Question #42030

Two parallel ideal conducting rails are placed at $y = a$ and $y = -a$ in the $xy$-plane, extending from $x=0$ to $x \to \infty$. They are connected at $x=0$ by a zero-resistance wire. A non-uniform magnetic field $\vec{B} = B_0 \frac{x}{a} \hat{k}$ exists in the region $x > 0$. Two rigid conducting wires, $W_1$ and $W_2$, each with uniform resistance $\rho$ per unit length, are placed across the rails. $W_1$ is a straight wire connecting the points $(x_0, a)$ and $(x_0, -a)$. $W_2$ is a semicircular wire bulging in the $+x$ direction, with its ends at $(x_0, a)$ and $(x_0, -a)$. Both wires are pulled independently in the $+x$ direction with a constant velocity $\vec{v} = v_0 \hat{i}$. At the instant $x_0 = a$, let $P_1$ and $P_2$ be the instantaneous thermal powers dissipated in $W_1$ and $W_2$ respectively. Neglecting self-inductance, what is the ratio $\frac{P_1}{P_2}$?