Question

A uniform but time-varying magnetic field $B(t)$ exists in a cylindrical 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

Answer: (B)

Draw 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, induced emf
$\therefore E\cdot \oint dl=\pi a^2\left|\frac{dB}{dt}\right|$
(but $\oint dl=2\pi r)$
$\therefore E\times (2\pi )=\pi a^2\left|\frac{dB}{dt}\right|$
$\therefore E=\frac{a^2}{2r}\left|\frac{dB}{dt}\right|$
$\Rightarrow E\propto \frac{1}{r}$