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 $P$ (outside the circular region) at a distance $r$ from the centre of the circular region

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

Answer 1

Answer: (B)

$\int \overrightarrow{E}.d\overrightarrow{l}=|\frac{d\varphi }{dt}|=A|\frac{dB}{dt}|$
$\Rightarrow$ $E\left(2\pi r\right)=\pi a^2\left|\frac{\text{d}\text{B}}{\text{d}\text{t}}\right|$ for $r\ge a$
Therefore, $E=\frac{a^2}{2r}\left|\frac{\text{d}\text{B}}{\text{d}\text{t}}\right|$
Therefore, the Induced electric field $\propto \frac{1}{\text{r}}$
For $r\le a$
$E(2\pi r)=\pi r^2|\frac{dB}{dt}|$
or
$\text{E}=\frac{r}{2}\left|\frac{\text{d}\text{B}}{\text{d}\text{t}}\right|$ or $E\propto r$
At r = a, $E=\frac{a}{2}\left|\frac{\text{d}\text{B}}{\text{d}\text{t}}\right|$
Therefore, the variation of $E$ with $r$ (distance from centre) will be as follows: