Question #43355

A small coil of $N=50$ turns, area $A=200 \text{ cm}^2$, and internal resistance $r=2 \text{ }\Omega$ moves with a constant velocity $\vec{v} = 5 \hat{i} \text{ m/s}$ along the x-axis. The coil is in a region of non-uniform magnetic field $\vec{B} = \beta x^2 \hat{k}$, where $\beta = 2 \text{ T/m}^2$ and $\hat{k}$ is the unit vector pointing out of the page. Assume the coil dimensions are small enough that the magnetic field is uniform over its area at any instant. The coil is connected in series with a variable external resistor $R$. The potential difference across $R$ is fed to a potentiometer circuit. The potentiometer consists of a wire of length $L=1 \text{ m}$ connected across an ideal $10 \text{ V}$ DC source. The connections to the potentiometer are made such that a null point can only be obtained if the induced current in the moving coil is in the clockwise direction. If $R$ is set to extract maximum electrical power from the coil, at what position $x$ of the coil will a null point be observed on the potentiometer wire at a balancing length of $l = 50 \text{ cm}$?