Question #43931

A circular conducting loop of radius $a$ (in $\text{m}$), self-inductance $L$ (in $\text{H}$), and negligible ohmic resistance is placed in a uniform magnetic field $B$ (in $\text{T}$) perpendicular to its plane. The loop is connected in series with an electrolytic cell of negligible EMF. The initial internal resistance of the cell is $R_0$ (in $\Omega$). At $t=0$, the magnetic field is suddenly switched off. As the induced transient current flows, the electrolyte heats up, causing its internal resistance to decrease exponentially with the total charge $q$ (in $\text{C}$) that has passed through it since $t=0$, given by $R(q) = R_0 e^{-\lambda q}$, where $\lambda$ is a positive constant in $\text{C}^{-1}$. Assuming $\lambda B \pi a^2 < R_0$, what is the total charge $Q$ (in $\text{C}$) that flows through the circuit until the current completely dies out?