Magnetic field for azimuthal current

[vibe3]

I am working with a current density defined in a spherical shell S(a, b) (inner radius a and outer radius b). The current density is completely in the azimuthal direction:

J = f(r,\theta) \hat{\phi}

I am trying to pick some simple (but non-trivial) f(r,\theta) so that the magnetic field has an analytical solution. All the functions I've tried lead to very complicated integrals in the Biot-Savart law, and I can't seem to find anything with enough symmetry to use an Ampere loop.

Does anyone know of any suggestions?

 

[kuruman]

Consider the shell carrying charge with uniform charge density $\rho$ and a volume element $dV=r^2dr~\sin\theta d\theta~ d\phi$. If the shell spins with angular speed $\omega$, the volume element produces an azimuthal current $$dI=\frac{dq}{dt}=\frac{\omega~dq}{2\pi}={\omega~\rho r^2~dr~\sin\theta d\theta};~~~~~(\omega=d\phi/dt)$$
The area element perpendicular to the azimuthal direction is $dA=r~d\theta~dr$ in which case you can write the current density as $$\vec J=\omega \rho r \sin\theta~\hat \phi=\rho~\vec {\omega} \times \vec r.$$
You get the magnetic fields both inside and outside the shell by using the current density to find the magnetic vector potential in the two regions (a) $r>r'$ outside and (b) $r<r'$ inside. $$\vec A(\vec r)=\frac{\mu_0}{4\pi}\int \frac{\vec J(\vec r')}{|\vec r-\vec r'|}~dV'$$Then $\vec B=\vec {\nabla}\times \vec A$.