A conducting circular loop is placed in a uniform magnetic field $$0.04\,T$$  with its plane perpendicular to the magnetic field. The radius of the loop starts shrinking at $$2\,mm\,{s^{ - 1}}.$$   The induced emf in the loop when the radius is $$2\,cm$$  is

Solution :
Magnetic field, $$B = 0.04\,T$$   and rate of change of radius of coil due to shrinkage, $$\frac{{ - dr}}{{dt}} = 2\,mm\,{s^{ - 1}}$$
Induced emf, $$e = \frac{{ - d\phi }}{{dt}} = - B\frac{{dA}}{{dt}} = - B\frac{{d\left( {\pi {r^2}} \right)}}{{dt}}$$
$$ = - B\pi 2r\frac{{dr}}{{dt}}$$
Now, if $$r = 2\,cm$$
$$\eqalign{ & e = - 0.04 \times \pi \times 2 \times 2 \times {10^{ - 2}} \times 2 \times {10^{ - 3}} \cr & = 3.2\,\pi \mu V \cr} $$