A thin wire loop of arbitrary shape is rotated around some axis with a stationary angular velocity $\omega$ in a homogeneous magnetic field of flux density $B$.

1 What is the induced voltage in the loop? Express it as a function of time in the simplest possible form.

2 Calculate the root mean square voltage as well. How should we adjust the orientation of the loop, the direction of the angular velocity and the magnetic field to make this minimal or maximal?

Consider the example depicted in the figure: $N$ turns of wire wound up on the surface of a sphere of radius $R$ between two opposite points $P$ and $Q$, closed by a straight segment connecting the poles. The pitch of the coil is uniform with respect to the polar angle of the $P Q$ axis.

3 Examine the questions of exercise 2 in this specific case. Discuss the result for different values of the number of turns.