#### Consider an infinitely long wire carrying a current I(t), with $\frac{dl}{dt}=\lambda =constant.$. Find the current produced in the rectangular loop of wire ABCD if its resistance is R (figure).

Let us consider a strip of length l and width dr at a distance r from an infinite long current carrying wire. The magnetic field in the strip is given by:

$\vec{B}(r)=\frac{\mu _{0}I}{2\pi r}$ (Out of paper)

Area of the elementary strip is, $dA=l.dr$

So, total flux through the loop is

$\phi _{m}=\vec{B}.\vec{A}=\frac{\mu _{0}I}{2\pi }l\int_{x_{0}}^{x}\frac{dr}{r}=\frac{\mu _{0}Il}{2\pi}ln\frac{x}{x_{0}}\; \; \; \; \; \; \; \; \; .....(i)$

The emf induced can be obtained by differentiating the eq. (i) w.r.t. t and then applying Ohm's law

$I=\frac{\varepsilon }{R}$      and $\left | \varepsilon \right |=\frac{d\phi }{dt}$

We have, induced current

$I=\frac{1}{R}\frac{d\phi }{dt}=\frac{\mu _{0}l}{2\pi}\frac{\lambda }{R}ln\frac{x}{x_{0}}\left ( \because \frac{dI}{dt} =\lambda \right )$