Use the superposition principle for the magnetic vector potential due to a continuous current distribution:

\begin{align} \vec{A}(\vec{r}) = \frac{\mu_0}{4\pi} \int \frac{\vec{J}'(\vec{r}^{\,\prime})}{\left| \vec{r}-\vec{r}'\right|}\, d\tau', \end{align}

to find the magnetic vector potential everywhere in space due to a spinning charged ring with radius \(R\), total charge \(Q\), and period \(T\).

Evaluate your expression for the special case that \(\vec{r}\) is on the \(z\)-axis.

Evaluate your expression for the special case that \(\vec{r}\) is on the \(x\)-axis.

Find a series expansion for the electrostatic potential at these special locations:

1. Near the center of the ring, in the plane of the ring;
2. Near the center of the ring, on the axis of the ring;
3. Far from the ring on the axis of symmetry;
4. Far from the ring, in the plane of the ring.