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Use metric signature $-+++$ and transverse-traceless gauge, $g_{ij}={\delta }_{ij}+h_{ij}^{\mathrm{TT}}$. To resolve the source's origin ambiguity, put the detector centre at $z=0$ and the source centre at $z=-r_0$, so the gravitational wave travels in the positive $z$ direction. Reversing that direction replaces $z$ by $-z$. Work in the weak-field approximation, with $\omega l\ll c$, $r_0\gg l,c/\omega$, and detector size $a\ll c/\omega ,r_0$. The radiation reaction is neglected, so the rod maintains the prescribed steady motion.

At the retarded source time $u=t-(r_0+z)/c$, take the source positions as ${\mathbf{x}}_{\pm }=\pm l(\cos (\omega u+\beta ),\sin (\omega u+\beta ),0)$. The arbitrary constant $\beta$ specifies the initial source phase. The second mass moment tensor $I_{ij}={\sum }_{A}M{x}_{Ai}{x}_{Aj}$ obeysThe quadrupole formula $h_{ij}^{\mathrm{TT}}=2G{\ddot{I}}_{ij}^{\mathrm{TT}}/(c^4{r}_0)$ projects away the transverse trace, which is already constant here. Differentiating twice therefore givesAll time and longitudinal metric perturbation components vanish in transverse-traceless gauge. The gravitational-wave frequency is $2\omega$ in angular units; $M$ is each source mass, not their sum, and the detector mass $m$ does not enter the strain. The plane wave approximation holds the amplitude fixed at $r_0$; retaining spherical spreading replaces its denominator by $r_0+z$ near the axis. The retarded quadrupole formula and the distinction between source and wave frequency are derived in ^[ref].

Put ${\omega }_0^2=k/m$, where $k$ is the binding spring constant rather than an optical wavenumber. The steady complex amplitude solution of the forced Lorentz oscillator model isThe acceleration complex amplitude is $-{\omega }^2{\mathbf{x}}_0$. HenceThis is the scattering cross-section of a harmonically bound charge. The displacement and induced electric dipole moment are parallel to the incident linear polarization, and integrating their dipole radiation over all directions gives the total cross section above.

For $\omega \ll {\omega }_0$, binding dominates andrecovering Rayleigh scattering with static electric polarizability $q^2/k$. For $\omega \gg {\omega }_0$, inertia dominates: ${\mathbf{x}}_0\simeq -q{\mathbf{E}}_0/(m{\omega }^2)$, so its second time derivative becomes frequency independent andThis is a classical Thomson scattering regime only while $\hslash \omega \ll m{c}^2$. Exactly at $\omega ={\omega }_0$, an undamped driven oscillator has no finite steady-state amplitude; the divergent formula must not be used. Finite damping and radiation reaction regulate the physical resonant response. Coherent contributions from many charges in an extended medium can also require a structure calculation rather than multiplying this single-charge result by a particle count.

After field electron emission, let $h$ denote the plate separation of the parallel-plate capacitor, which the question does not specify. The uniform capacitor electric field has magnitude $E_c=|V|/h$, and a nonrelativistic electron of mass $m_e$ has constant acceleration magnitude $a=e{E}_c/m_e$. The sign of its charge affects the direction of the radiation field but not its radiated power.

In the supplied nonrelativistic Liénard-Wiechert field, $\dot{\mathbf{\beta }}=\mathbf{a}/c$. At polar angle $\theta$ to the acceleration, $|\hat{\mathbf{n}}\times (\hat{\mathbf{n}}\times \mathbf{a})|=a\sin \theta$, henceThe outgoing Poynting vector has magnitude $S={ϵ}_{0}c{E}_{\mathrm{rad}}^2$. Its radiated power per solid angle is thereforeUsing $\int {\sin }^2\theta d\Omega =8\pi /3$ gives the Larmor formula and the requested constant power during the uniform-field part of the flight:The brackets in the given field require the acceleration at retarded time; the radiation reaches the far observer after the propagation delay. This derivation assumes that radiation reaction is a small correction to the acceleration and that the flight is nonrelativistic, for example $e|V|\ll m_e{c}^2$ for release from rest. Entry, emission and impact transients are not described by the uniform-field model. A result solely in terms of $V$ cannot be obtained without a plate spacing or a specified internal field.

Let $\mathcal{I}$ be the sphere's mechanical moment of inertia about its rotation axis, and let ${\omega }_i=\omega (0)$. For an undriven rigid sphere the rotational energy is $E_{\mathrm{rot}}=\mathcal{I}{\omega }^2/2$. In the weak-radiation-loss approximation, conservation of energy givesIntegrating $d({\omega }^{-2})/dt=2K/\mathcal{I}$ givesEquivalently, the characteristic spin-down time isThe radiative spin-down of a rotating electric dipole therefore has angular frequency that decays as $t^{-1/2}$ at late times within this model, rather than exponentially. The original paper gives no mass or mass-density profile, so $\mathcal{I}$ cannot be determined from $R$ and the relative permittivity. If a homogeneous solid sphere of mass $M$ is additionally assumed, insert $\mathcal{I}=2M{R}^2/5$; uniform dielectric response alone does not imply this mechanical mass distribution.

The approximation is important. If the dipole angle is $\varphi (t)$ with $\dot{\varphi }=\omega (t)$, direct differentiation gives $|\ddot{\mathbf{p}}{|}^2=p_0^2({\omega }^4+{\dot{\omega }}^2)$. Using the constant-frequency radiation law instantaneously neglects the second term and assumes $|\dot{\omega }|\ll {\omega }^2$. This gives the standard leading slow spin-down solution above. An exact nonadiabatic radiation reaction evolution would require the mechanical and near-field dynamics beyond the information supplied, rather than treating this leading law as exact at arbitrarily rapid damping.

In a Cartesian coordinate system the leading electric dipole field givesThereforeThe force is downwards on the equator and upwards on the negative axial side. A free positive electric charge released from rest at the left-hand point initially accelerates downwards. As it moves into $x<0,y<0$, the force also acquires a positive $x$ component. Its path bends toward the right; the force continually changes as its position changes. An initial velocity was not specified, so the following more precise trajectory assumes release from rest and neglects radiation reaction.

There is a useful exact result within the point-electric dipole approximation. Put $A=k_{e}pQ>0$; the potential energy is $U=Ay/r^3$, a homogeneous function of degree $-2$. Thus $\mathbf{r}\cdot \mathbf{F}=2U$. For a particle of mass $m$, Newton's second law and conservation of energy implyAt release, both kinetic energy and potential energy vanish, so $E=0$; also $d(r^2)/dt=0$. Consequently $r=d$ throughout the motion. Parametrize the lower semicircle by $x=-d\cos \alpha$, $y=-d\sin \alpha$. ThenThe electric charge gains speed until the bottom point, where the force is inward and the speed is $\sqrt{2A/(m{d}^2)}$. It then slows, reaches $(d,0)$ at rest, and reverses. Thus, under these assumptions,Its trajectory is not an electric field line: the force determines acceleration, rather than velocity direction. For the finite two-charge electric dipole, corrections of order $a^2/d^2$ perturb the circular trajectory; an arbitrary nonzero initial velocity also changes it.

Source: www.whatdotheyknow.com/request/oxford_physics_past_papers_2/response/2340696/attach/9/Physics%20Papers%202020.pdf?cookie_passthrough=1

Source bundle pages 70–73; the cover identifies Trinity Term 2021 despite the attachment filename. Use coordinates $x^{\mu }=(ct,x,y,z)$, metric signature $(-,+,+,+)$, and $A^{\mu }=(\varphi /c,\mathbf{A})$. For the electromagnetic field tensor use $F^{0i}=E_i/c$ and $F^{ij}={ϵ}_{ijk}{B}_k$. Prescribed external fields are treated classically, neglecting radiation reaction in the trajectories; radiation losses are computed separately where requested.

Radiation reaction is the influence on a source's motion of its own electromagnetic field, including the balance of radiated energy and momentum with changes of the near field. For sufficiently weak radiation loss, an averaged power balance can give the leading damping law without solving the full self-field dynamics. Replacing a constant-frequency radiation formula by its instantaneous slowly varying value requires an additional time-scale assumption; it is not an exact treatment of arbitrarily rapid motion changes.

For a charged particle in a uniform magnetic field $\mathbf{B}=B\hat{z}$ with no electric field, the relativistic Lorentz force preserves energy, so the Lorentz factor $\gamma$ is constant. Decompose the velocity into $v_{\parallel }$ and ${\mathbf{v}}_{\perp }$. The parallel component remains constant; the perpendicular component rotates. The generic orbit is a helix, becoming a circle only when $v_{\parallel }=0$.

Writing proper time as $\tau$ and spatial four-velocity components as $u_x,u_y$, the force equation givesThus the signed cyclotron parameter is ${\omega }_{\tau }=qB/m$; the counterclockwise phase of $u_x+i{u}_y$ changes at $-{\omega }_{\tau }$. The magnitude of the coordinate-time angular frequency is $|qB|/(\gamma m)$.

The orbit radius depends on transverse momentum,not on the total speed unless the motion is perpendicular to the field. At fixed $p_{\perp }$, adding parallel momentum leaves $R$ unchanged but increases $\gamma$ and the coordinate-time orbit period. These formulas neglect radiation reaction and collisions.^[ref]

For a charge $q$ of mass $m$ bound by spring constant $k=m{\omega }_0^2$, the undamped equation $m\ddot{x}+kx=q{E}_0{e}^{-i\omega t}$ has complex amplitude $x_0=q{E}_0/[m({\omega }_0^2-{\omega }^2)]$. The induced electric dipole moment is $p_0=q{x}_0$. Averaged electric dipole radiation power, divided by incident light intensity, givesThe Rayleigh scattering limit $\omega \ll {\omega }_0$ gives $\sigma \sim {\sigma }_T(\omega /{\omega }_0{)}^4$, proportional to ${\lambda }^{-4}$. For ${\omega }_0\ll \omega \ll m{c}^2/\hslash$, the charge behaves as free and the Thomson cross-section is recovered. At resonance the undamped steady-state formula is invalid; a finite damping mechanism or radiation reaction must be included. The formula describes independent weakly driven scatterers, not arbitrary coherent scattering from an extended medium.