Question #43357

Consider a rectangular primary loop $ACDB$ in the $xy$-plane with vertices $A(0,0)$, $B(L,0)$, $D(L,w)$, and $C(0,w)$. The side $AB$ is a uniform potentiometer wire of length $L$ and resistance $R$. The other sides are made of ideal resistanceless wires. An ideal primary cell of EMF $V_0$ is connected in the side $CD$ such that it drives a steady current in the counter-clockwise direction. A uniform magnetic field $\vec{B} = \alpha t \hat{k}$ exists everywhere, where $\hat{k}$ is the unit vector in the $+z$ direction (out of the page) and $\alpha > 0$ is a constant. A secondary circuit consists of an ideal cell of EMF $E_s$ and a galvanometer. It connects point $A$ to a movable jockey $J$ at position $(x,0)$ on the wire $AB$. The connecting wires of the secondary circuit form a smaller rectangular loop $A J J' C'$ with vertices $A(0,0)$, $J(x,0)$, $J'(x, w/2)$, and $C'(0, w/2)$. The cell $E_s$ and the galvanometer are located in the branch $C'A$, with the positive terminal of $E_s$ connected to $A$. Given $V_0 > \alpha L w$, find the balancing length $x$ for which the galvanometer shows zero deflection.