Question

A conducting circular loop is placed in a uniform magnetic field $B=0.025T$ with its plane perpendicular to the magnetic field. The radius of the loop starts shrinking at $1mm/s$. What is the induced emf in the loop if the radius is $2cm$?

Answer 1

Step 1: Given

Magnetic field $B=0.025T$

The radius of the loop $r=2cm=2$$\times$${10}^{-2}m$

the constant rate at which the loop shrinks =$\frac{dr}{dt}=1\times {10}^{-3}m{s}^{-1}$

Step 2: Formulae used

Magnetic flux linked with the loop $\varphi =BA\cos \theta =B{\mathrm{\pi r}}^2\cos 0^{\circ }={\mathrm{B\pi r}}^2(\cos 0^{\circ }=1)$

Step 3: Solution

Induced emf will be:

$$\begin{gathered} \frac{d\varphi }{dt}=\frac{d\left(B\pi r^2\right)}{dt}=B\pi 2r\frac{dr}{dt} \\ =0.025\times \mathrm{\pi }\times 2\times 2\times {10}^{-2}\times 1\times {10}^{-3} \\ =\pi \times {10}^{-6}V=\pi \mu V \end{gathered}$$

The induced emf in the loop is$\pi \mu V$.