Question

A conducting circular loop is placed in a uniform magnetic field $0.04\text{T}$ with its plane perpendicular to the magnetic field. The radius of the loop starts shrinking at a uniform rate of $2\text{mm/s}$. The induced emf in the loop, when the radius is $2\text{cm}$, is:

• A. $1.6\pi \mu \text{V}$
• B. $3.2\pi \mu \text{V}$
• C. $4.8\pi \mu \text{V}$
• D. $0.8\pi \mu \text{V}$

Answer 1

The correct option is B $3.2\pi \mu \text{V}$

Given,

$B=0.04\text{T}$

$\theta =0^{\circ }$

$\left|\frac{dR}{dt}\right|=2\text{mm/s}=2\times {10}^{-3}\text{m/s}$

Induced emf in the coil is given by,

$E=\left|\frac{d\varphi }{dt}\right|=\frac{d(\overrightarrow{B}.\overrightarrow{A})}{dt}$

$=\left|\frac{d\left(BA\cos 0^{\circ }\right)}{dt}\right|$

$=\left|B\left(\frac{d\left(\pi R^2\right)}{dt}\right)\right|$

$\Rightarrow E=B\times 2\pi R\left|\frac{dR}{dt}\right|$

From the valves given in question, emf in the coil at $R=2\text{cm}$ is,

$E=0.04\times 2\pi \times 2\times {10}^{-2}\times 2\times {10}^{-3}$

$E=3.2\pi \times {10}^{-6}\text{V}$

$\therefore E=3.2\pi \mu \text{V}$

Hence, option ($B$) is the correct answer.