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.