= Solution
Ignoring magnetization, <electric polarization> charge transport gives <bound current> $\mathbf J_{\rm b}=\partial_t\mathbf P$. The assumed weak dielectric contrast therefore gives $\mathbf J_{\rm b}\simeq-i\omega\epsilon_0(\epsilon_r-1)\mathbf E_0e^{i(\mathbf k\cdot\mathbf x-\omega t)}$ inside the <sphere>. This is the first <Born approximation>: the incident field supplies the <electric polarization> to leading order in $\epsilon_r-1$.
For $\mathbf x=r\mathbf n$, the far-distance expansions are $|\mathbf x-\mathbf x'|=r-\mathbf n\cdot\mathbf x'+O(a^2/r)$ and $|\mathbf x-\mathbf x'|^{-1}\simeq r^{-1}$. Substitute the <retarded time> into the phase in the <Lorenz gauge> <vector potential>. The phase becomes $kr-\omega t+(\mathbf k-k\mathbf n)\cdot\mathbf x'$, so, using $\mu_0\epsilon_0=c^{-2}$,
$$
\boxed{\mathbf A_{\rm scatt}\simeq-\frac{i\omega(\epsilon_r-1)}{4\pi c^2r}\mathbf E_0e^{i(kr-\omega t)}I(q),\qquad
I(q)=\int_{|\mathbf x'|\leq a}e^{i\mathbf q\cdot\mathbf x'}d^3x',\quad\mathbf q=\mathbf k-k\mathbf n.}
$$
The assumptions $r\gg a$ and $ka^2/r\ll2\pi$ control amplitude and phase errors respectively. In the <radiation zone> the supplied electric and magnetic fields satisfy $\mathbf B_{\rm scatt}=c^{-1}\mathbf n\times\mathbf E_{\rm scatt}$. Complex phasors therefore give the time-averaged radial Poynting flux $|\mathbf E_{\rm scatt}|^2/(2\mu_0c)$. The incident flux is $|\mathbf E_0|^2/(2\mu_0c)$, so
$$
\frac{d\sigma}{d\Omega}=r^2\frac{|\mathbf E_{\rm scatt}|^2}{|\mathbf E_0|^2}
=\frac{(\epsilon_r-1)^2k^4}{16\pi^2}|I(q)|^2|\mathbf n\times\widehat{\mathbf E}_0|^2.
$$
Orient <spherical coordinates> along $\mathbf q$ to evaluate the form factor:
$$
I(q)=4\pi\int_0^a s^2\frac{\sin(qs)}{qs}\,ds=4\pi\frac{\sin(qa)-qa\cos(qa)}{q^3}.
$$
Consequently
$$
\boxed{\frac{d\sigma}{d\Omega}=(\epsilon_r-1)^2k^4\left[\frac{\sin(qa)-qa\cos(qa)}{q^3}\right]^2|\mathbf n\times\widehat{\mathbf E}_0|^2.}
$$
At $q=0$ the bracket is interpreted by its <limit> $a^3/3$, not a singularity. The transverse factor accounts for <polarization of an electromagnetic wave>; for complex <polarization of an electromagnetic wave> its squared <norm> means the Hermitian <norm>.
Back to article page
