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
$\therefore E \cdot \oint d l=\pi a ^{2}\left|\frac{ dB }{ dt }\right|$
(but $\oint d l=2 \pi r )$
$\therefore E \times(2 \pi)=\pi a^{2}\left|\frac{ dB }{ dt }\right|$
$\therefore E =\frac{ a ^{2}}{2 r }\left|\frac{ dB }{ dt }\right|$
$\Rightarrow E \propto \frac{1}{ r}$