Question : Consider a long infinite wire carrying a time varying current i = kt (k > 0). A circular loop of radius a and resistance R is placed with its centre at a distance d from the wire (a < < d). Find out the induced current in the loop ?
Solution : Since current in the wire is continuously increasing therefore we conclude that magnetic field due to this wire in the region is also increasing.Magnetic field B due to wire is
$\displaystyle B = \frac{\mu_0}{2\pi}\frac{i}{d}$ ; going into and perpendicular to the plane of the paper
Flux through the circular loop,
$\displaystyle \phi = \frac{\mu_0 i}{2\pi d} \times \pi a^2$
$\displaystyle \phi = \frac{\mu_0 a^2 k t}{2 d} $
Induced e.m.f. in the loop
$\displaystyle e = – \frac{d \phi}{dt} = – \frac{\mu_0 a^2 k}{2 d} $
Induced current in the loop is
$\displaystyle i = \frac{e}{R} = \frac{\mu_0 a^2 k}{2 d R}$
Direction of induced current in the loop is anticlokwise.