question_answer

A uniform but time-varying magnetic field B(t) exists in a circular region of radius a and is directed into the plane of the paper, as shown. The magnitude of the induced electric field at point P at a distance r from the centre of the circular region                                              [IIT-JEE (Screening) 2000]

A)            Is zero

B)            Decreases as \[\frac{1}{r}\]

C)            Increases as r

D)            Decreases as \[\frac{1}{{{r}^{2}}}\]

Correct Answer: B

Solution :

                   Construct a concentric circle of radius r. The induced electric field (E) at any point on the circle is equal to that at P. For this circle \[\oint{\vec{E}.d\vec{l}=\left| \frac{d\varphi }{dt} \right|=A\ \left| \frac{dB}{dt} \right|}\]                    or \[E\times (2\pi r)=\pi {{a}^{2}}.\left| \frac{dB}{dt} \right|\]                    \[\Rightarrow E=\frac{{{a}^{2}}}{2r}\left| \frac{dB}{dt} \right|\Rightarrow E\propto \frac{1}{r}\]