Two conductors of infinite length carry a current i each, which varies with time according to relation $\mathrm{i}={\mathrm{i}}_0{\mathrm{e}}^{\left(\frac{2}{{\mathrm{\mu }}_0}\right)\mathrm{t}}$ . They are parallel and separated by a distance $2a$. A circular conducting ring of radius $a$ and having a resistance 1$\mathrm{\Omega }/$length, in the plane of wires lies between the two straight conductors and insulated from them as shown :-

 

Flux through the strip, $d\varphi =\left[\frac{{\mu }_0}{2\pi }\frac{i}{(a-x)}+\frac{{\mu }_0}{2\pi }\frac{i}{(a+x)}\right]\left(2\sqrt{a^2-x^2}\right)dx$
$=\frac{{\mu }_0}{2\pi 0}i\left[\frac{1}{a-x}+\frac{1}{a+x}\right]\times 2\sqrt{a^2-x^2}dx$

$\mathrm{d}\varphi =\frac{2{\mu }_0\mathrm{ia}}{\pi }\cdot \frac{\mathrm{dx}}{\sqrt{{\mathrm{a}}^2-{\mathrm{x}}^2}}$
Total flux through the ring $\varphi =2\left[\frac{2{\mu }_{0}ia}{\pi }{\int }_0^a \frac{dx}{\sqrt{a^2-x^2}}\right]$
$\begin{array}{r}=\frac{4{\mu }_0\mathrm{ia}}{\pi }{\left[{\sin }^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right)\right]}_0^{\mathrm{a}}\\ =\frac{4{\mu }_0\mathrm{ia}}{\pi }\left[\frac{\pi }{2}\right]\\ \varphi =2{\mu }_0\mathrm{ia}\end{array}$
Mutual inductance $\mathrm{M}=\frac{{\mathrm{Q}}_2}{{\mathrm{i}}_1}=\frac{2{\mu }_0\mathrm{ia}}{\mathrm{i}}=2{\mu }_0\mathrm{a}$
$\begin{array}{l}\text{ Also, }\mathrm{M}=\frac{{\mathrm{\epsilon }}_2}{{\mathrm{di}}_1/\mathrm{dt}}\Rightarrow {\mathrm{\epsilon }}_2=\frac{{\mathrm{Md}}_1}{\mathrm{dt}}\\ {\mathrm{\epsilon }}_2=2{\mathrm{\mu }}_0{\text{ ai.e }}^{\left(\frac{2}{{\mathrm{\mu }}_0}\right)\mathrm{t}}\cdot \frac{2}{{\mathrm{\mu }}_0}\\ {\mathrm{\epsilon }}_2=4\mathrm{ai}\end{array}$
Induced current in the ring ${\mathrm{i}}_{\mathrm{R}}=\frac{{\epsilon }_2}{\mathrm{R}}$
(R resistance)
$i_R=\frac{4ai}{2\pi a}=\frac{2i}{\pi }$
Magnetic field at the centre,
$\begin{array}{l}{\mathrm{B}}_{\text{net }}=\frac{{\mathrm{\mu }}_0}{2\mathrm{\pi }}\cdot \frac{\mathrm{i}}{\mathrm{a}}+\frac{{\mathrm{\mu }}_0}{2\mathrm{\pi }}\cdot \frac{\mathrm{i}}{\mathrm{a}}-\frac{{\mathrm{\mu }}_0}{2}\cdot \frac{(2\mathrm{i}/\mathrm{\pi })}{\mathrm{a}}\\ \frac{{\mathrm{\mu }}_0}{\mathrm{\pi }}\cdot \frac{\mathrm{i}}{\mathrm{a}}-\frac{{\mathrm{\mu }}_0\mathrm{i}}{\mathrm{\pi a}}\\ =0\end{array}$