Question #42981

A rigid conducting wire of arbitrary shape and unknown mass connects the center $O$ of a stationary vertical circular conducting ring of radius $a = 2.0 \text{ m}$ to a point on its rim. The ring lies in the vertical $xy$-plane, with uniform gravity acting in the $-y$ direction. The center $O$ and the rim are connected via ideal slip rings to an external stationary resistor $R = 10 \ \Omega$. The electrical resistance of the wire and the ring are negligible. A magnetic field $\vec{B} = B_0 \left( \frac{r}{a} \right)^2 \hat{k}$ exists in the region, where $r$ is the radial distance from $O$, $\hat{k}$ is the unit vector normal to the $xy$-plane, and $B_0 = 4.0 \text{ T}$. An external motor rotates the wire about the $z$-axis with a constant angular velocity $\omega = 5.0 \text{ rad s}^{-1}$. Calculate the total work done by the motor during exactly one complete revolution of the wire.