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

Question

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

Infinity Learn · Accepted Answer

Volume of the balloon at any instant, when radius is r,$$V=43$$$$π$$$$r3Time$$ rate of change of volume,$$dVdt=4$$$$π$$$$r2drdtTime$$ rate of change of radius of balloon,$$drdt=14$$$$π$$$$r2dVdtFlux$$ through rubber band at the given instant,ϕ$$=B$$$$π$$$$r2Induced$$ emf $$=-d$$ϕ$$dt=-ddtB$$$$π$$$$r2=-2$$$$π$$$$rBdrdt=-2$$$$π$$$$rB14$$$$π$$$$r2dVdt=-B2rdVdtAs$$ volume of the balloon is decreasing, dVdt is $$negativeEinduced=-0.042$$×$$10$$×$$10-2$$×$$-100$$×$$10-6=20$$μV

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