Question

Radius of a circular loop placed in a perpendicular uniform magnetic field is increasing at a constant rate at \[{r_0} ms^{-1}\] . If at any instant radius of the loop is r, then emf induced in the loop at that instant will be
A. $ - 2Br{r_0}$
B. $ - 2B\pi r$
C. $ - B\pi r{r_0}$
D. $ - 2B\pi r{r_0}$

Answer 1

Hint: See that we are given that the circular loop is placed in a plane perpendicular to a uniform magnetic field and we are expected to find the emf, thus we can understand that this is related to electromagnetic induction. We are given the rate of change of radius, thus using it we can get rate electric flux and can calculate emf.

Complete step by step answer:
We are given the rate of change of radius as :
\[\dfrac{{dr(t)}}{{dt}} = {r_0} \] - - - - (1)
The surface area of the coil is given by $\pi {r^2}$, where $r$ is the radius at that time $t$. Now from Faraday’s law of electromagnetic induction we get the formula of emf , given by emf = -$\dfrac{{d\phi }}{{dt}}$ , where $\phi $ is the magnetic flux linked with the coil .
Now the value of magnetic flux $\phi $ is : $\overrightarrow {B.} \overrightarrow A $
Where B is the magnetic flux linked with the coil and A is the area vector of the coil whose direction is normal to the plane of the coil.

It is given that the direction of the magnetic field is normal to the plane of the coil, hence the angle between the magnetic field and the area vector will be $0^\circ $ .
Thus emf = $\dfrac{{ - d\phi (t)}}{{dt}}$
Substituting the value of the given equation
$ \Rightarrow emf = - \dfrac{{d(B.\pi {r^2}(t))}}{{dt}}$
$ \Rightarrow emf = - B.\dfrac{{d(\pi {r^2}(t))}}{{dt}}$
As B is constant for the given problem and angle is $0^\circ $ thus cos value is $1$ .
$ \Rightarrow emf = - B.2\pi r\dfrac{{d(r(t))}}{{dt}}$
We know from $1$ that value of \[\dfrac{{dr(t)}}{{dt}}\] , thus
\[ \therefore emf = - B.2\pi r{r_0}\] .

Hence, the correct answer is option D.

Note:A current in the wire would be caused if a coil of wire is put in a shifting magnetic field. Therefore a shifting magnetic field in a coil of wire would induce an emf in the coil which in turn induces the flow of current.Because of the variations in the magnetic flux through it, it can be described as the generation of a potential difference in a coil. In simpler terms, as the flux connected to a conductor or coil increases, the electromotive force or EMF is said to be caused.