame of the medium

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2007

• 1.II.34E

  Electrodynamics | Part II, 2007

  Frame $\mathcal{S}^{\prime}$ is moving with uniform speed $v$ in the $z$-direction relative to a laboratory frame $\mathcal{S}$. Using Cartesian coordinates and units such that $c=1$, the relevant Lorentz transformation is

  $$t^{\prime}=\gamma(t-v z), \quad x^{\prime}=x, \quad y^{\prime}=y, \quad z^{\prime}=\gamma(z-v t)$$

  where $\gamma=1 / \sqrt{1-v^{2}}$. A straight thin wire of infinite extent lies along the $z$-axis and carries charge and current line densities $\sigma$ and $J$ per unit length, as measured in $\mathcal{S}$. Stating carefully your assumptions show that the corresponding quantities in $\mathcal{S}^{\prime}$ are given by

  $$\sigma^{\prime}=\gamma(\sigma-v J), \quad J^{\prime}=\gamma(J-v \sigma)$$

  Using cylindrical polar coordinates, and the integral forms of the Maxwell equations $\nabla \cdot \mathbf{E}=\mu_{0} \rho$ and $\nabla \times \mathbf{B}=\mu_{0} \mathbf{j}$, derive the electric and magnetic fields outside the wire in both frames.

  In a standard notation the Lorentz transformation for the electric and magnetic fields is

  $$\mathbf{E}_{\|}^{\prime}=\mathbf{E}_{\|}, \quad \mathbf{B}_{\|}^{\prime}=\mathbf{B}_{\|}, \quad \quad \mathbf{E}_{\perp}^{\prime}=\gamma\left(\mathbf{E}_{\perp}+\mathbf{v} \times \mathbf{B}_{\perp}\right), \quad \mathbf{B}_{\perp}^{\prime}=\gamma\left(\mathbf{B}_{\perp}-\mathbf{v} \times \mathbf{E}_{\perp}\right)$$

  Is your result consistent with this?

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• 3.II $. 35 \mathrm{E} \quad$

  Electrodynamics | Part II, 2007

  Consider a particle of charge $q$ moving with 3 -velocity $\mathbf{v}$. If the particle is moving slowly then Larmor's formula asserts that the instantaneous radiated power is

  $$\mathcal{P}=\frac{\mu_{0}}{6 \pi} q^{2}\left|\frac{d \mathbf{v}}{d t}\right|^{2}$$

  Suppose, however, that the particle is moving relativistically. Give reasons why one should conclude that $\mathcal{P}$ is a Lorentz invariant. Writing the 4-velocity as $U^{a}=(\gamma, \gamma \mathbf{v})$ where $\gamma=1 / \sqrt{1-|\mathbf{v}|^{2}}$ and $c=1$, show that

  $$\dot{U}^{a}=\left(\gamma^{3} \alpha, \gamma^{3} \alpha \mathbf{v}+\gamma \dot{\mathbf{v}}\right)$$

  where $\alpha=\mathbf{v} \cdot \dot{\mathbf{v}}$ and $\dot{f}=d f / d s$ where $s$ is the particle's proper time. Show also that

  $$\dot{U}^{a} \dot{U}_{a}=-\gamma^{4} \alpha^{2}-\gamma^{2}|\dot{\mathbf{v}}|^{2} .$$

  Deduce the relativistic version of Larmor's formula.

  Suppose the particle moves in a circular orbit perpendicular to a uniform magnetic field $\mathbf{B}$. Show that

  $$\mathcal{P}=\frac{\mu_{0}}{6 \pi} \frac{q^{4}}{m^{2}}\left(\gamma^{2}-1\right)|\mathbf{B}|^{2}$$

  where $m$ is the mass of the particle, and comment briefly on the slow motion limit.

  comment

• 4.II.35E

  Electrodynamics | Part II, 2007

  An action

  $$S[\varphi]=\int d^{4} x L(\varphi, \varphi, a)$$

  is given, where $\varphi(x)$ is a scalar field. Explain heuristically how to compute the functional derivative $\delta S / \delta \varphi$.

  Consider the action for electromagnetism,

  $$S\left[A_{a}\right]=-\int d^{4} x\left\{\frac{1}{4 \mu_{0}} F^{a b} F_{a b}+J^{a} A_{a}\right\} .$$

  Here $J^{a}$ is the 4-current density, $A_{a}$ is the 4-potential and $F_{a b}=A_{b, a}-A_{a, b}$ is the Maxwell field tensor. Obtain Maxwell's equations in 4-vector form.

  Another action that is sometimes suggested is

  $$\widehat{S}\left[A_{a}\right]=-\int d^{4} x\left\{\frac{1}{2 \mu_{0}} A^{a, b} A_{a, b}+J^{a} A_{a}\right\} .$$

  Under which additional assumption can Maxwell's equations be obtained using this action?

  Using this additional assumption establish the relationship between the actions $S$ and $\widehat{S}$.

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2006

• $3 . \mathrm{II} . 35 \mathrm{E} \quad$

  Electrodynamics | Part II, 2006

  A particle of rest mass $m$ and charge $q$ is moving along a trajectory $x^{a}(s)$, where $s$ is the particle's proper time, in a given external electromagnetic field with 4-potential $A^{a}\left(x^{c}\right)$. Consider the action principle $\delta S=0$ where the action is $S=\int L d s$ and

  $$L\left(s, x^{a}, \dot{x}^{a}\right)=-m \sqrt{\eta_{a b} \dot{x}^{a} \dot{x}^{b}}-q A_{a}\left(x^{c}\right) \dot{x}^{a},$$

  and variations are taken with fixed endpoints.

  Show first that the action is invariant both under reparametrizations $s \rightarrow \alpha s+\beta$ where $\alpha$ and $\beta$ are constants and also under a change of electromagnetic gauge. Next define the generalized momentum $P_{a}=\partial L / \partial \dot{x}^{a}$, and obtain the equation of motion

  $$m \ddot{x}^{a}=q F_{b}^{a} \dot{x}^{b},$$

  where the tensor $F^{a}{ }_{b}$ should be defined and you may assume that $d / d s\left(\eta_{a b} \dot{x}^{a} \dot{x}^{b}\right)=0$. Then verify from $(*)$ that indeed $d / d s\left(\eta_{a b} \dot{x}^{a} \dot{x}^{b}\right)=0$.

  How does $P_{a}$ differ from the momentum $p_{a}$ of an uncharged particle? Comment briefly on the principle of minimal coupling.

  comment

• 1.II $. 34 \mathrm{E} \quad$

  Electrodynamics | Part II, 2006

  $\mathcal{S}$ and $\mathcal{S}^{\prime}$ are two reference frames with $\mathcal{S}^{\prime}$ moving with constant speed $v$ in the $x$-direction relative to $\mathcal{S}$. The co-ordinates $x^{a}$ and ${x^{\prime}}^{a}$ are related by $d x^{\prime a}=L^{a}{ }_{b} d x^{b}$ where

  $$L_{b}^{a}=\left(\begin{array}{cccc} \gamma & -\gamma v & 0 & 0 \\ -\gamma v & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$$

  and $\gamma=\left(1-v^{2}\right)^{-1 / 2}$. What is the transformation rule for the scalar potential $\varphi$ and vector potential A between the two frames?

  As seen in $\mathcal{S}^{\prime}$ there is an infinite uniform stationary distribution of charge along the $x$-axis with uniform line density $\sigma$. Determine the electric and magnetic fields $\mathbf{E}$ and B both in $\mathcal{S}^{\prime}$ and $\mathcal{S}$. Check your answer by verifying explicitly the invariance of the two quadratic Lorentz invariants.

  Comment briefly on the limit $|v| \ll 1$.

  comment

• 4.II $. 35 \mathrm{E} \quad$

  Electrodynamics | Part II, 2006

  The retarded scalar potential produced by a charge distribution $\rho\left(t^{\prime}, \mathbf{x}^{\prime}\right)$ is

  $$\varphi(t, \mathbf{x})=\frac{1}{4 \pi \epsilon_{0}} \int d^{3} x^{\prime} \frac{\rho\left(t-R, \mathbf{x}^{\prime}\right)}{R},$$

  where $R=|\mathbf{R}|$ and $\mathbf{R}=\mathbf{x}-\mathbf{x}^{\prime}$. By use of an appropriate delta function rewrite the integral as an integral over both $d^{3} x^{\prime}$ and $d t^{\prime}$ involving $\rho\left(t^{\prime}, \mathbf{x}^{\prime}\right)$.

  Now specialize to a point charge $q$ moving on a path $\mathbf{x}^{\prime}=\mathbf{x}_{0}\left(t^{\prime}\right)$ so that we may set

  $$\rho\left(t^{\prime}, \mathbf{x}^{\prime}\right)=q \delta^{(3)}\left(\mathbf{x}^{\prime}-\mathbf{x}_{0}\left(t^{\prime}\right)\right) .$$

  By performing the volume integral first obtain the Liénard-Wiechert potential

  $$\varphi(t, \mathbf{x})=\frac{q}{4 \pi \epsilon_{0}} \frac{1}{\left(R^{*}-\mathbf{v} \cdot \mathbf{R}^{*}\right)},$$

  where $\mathbf{R}^{*}$ and $\mathbf{v}$ should be specified.

  Obtain the corresponding result for the magnetic potential.

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2005

• 1.II $. 34 \mathrm{~B} \quad$

  Electrodynamics | Part II, 2005

  In a frame $\mathcal{F}$ the electromagnetic fields $(\mathbf{E}, \mathbf{B})$ are encoded into the Maxwell field 4-tensor $F^{a b}$ and its dual ${ }^{*} F^{a b}$, where

  $$F^{a b}=\left(\begin{array}{cccc} 0 & -E_{x} & -E_{y} & -E_{z} \\ E_{x} & 0 & -B_{z} & B_{y} \\ E_{y} & B_{z} & 0 & -B_{x} \\ E_{z} & -B_{y} & B_{x} & 0 \end{array}\right)$$

  and

  $${ }^{*} F^{a b}=\left(\begin{array}{cccc} 0 & -B_{x} & -B_{y} & -B_{z} \\ B_{x} & 0 & E_{z} & -E_{y} \\ B_{y} & -E_{z} & 0 & E_{x} \\ B_{z} & E_{y} & -E_{x} & 0 \end{array}\right)$$

  [Here the signature is $(+---)$ and units are chosen so that $c=1$.] Obtain two independent Lorentz scalars of the electromagnetic field in terms of $\mathbf{E}$ and $\mathbf{B}$.

  Suppose that $\mathbf{E} \cdot \mathbf{B}>0$ in the frame $\mathcal{F}$. Given that there exists a frame $\mathcal{F}^{\prime}$ in which $\mathbf{E}^{\prime} \times \mathbf{B}^{\prime}=\mathbf{0}$, show that

  $$E^{\prime}=\left[(\mathbf{E} \cdot \mathbf{B})\left(K+\sqrt{1+K^{2}}\right)\right]^{1 / 2}, \quad B^{\prime}=\left[\frac{\mathbf{E} \cdot \mathbf{B}}{K+\sqrt{1+K^{2}}}\right]^{1 / 2},$$

  where $\left(E^{\prime}, B^{\prime}\right)$ are the magnitudes of $\left(\mathbf{E}^{\prime}, \mathbf{B}^{\prime}\right)$, and

  $$K=\frac{1}{2}\left(|\mathbf{E}|^{2}-|\mathbf{B}|^{2}\right) /(\mathbf{E} \cdot \mathbf{B})$$

  [Hint: there is no need to consider the Lorentz transformations for $\mathbf{E}^{\prime}$ and $\mathbf{B}^{\prime}$.]

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• 3.II $. 35 \mathrm{~B} \quad$

  Electrodynamics | Part II, 2005

  A non-relativistic particle of rest mass $m$ and charge $q$ is moving slowly with velocity $\mathbf{v}(t)$. The power $d P / d \Omega$ radiated per unit solid angle in the direction of a unit vector $\mathbf{n}$ is

  $$\frac{d P}{d \Omega}=\frac{\mu_{0}}{16 \pi^{2}}|\mathbf{n} \times q \dot{\mathbf{v}}|^{2} .$$

  Obtain Larmor's formula

  $$P=\frac{\mu_{0} q^{2}}{6 \pi}|\dot{\mathbf{v}}|^{2} .$$

  The particle has energy $\mathcal{E}$ and, starting from afar, makes a head-on collision with a fixed central force described by a potential $V(r)$, where $V(r)>\mathcal{E}$ for $r<r_{0}$ and $V(r)<\mathcal{E}$ for $r>r_{0}$. Let $W$ be the total energy radiated by the particle. Given that $W \ll \mathcal{E}$, show that

  $$W \approx \frac{\mu_{0} q^{2}}{3 \pi m^{2}} \sqrt{\frac{m}{2}} \int_{r_{0}}^{\infty}\left(\frac{d V}{d r}\right)^{2} \frac{d r}{\sqrt{V\left(r_{0}\right)-V(r)}}$$

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• 4.II $. 35 \mathrm{~B} \quad$

  Electrodynamics | Part II, 2005

  In Ginzburg-Landau theory, superconductivity is due to "supercarriers" of mass $m_{s}$ and charge $q_{s}$, which are described by a macroscopic wavefunction $\psi$ with "Mexican hat" potential

  $$V=\alpha(T)|\psi|^{2}+\frac{1}{2} \beta|\psi|^{4}$$

  Here, $\beta>0$ is constant and $\alpha(T)$ is a function of temperature $T$ such that $\alpha(T)>0$ for $T>T_{c}$ but $\alpha(T)<0$ for $T<T_{c}$, where $T_{c}$ is a critical temperature. In the presence of a magnetic field $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$, the total energy of the superconducting system is

  $$E\left[\psi, \psi^{*}, \mathbf{A}\right]=\int d^{3} x\left[\frac{1}{2 \mu_{0}} A_{k, j}\left(A_{k, j}-A_{j, k}\right)+\frac{\hbar^{2}}{2 m_{s}}\left|\psi_{, k}+i \frac{q_{s}}{\hbar} A_{k} \psi\right|^{2}+V\right]$$

  Use this to derive the equations

  $$-\frac{\hbar^{2}}{2 m_{s}}\left(\nabla-i \frac{q_{s}}{\hbar} \mathbf{A}\right)^{2} \psi+\left(\alpha+\beta|\psi|^{2}\right) \psi=0$$

  and

  $$\boldsymbol{\nabla} \times \mathbf{B} \equiv \boldsymbol{\nabla}(\boldsymbol{\nabla} \cdot \mathbf{A})-\nabla^{2} \mathbf{A}=\mu_{0} \mathbf{j}$$

  where

  $$\begin{aligned} \mathbf{j} &=-\frac{i q_{s} \hbar}{2 m_{s}}\left(\psi^{*} \boldsymbol{\nabla} \psi-\psi \boldsymbol{\nabla} \psi^{*}\right)-\frac{q_{s}^{2}}{m_{s}}|\psi|^{2} \mathbf{A} \\ &=\frac{q_{s}}{2 m_{s}}\left[\psi^{*}\left(-i \hbar \boldsymbol{\nabla}-q_{s} \mathbf{A}\right) \psi+\psi\left(i \hbar \boldsymbol{\nabla}-q_{s} \mathbf{A}\right) \psi^{*}\right] \end{aligned}$$

  Suppose that we write the wavefunction as

  $$\psi=\sqrt{n_{s}} e^{i \theta}$$

  where $n_{s}$ is the (real positive) supercarrier density and $\theta$ is a real phase function. Given that

  $$\left(\boldsymbol{\nabla}-\frac{i q_{s}}{\hbar} \mathbf{A}\right) \psi=0$$

  show that $n_{s}$ is constant and that $\hbar \boldsymbol{\nabla} \theta=q_{s} \mathbf{A}$. Given also that $T<T_{c}$, deduce that (*) allows such solutions for a certain choice of $n_{s}$, which should be determined. Verify that your results imply $\mathbf{j}=\mathbf{0}$. Show also that $\mathbf{B}=\mathbf{0}$ and hence that ( $\dagger$ ) is solved.

  Let $\mathcal{S}$ be a surface within the superconductor with closed boundary $\mathcal{C}$. Show that the magnetic flux through $\mathcal{S}$ is

  $$\Phi \equiv \int_{\mathcal{S}} \mathbf{B} \cdot \mathbf{d} \mathbf{S}=\frac{\hbar}{q_{s}}[\theta]_{\mathcal{C}}$$

  Discuss, briefly, flux quantization.

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2004

• B1.21

  Electrodynamics | Part II, 2004

  The Maxwell field tensor is

  $$F^{a b}=\left(\begin{array}{cccc} 0 & -E_{x} & -E_{y} & -E_{z} \\ E_{x} & 0 & -B_{z} & B_{y} \\ E_{y} & B_{z} & 0 & -B_{x} \\ E_{z} & -B_{y} & B_{x} & 0 \end{array}\right)$$

  and the 4-current density is $J^{a}=(\rho, \mathbf{j})$. Write down the 3-vector form of Maxwell's equations and the continuity equation, and obtain the equivalent 4-vector equations.

  Consider a Lorentz transformation from a frame $\mathcal{F}$ to a frame $\mathcal{F}^{\prime}$ moving with relative (coordinate) velocity $v$ in the $x$-direction

  $$L_{b}^{a}=\left(\begin{array}{cccc} \gamma & \gamma v & 0 & 0 \\ \gamma v & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$$

  where $\gamma=1 / \sqrt{1-v^{2}}$. Obtain the transformation laws for $\mathbf{E}$ and $\mathbf{B}$. Which quantities, quadratic in $\mathbf{E}$ and $\mathbf{B}$, are Lorentz scalars?

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• B2.21

  Electrodynamics | Part II, 2004

  A particle of rest mass $m$ and charge $q$ moves along a path $x^{a}(s)$, where $s$ is the particle's proper time. The equation of motion is

  $$m \ddot{x}^{a}=q F^{a b} \eta_{b c} \dot{x}^{c}$$

  where $\dot{x}^{a}=d x^{a} / d s$ etc., $F^{a b}$ is the Maxwell field tensor $\left(F^{01}=-E_{x}, F^{23}=-B_{x}\right.$, where $E_{x}$ and $B_{x}$ are the $x$-components of the electric and magnetic fields) and $\eta_{b c}$ is the Minkowski metric tensor. Show that $\dot{x}_{a} \ddot{x}^{a}=0$ and interpret both the equation of motion and this equation in the classical limit.

  The electromagnetic field is given in cartesian coordinates by $\mathbf{E}=(0, E, 0)$ and $\mathbf{B}=(0,0, E)$, where $E$ is constant and uniform. The particle starts from rest at the origin. Show that the orbit is given by

  $$9 x^{2}=2 \alpha y^{3}, \quad z=0$$

  where $\alpha=q E / m$.

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• B4.21

  Electrodynamics | Part II, 2004

  Using Lorentz gauge, $A_{, a}^{a}=0$, Maxwell's equations for a current distribution $J^{a}$ can be reduced to $\square A^{a}(x)=\mu_{0} J^{a}(x)$. The retarded solution is

  $$A^{a}(x)=\frac{\mu_{0}}{2 \pi} \int d^{4} y \theta\left(z^{0}\right) \delta\left(z_{c} z^{c}\right) J^{a}(y)$$

  where $z^{a}=x^{a}-y^{a}$. Explain, heuristically, the rôle of the $\delta$-function and Heaviside step function $\theta$ in this formula.

  The current distribution is produced by a point particle of charge $q$ moving on a world line $r^{a}(s)$, where $s$ is the particle's proper time, so that

  $$J^{a}(y)=q \int d s V^{a}(s) \delta^{(4)}(y-r(s))$$

  where $V^{a}=\dot{r}^{a}(s)=d r^{a} / d s$. Show that

  $$A^{a}(x)=\frac{\mu_{0} q}{2 \pi} \int d s \theta\left(X^{0}\right) \delta\left(X_{c} X^{c}\right) V^{a}(s),$$

  where $X^{a}=x^{a}-r^{a}(s)$, and further that, setting $\alpha=X_{c} V^{c}$,

  $$A^{a}(x)=\frac{\mu_{0} q}{4 \pi}\left[\frac{V^{a}}{\alpha}\right]_{s=s^{*}}$$

  where $s^{*}$ should be defined. Verify that

  $$s^{*}, a=\left[\frac{X_{a}}{\alpha}\right]_{s=s^{*}} .$$

  Evaluating quantities at $s=s^{*}$ show that

  $$\left[\frac{V^{a}}{\alpha}\right]_{, b}=\frac{1}{\alpha^{2}}\left[-V^{a} V_{b}+S^{a} X_{b}\right]$$

  where $S^{a}=\dot{V}^{a}+V^{a}\left(1-X_{c} \dot{V}^{c}\right) / \alpha$. Hence verify that $A^{a}{ }_{, a}(x)=0$ and

  $$F_{a b}=\frac{\mu_{0} q}{4 \pi \alpha^{2}}\left(S_{a} X_{b}-S_{b} X_{a}\right) .$$

  Verify this formula for a stationary point charge at the origin.

  [Hint: If $f(s)$ has simple zeros at $s_{i}, i=1,2, \ldots$ then

  $$\delta(f(s))=\sum_{i} \frac{\delta\left(s_{i}\right)}{\left|f^{\prime}\left(s_{i}\right)\right|}$$

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2003

• B1.21

  Electrodynamics | Part II, 2003

  A particle of charge $q$ and mass $m$ moves non-relativistically with 4 -velocity $u^{a}(t)$ along a trajectory $x^{a}(t)$. Its electromagnetic field is determined by the Liénard-Wiechert potential

  $$A^{a}\left(\mathbf{x}^{\prime}, t^{\prime}\right)=\frac{q}{4 \pi \epsilon_{0}} \frac{u^{a}(t)}{u_{b}(t)\left(x^{\prime}-x(t)\right)^{b}}$$

  where $t^{\prime}=t+\left|\mathbf{x}-\mathbf{x}^{\prime}\right|$ and $\mathbf{x}$ denotes the spatial part of the 4 -vector $x^{a}$.

  Derive a formula for the Poynting vector at very large distances from the particle. Hence deduce Larmor's formula for the rate of loss of energy due to electromagnetic radiation by the particle.

  A particle moves in the $(x, y)$ plane in a constant magnetic field $\mathbf{B}=(0,0, B)$. Initially it has kinetic energy $E_{0}$; derive a formula for the kinetic energy of this particle as a function of time.

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• B2.20

  Electrodynamics | Part II, 2003

  A plane electromagnetic wave of frequency $\omega$ and wavevector $\mathbf{k}$ has an electromagnetic potential given by

  $$A^{a}=A \epsilon^{a} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}$$

  where $A$ is the amplitude of the wave and $\epsilon^{a}$ is the polarization vector. Explain carefully why there are two independent polarization states for such a wave, and why $|\mathbf{k}|^{2}=\omega^{2}$.

  A wave travels in the positive $z$-direction with polarization vector $\epsilon^{a}=(0,1, i, 0)$. It is incident at $z=0$ on a plane surface which conducts perfectly in the $x$-direction, but not at all in the $y$-direction. Find an expression for the electromagnetic potential of the radiation that is reflected from this surface.

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• B4.21

  Electrodynamics | Part II, 2003

  Describe the physical meaning of the various components of the stress-energy tensor $T^{a b}$ of the electromagnetic field.

  Suppose that one is given an electric field $\mathbf{E}(\mathbf{x})$ and a magnetic field $\mathbf{B}(\mathbf{x})$. Show that the angular momentum about the origin of these fields is

  $$\mathbf{J}=\frac{1}{\mu_{0}} \int \mathbf{x} \times(\mathbf{E} \times \mathbf{B}) d^{3} \mathbf{x}$$

  where the integral is taken over all space.

  A point electric charge $Q$ is at the origin, and has electric field

  $$\mathbf{E}=\frac{Q}{4 \pi \epsilon_{0}} \frac{\mathbf{x}}{|\mathbf{x}|^{3}}$$

  A point magnetic monopole of strength $P$ is at $\mathbf{y}$ and has magnetic field

  $$\mathbf{B}=\frac{\mu_{0} P}{4 \pi} \frac{\mathbf{x}-\mathbf{y}}{|\mathbf{x}-\mathbf{y}|^{3}}$$

  Find the component, along the axis between the electric charge and the magnetic monopole, of the angular momentum of the electromagnetic field about the origin.

  [Hint: You may find it helpful to express both $\mathbf{E}$ and $\mathbf{B}$ as gradients of scalar potentials.]

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2002

• B1.21

  Electrodynamics | Part II, 2002

  Explain how one can write Maxwell's equations in relativistic form by introducing an antisymmetric field strength tensor $F_{a b}$.

  In an inertial frame $S$, the electric and magnetic fields are $\mathbf{E}$ and $\mathbf{B}$. Suppose that there is a second inertial frame $S^{\prime}$ moving with velocity $v$ along the $x$-axis relative to $S$. Derive the rules for finding the electric and magnetic fields $\mathbf{E}^{\prime}$ and $\mathbf{B}^{\prime}$ in the frame $S^{\prime}$. Show that $|\mathbf{E}|^{2}-|\mathbf{B}|^{2}$ and $\mathbf{E} \cdot \mathbf{B}$ are invariant under Lorentz transformations.

  Suppose that $\mathbf{E}=E_{0}(0,1,0)$ and $\mathbf{B}=E_{0}(0, \cos \theta, \sin \theta)$, where $0 \leq \theta<\pi / 2$. At what velocity must an observer be moving in the frame $S$ for the electric and magnetic fields to appear to be parallel?

  Comment on the case $\theta=\pi / 2$.

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• B2.20

  Electrodynamics | Part II, 2002

  A particle of rest mass $m$ and charge $q$ moves in an electromagnetic field given by a potential $A_{a}$ along a trajectory $x^{a}(\tau)$, where $\tau$ is the proper time along the particle's worldline. The action for such a particle is

  $$I=\int\left(m \sqrt{-\eta_{a b} \dot{x}^{a} \dot{x}^{b}}-q A_{a} \dot{x}^{a}\right) d \tau .$$

  Show that the Euler-Lagrange equations resulting from this action reproduce the relativistic equation of motion for the particle.

  Suppose that the particle is moving in the electrostatic field of a fixed point charge $Q$ with radial electric field $E_{r}$ given by

  $$E_{r}=\frac{Q}{4 \pi \epsilon_{0} r^{2}} .$$

  Show that one can choose a gauge such that $A_{i}=0$ and only $A_{0} \neq 0$. Find $A_{0}$.

  Assume that the particle executes planar motion, which in spherical polar coordinates $(r, \theta, \phi)$ can be taken to be in the plane $\theta=\pi / 2$. Derive the equations of motion for $t$ and $\phi$.

  By using the fact that $\eta_{a b} \dot{x}^{a} \dot{x}^{b}=-1$, find the equation of motion for $r$, and hence show that the shape of the orbit is described by

  $$\frac{d r}{d \phi}=\pm \frac{r^{2}}{\ell} \sqrt{\left(E+\frac{\gamma}{r}\right)^{2}-1-\frac{\ell^{2}}{r^{2}}}$$

  where $E(>1)$ and $\ell$ are constants of integration and $\gamma$ is to be determined.

  By putting $u=1 / r$ or otherwise, show that if $\gamma^{2}<\ell^{2}$ then the orbits are bounded and generally not closed, and show that the angle between successive minimal values of $r$ is $2 \pi\left(1-\gamma^{2} / \ell^{2}\right)^{-1 / 2}$.

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• B4.21

  Electrodynamics | Part II, 2002

  Derive Larmor's formula for the rate at which radiation is produced by a particle of charge $q$ moving along a trajectory $\mathbf{x}(t)$.

  A non-relativistic particle of mass $m$, charge $q$ and energy $E$ is incident along a radial line in a central potential $V(r)$. The potential is vanishingly small for $r$ very large, but increases without bound as $r \rightarrow 0$. Show that the total amount of energy $\mathcal{E}$ radiated by the particle is

  $$\mathcal{E}=\frac{\mu_{0} q^{2}}{3 \pi m^{2}} \sqrt{\frac{m}{2}} \int_{r_{0}}^{\infty} \frac{1}{\sqrt{E-V(r)}}\left(\frac{d V}{d r}\right)^{2} d r$$

  where $V\left(r_{0}\right)=E$.

  Suppose that $V$ is the Coulomb potential $V(r)=A / r$. Evaluate $\mathcal{E}$.

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2001

• B1.21

  Electrodynamics | Part II, 2001

  Explain the multipole expansion in electrostatics, and devise formulae for the total charge, dipole moments and quadrupole moments given by a static charge distribution $\rho(\mathbf{r})$.

  A nucleus is modelled as a uniform distribution of charge inside the ellipsoid

  $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}+\frac{z^{2}}{c^{2}}=1$$

  The total charge of the nucleus is $Q$. What are the dipole moments and quadrupole moments of this distribution?

  Describe qualitatively what happens if the nucleus starts to oscillate.

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• B2.20

  Electrodynamics | Part II, 2001

  In a superconductor, there are superconducting charge carriers with number density $n$, mass $m$ and charge $q$. Starting from the quantum mechanical wavefunction $\Psi=R e^{i \Phi}$ (with real $R$ and $\Phi$ ), construct a formula for the electric current and explain carefully why your result is gauge invariant.

  Now show that inside a superconductor a static magnetic field obeys the equation

  $$\nabla^{2} \mathbf{B}=\frac{\mu_{0} n q^{2}}{m} \mathbf{B}$$

  A superconductor occupies the region $z>0$, while for $z<0$ there is a vacuum with a constant magnetic field in the $x$ direction. Show that the magnetic field cannot penetrate deep into the superconductor.

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• B4.21

  Electrodynamics | Part II, 2001

  The Liénard-Wiechert potential for a particle of charge $q$, assumed to be moving non-relativistically along the trajectory $y^{\mu}(\tau), \tau$ being the proper time along the trajectory,

  $$A^{\mu}(x, t)=\left.\frac{\mu_{0} q}{4 \pi} \frac{d y^{\mu} / d \tau}{(x-y(\tau))_{\nu} d y^{\nu} / d \tau}\right|_{\tau=\tau_{0}} .$$

  Explain how to calculate $\tau_{0}$ given $x^{\mu}=(x, t)$ and $y^{\mu}=\left(y, t^{\prime}\right)$.

  Derive Larmor's formula for the rate at which electromagnetic energy is radiated from a particle of charge $q$ undergoing an acceleration $a$.

  Suppose that one considers the classical non-relativistic hydrogen atom with an electron of mass $m$ and charge $-e$ orbiting a fixed proton of charge $+e$, in a circular orbit of radius $r_{0}$. What is the total energy of the electron? As the electron is accelerated towards the proton it will radiate, thereby losing energy and causing the orbit to decay. Derive a formula for the lifetime of the orbit.

  Part II

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