 ## Question:

A charged spherical capacitor slowly discharges as a result of the slight conductivity of the dielectric between its concentric plates. What are the magnitude and direction of the magnetic field caused by the resulting electric current?

## Step-by-step

The system described in the problem is spherically symmetrical. Therefore the magnetic field that is built up has to be spherically symmetrical as well. A spherically symmetrical vector field has to be radial everywhere and its magnitude can depend only on the distance from the origin: ${{B}_{(r)}}={{B}_{(r)}}{{r}/\left | {r} \right |}.$
On the other hand, a magnetic field contains no sources (magnetic monopoles), and the magnetic flux crossing any closed surface has to be zero at any given moment. In particular, consider a spherical surface of radius r around the capacitor. The consequences of sourcelessness can only be met if ${{B}_{(r)}}=0$ for any r. This means that the current described in the problem builds up no magnetic field, either inside or outside the spherical capacitor.
Note. It is worth examining how the basic laws of electrodynamics are satisfied between the plates of the spherical capacitor. Is it true that a magnetic field builds up around a current flowing in a conductor, and that the rotation (curl) of this field is proportional to the current?