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

If $(t)$ is instantaneous current then,

$I(t)=\frac{1}{R}\frac{d\varphi }{dt}....(I)$

If $Q$ is charge passing in time $t$

$\therefore I(t)=\frac{dQ}{dt}....(II)$

From (I) and (II)

$\frac{dQ}{dt}=\frac{1}{R}\cdot \frac{d\varphi }{dt}$

Or $dQ=\frac{1}{R}.d\varphi ....(III)$

Integrating both sides,

${\int }_{Q_1}^{Q_2}dQ=\frac{1}{R}{\int }_{{\varphi }_1}^{{\varphi }_2}d\varphi$

$Q_2(t)-Q_1(t)=\frac{1}{R}[{\varphi }_2(t)-{\varphi }_1(t)]$

For magnetic flux in rectangle:

Magnetic flux due to current carrying conductor at a distance $x$'

$Q(t)=\frac{{\mu }_{0}I(t)}{2\pi x^{\prime }}$

If length of strip is $L_1$ so total flux on strip of length $L_1$ at distance $x^{\prime }$ is

$Q(t)=\frac{{\mu }_{0}I(t)}{2\pi x^{\prime }}{L}_1$

$X^{\prime }$ varies from $x$ to $(x+L_2)$ so total flux in strip

$\varphi (t)=\frac{{\mu }_0}{2\pi }{\int }_x^{x-L_2}\frac{dx}{x^{\prime }}I(t)=\frac{{\mu }_0{L}_2}{2\pi }.I(t_1){\log }_e\frac{(L_2+x)}{x}$

The magnitude of charge is given on length $L_1$

${\int }_{Q_1}dQ=\frac{1}{R}\int d\varphi$ [from (III)]

${\int }_0^{Q}dQ=\frac{1}{R}\cdot \frac{{\mu }_0{L}_1}{2\pi }{\log }_e\left(\frac{L_2+x}{x}\right){\int }_0^{1}I(t_1)$

$Q=\frac{{\mu }_0{L}_1}{R2\pi }{\log }_e\left(\frac{L_2+x}{x}\right)(I-0)=\frac{{\mu }_0{L}_1{L}_1}{2\pi R}\log \left(\frac{L_2+x}{x}\right)$.