= Solution
Let $R=|\mathbf x|$, $\widehat{\mathbf R}=\mathbf x/R$, and $t_R=t-R/c$. To leading nonzero <electromagnetic multipole expansion>[multipole expansion] order, the loop is a time-dependent <magnetic dipole moment>[magnetic dipole] with
$$
\mathbf m(t)=\pi r^2I(t)\mathbf n.
$$
Expanding the <retarded electromagnetic potential> across the small loop gives
$$
\mathbf A(\mathbf x,t)
=\frac{\mu_0}{4\pi}
\left[
\frac{\mathbf m(t_R)\times\widehat{\mathbf R}}{R^2}
+\frac{\dot{\mathbf m}(t_R)\times\widehat{\mathbf R}}{cR}
\right]
+O\!\left(\frac{r^3}{R^3}\right).
$$
The $R^{-2}$ term is the near magnetic-dipole field and the $R^{-1}$ term is the <magnetic dipole radiation> field. Substituting the Fourier series for the current gives
$$
\boxed{
\mathbf A(\mathbf x,t)
=\frac{\mu_0r^2}{4}
(\mathbf n\times\widehat{\mathbf R})
\sum_{n=0}^{\infty}I_n
\left[
\frac{\sin(n\omega t_R)}{R^2}
+\frac{n\omega}{cR}\cos(n\omega t_R)
\right]
}.
$$
The $n=0$ summand vanishes for the stated sine series.
Back to article page
