Q: A wire in the form of a sector of radius l and of angle (θ = π/4) having a resistance R is free to rotate about an axis passing through point O and perpendicular to horizontal plane. A vertical magnetic field $B = -B_0 \hat{k}$ exists in the space. If the sector rotates with constant angular velocity so that Q Joules of heat is produced per revolution, find the constant angular velocity.
Solution : Induced current will flow while the loop enters in to field and again while it comes out. There will be no current when the loop is completely in the magnetic field.
$\displaystyle e = \frac{1}{2} B \omega l^2 $
$\displaystyle I = \frac{e}{R} = \frac{1}{2R} B \omega l^2 $
$\displaystyle H = I^2 R t $
$\displaystyle H = (\frac{B \omega l^2)}{2R})^2 R t $
$\displaystyle t = \frac{\pi}{4 \omega} \times 2$
$\displaystyle Q = (\frac{B \omega l^2)}{2R})^2 R \times \frac{2 \pi}{4 \omega} $
$\displaystyle \omega = \frac{8 Q R}{\pi B^2 l^2}$