A rotating current distribution infinitely extended in z directionAmpère's law is hard to solve in general. What if certain symmetries of a current distribution are present? We may find much simpler differential equations which might be integrated directly. Find out what to do in rotational and translational symmetry!

Problem Statement

Schematic of a rotating current distribution infinitely extended in z direction with coordinate system.Let us assume a "cylinder shell current distribution" of the following form:\[\begin{eqnarray*}  \mathbf{j}\left(\mathbf{r}\right)&=&\begin{cases}
j_{0}\mathbf{e}_{\varphi} & \rho\in\left[R_{1},R_{2}\right]\\0 & \text{else}\end{cases}\ .  \end{eqnarray*}\]This current distribution is rotationally symmetric which means that it does not vary in \(\varphi\). Furthermore, the current distribution is infinitely extended in the \(z\)-direction. This is called a translational symmetry.

Use Ampère's law to determine the magnetic field \(\mathbf{H}\left(\mathbf{r}\right)\) everywhere for the given current distribution.

If you found a solution for the cylinder shell current distribution, how would you determine the magnetic field of a more general rotationally and translationally invariant current distribution, i.e. \[\mathbf{j}\left(\mathbf{r}\right)    =    j\left(\rho\right)\mathbf{e}_{\varphi}\ ?\]

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