Question

A sphercial conducting shell is placed in a time-varying magnetic field. Is there an induced current along the equator ?

Answer 1

I believe, it depend on the angle between the equator and the field.
\[{rot} \vec E=-\frac{\partial \vec B}{\partial t}\]
So the current is perpendicular to the magnetic field. If the equator is not parallel to the field there ist a current along it. To calculate the current along the equator you only take the part of the field being perpendicular to the equator.
\[\vec B=\vec B_{\perp}+\vec B_{\parallel}\]
\[{rot} \vec E_{equator}=-\frac{\partial \vec B_{\perp}}{\partial t}\]
Obviously \[\vec B_{\perp}\] is 0 if the field is parallel to the equator.