A conducting ring of radius r and resistance R is placed in a...

A conducting ring of radius $r$ and resistance $R$ is placed in a region of uniform time varying magnetic field $B$ which is perpendicular to the plane of the ring. If the magnetic field is changing at a rate $\alpha$ , then the current induced in the ring is

Options

A $\dfrac{\pir^{2}\alpha}{2R}$
B $\dfrac{2\pir\alpha}{R}$
C $\dfrac{\pir\alpha}{R}$
D $\dfrac{\pir^{2}\alpha}{R}$

Correct answer

D. $\dfrac{\pir^{2}\alpha}{R}$

Step-by-step solution

The magnitude of the induced emf in the ring is

\left|ε\right| = \dfrac{dϕ}{dt} = \dfrac{d}{dt}\left(BA\right) = A\dfrac{dB}{dt} = \pir^{2}\alpha

The induced current in the ring,

I = \dfrac{\left|ε\right|}{R} = \dfrac{\pir^{2}\alpha}{R}

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