An elasticized conducting band r is around a spherical balloon T (figure). Its plane passes through the center of the balloon. A uniform magnetic field of magnitude 0.04 T  is directed perpendicular to the plane of the band. Air is let out of the balloon at 100 cm³s^-1 at an instant when the radius of the balloon is 10 cm. The induced emf in the band is

Volume of the balloon at any instant, when radius is r,

$V=\frac{4}{3}\pi r^3$

Time rate of change of volume,

$\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$

Time rate of change of radius of balloon,

$\frac{dr}{dt}=\frac{1}{4\pi r^2}\frac{dV}{dt}$

Flux through rubber band at the given instant,

$\varphi =B\left(\pi r^2\right)$

Induced emf $=-\frac{d\varphi }{dt}=-\frac{d}{dt}\left(B\pi r^2\right)=-2\pi rB\frac{dr}{dt}$

$=-2\pi rB\left(\frac{1}{4\pi r^2}\frac{dV}{dt}\right)=-\frac{B}{2r}\frac{dV}{dt}$

As volume of the balloon is decreasing, $\frac{dV}{dt}$ is negative

$E_{induced}=-\frac{\left(0.04\right)}{2\times 10\times {10}^{-2}}\times \left(-100\times {10}^{-6}\right)=20\mu V$