Question

A uniform but time-varying magnetic field B(t) exists in a circular region of radius a and is directed into the plane of the paper, as shown. The magnitude of the induced electric field at point P at a distance r from the centre of the circular region

• A. Is zero
• B. Decreases as $\frac{1}{r}$
• C. Increases as r
• D. Decreases as $\frac{1}{r^2}$

Answer 1

The correct option is B Decreases as $\frac{1}{r}$

Construct a concentric circle of radius r. The induced electric field (E) at any point on the circle is equal to that at P. For this circle





$\oint \overrightarrow{E}.d\overrightarrow{l}=\left|\frac{d\varphi }{dt}\right|=A\left|\frac{dB}{dt}\right|$
Or $E\times \left(2\pi r\right)=\pi a^2.\left|\frac{dB}{dt}\right|$
$\Rightarrow E=\frac{a^2}{2r}\left|\frac{dB}{dt}\right|\Rightarrow E\propto \frac{1}{r}$