$A$ rectangular loop of wire $ABCD$ is kept close to an infinitely long wire carrying a current $I(t) = I_0(1 - t/T)$ for $0 \le t \le T$ and $I(t) = 0$ for $t > T$ as shown in the figure. Find the total charge passing through a given point in the loop in time $T$. The resistance of the loop is $R$.

(N/A) The magnetic field $B$ at a distance $r$ from an infinitely long wire carrying current $I$ is $B = \frac{\mu_0 I}{2 \pi r}$.
Consider a strip of thickness $dr$ and length $L_1$ at a distance $r$ from the wire.
The magnetic flux $d\phi$ linked with this strip is $d\phi = B \cdot dA = \frac{\mu_0 I}{2 \pi r} (L_1 dr)$.
The total magnetic flux $\phi$ linked with the rectangular loop $ABCD$ is the integral of $d\phi$ from $r = x$ to $r = x + L_2$:
$\phi = \int_{x}^{x+L_2} \frac{\mu_0 I L_1}{2 \pi r} dr = \frac{\mu_0 I L_1}{2 \pi} [\ln r]_{x}^{x+L_2} = \frac{\mu_0 I L_1}{2 \pi} \ln \left( \frac{x + L_2}{x} \right)$.
The induced current is $i = \frac{|d\phi/dt|}{R}$. The total charge $Q$ passing through a point is $Q = \int i dt = \frac{1}{R} \int |d\phi| = \frac{|\Delta \phi|}{R}$.
At $t = 0$,$I(0) = I_0$,so $\phi_i = \frac{\mu_0 I_0 L_1}{2 \pi} \ln \left( \frac{x + L_2}{x} \right)$.
At $t = T$,$I(T) = 0$,so $\phi_f = 0$.
The magnitude of the change in flux is $|\Delta \phi| = |\phi_f - \phi_i| = \frac{\mu_0 I_0 L_1}{2 \pi} \ln \left( \frac{x + L_2}{x} \right)$.
Therefore,the total charge $Q$ is $\frac{\mu_0 I_0 L_1}{2 \pi R} \ln \left( 1 + \frac{L_2}{x} \right)$.