(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 $\lambda$ ?