Question

A conducting circular loop is placed in a uniform magnetic field B= 0.025 T with its plane perpendicular to the magnetic field. The radius of the loop starts shrinking at 1 mm/s. The induced emf in the loop when the radius is 2 cm?
A. $2\pi \mu$ V
B. $\pi \mu$V
C. $\dfrac{\pi }{2}\mu$V
D. $3.2\pi \mu$V

Answer 1

The magnetic field is perpendicular to the plane of the coil. The angle between the area vector and the field is thus ${{0}^{0}}$. Also, the magnetic field is constant and not varying with time. We know for induced emf to take place, the flux associated with the coil must vary with time.

Complete step by step answer:
There can be three ways for the flux associated with the coil to change. This could happen if the field is changing continuously, or the area of the coil is increasing or decreasing continuously and either the angle between the area vector and the field changes continuously.In this problem the area is changing, hence the induced emf will be generated.
Magnetic flux is given as,

\Rightarrow{{\phi }_{1}} =BA\cos \alpha \\\ \Rightarrow{{\phi }_{1}} =BA\cos {{0}^{0}}\\\ \Rightarrow{{\phi }_{1}} =BA$$ From Faraday’s law of electromagnetic induction, induced emf is given by, Now, using faraday’s law we get: $e=\dfrac{d\phi }{dt} \\\ \Rightarrow e=\dfrac{d[BA]}{dt} \\\ \Rightarrow e=B\dfrac{dA}{dt} \\\ $ This is a circular loop, so area is given by: $$A=\pi {{r}^{2}}$$ $ e=B\dfrac{d(\pi {{r}^{2}})}{dt} \\\ \Rightarrow e=\pi B\dfrac{d({{r}^{2}})}{dt} \\\ \Rightarrow e=2\pi rB\dfrac{dr}{dt} \\\ $ Now given values are: B= 0.025 T, r= 2cm= 0.02m and $$\dfrac{dr}{dt}=1mm/s$$ or $$\dfrac{dr}{dt}=0.001m/s$$ $\Rightarrow e=2\times \pi \times 0.02\times 0.025\times 0.001 \\\ \Rightarrow e=2\pi \times {{10}^{-6}}V \\\ \therefore e=2\pi \mu V $ **So, the correct option is A.** **Note:** In this problem, we have to use Faraday’s law of electromagnetic induction. It says a changing magnetic field induces potential difference inside a conductor placed in it. Faraday’s law states that a current will be induced in a conductor which is exposed to a changing magnetic field. while solving such problems we have to keep in mind that while finding out the angle between the field and the plane we have to take the area vector and area vector is always perpendicular to the plane.