Question #42740

A rigid rectangular wire loop is placed in a uniform magnetic field $\vec{B}$ perpendicular to its plane, pointing into the page. The loop is divided into two identical adjacent square loops, Loop 1 (left) and Loop 2 (right), by a central straight wire. The side length of each square is $a$ (in $\text{m}$). The magnetic field magnitude increases with time as $B(t) = \beta t$, where $\beta > 0$ is a constant in $\text{T/s}$. The left-most vertical branch has a resistance $R$ (in $\Omega$), and the right-most vertical branch has a resistance $2R$ (in $\Omega$). The top and bottom horizontal branches have negligible resistance. The central vertical branch contains an ideal inductor of inductance $L$ (in $\text{H}$) and has negligible resistance. Assume the self-inductance of the square loops is negligible compared to $L$. At $t=0$, the magnetic field starts increasing. Find the magnitude of the current flowing through the central branch as a function of time $t$ (in $\text{s}$).