In the fig, XY is an infinitely long wire carrying current $ I (t) = \frac {I_0}{1 - \frac{t}{T}}$ for $ 0 \leq t \leq T$
If I(t) is current induced in the loop, then;
$ I(t) = \frac {d\phi / dt}{R} = \frac {dQ}{dt}$
Integrating on both sides;
$ Q (L_1) - Q (t_2) = \frac {1}{R} [ \phi (t_1) - \phi (t_2)]$
$ \phi (T_1) = L_1 \frac {\mu_0}{2\pi} \; \int\limits_x^{L_2 + x} \; \frac {dx^1}{x^1} \; I (t_1)$
$ = \frac {\mu_0 L_1}{2\pi}\; I (t_1) \; log_e \; (\frac {L_2 + x}{x})$
The magnitude of charge would be;
$ Q = \frac {\mu_0 L_1}{2\pi} \; log_e \; (\frac {L_2 + x}{x}) [I_1 - 0]$
$ = \frac {\mu_0 L_1 I_1}{2\pi} \; log_e \; (\frac {L_2 + x}{x})$