Question #43352

A potentiometer wire $AB$ of length $L = 10 \text{ m}$ and resistance $R_w = 10 \ \Omega$ is connected in a primary circuit with a DC battery of EMF $V_0 = 20 \text{ V}$ and an external resistance $R_0 = 10 \ \Omega$. The primary circuit also contains an ideal coil (zero resistance and self-inductance) of $N = 100$ turns and area $A = 0.1 \text{ m}^2$. A uniform magnetic field perpendicular to the coil's plane varies with time as $B(t) = \alpha t^2$, where $\alpha = 0.5 \text{ T s}^{-2}$. By Lenz's law, the induced EMF in the coil opposes the battery $V_0$.

A secondary cell of constant EMF $E_s = 2 \text{ V}$ is connected to find the null point $x(t)$ from end $A$. Due to electromagnetic interference, a time-varying zero error $e(t)$ appears at end $A$, such that the actual potential difference balanced by the secondary cell is $V = \lambda(t) x(t) + e(t)$, where $\lambda(t)$ is the instantaneous potential gradient of the wire. If $e(t) = \gamma |\varepsilon_{ind}|$, where $\varepsilon_{ind}$ is the induced EMF in the coil and $\gamma = 0.05$, find the velocity of the null point $\frac{dx}{dt}$ at $t = 1 \text{ s}$.