Question

Radius of a circular loop placed in a perpendicular uniform magnetic field is increasing at a constant rate of r metre per second if at any instant radius of the loop is our then emf induced in the loop at that instant will be

Answer 1

hope it will help you..

Answer 2

Answer:

$-B\pi 2r.$

Explanation:

Given that the radius of the circular loop is increasing at a constant rate of r.

Let the radius of the loop at time t be R(t), then,

$\rm \dfrac{dR(t)}{dt}=r\ m/s.$

The surface area of the coil at time t is given by,

$\rm A(t) = \pi R(t)^2.$

The magnetic flux linked with the coil at time t is given by

$\phi (t) = \vec B \cdot \vec A.$

where,

• $\vec B$ = magnetic field.

• $\vec A$ = area vector of the coil whose direction is along the normal to the plane of the coil.

The magnetic field is perpendicular to the plane of the coil therefore the direction between the magnetic field and the area vector is 0$^\circ.$

$\phi(t)=\rm BA(t)\cos\theta\\ = BA(t)\cos0^\circ\\ = BA(t).\\=B\pi R^2(t).$

According to the Faraday's law of electromagnetic induction, the emf induced in the coil is given by

$\rm e=-\dfrac{d\phi(t)}{dt}\\=-\dfrac{d}{dt}\left ( B\pi R^2(t)\right )\\=-B\pi\dfrac{dR^2(t)}{dt}\\=-B\pi 2\dfrac{dR(t)}{dt}\\=-B\pi 2r.$