Question

A conducting ring of radius $r$ is placed in a varying magnetic field perpendicular to the plane of the ring. If the rate at which the magnetic field varies is $x$, the electric field intensity at any point of the ring is

• A. r x
• B. $\frac{rx}{2}$
• C. 2 r x
• D. $\frac{4r}{x}$

Answer 1

Answer: (B)

Let $\vec{E}$ be the electric field intensity at a point on the circumference of the ring. Then, the emf induced $\epsilon =\oint \vec{E}\cdot d\vec{l}$
where $d\vec{l}$ is a length element of the ring. Since $|\vec{E}|$ is constant and $\vec{E}\Vert d\vec{l}$,
$\therefore \epsilon =E(2\pi r)$...(i)
Also, the induced emf is
$\epsilon =\frac{d\varphi }{dt}=\pi r^2\frac{dB}{dt}=\pi r^{2}x$...(ii)
Equating (i) and (ii), we get $E=\frac{rx}{2}$