Question

Figure shows a circular region of radius $R$ in which uniform magnetic field $B$ exists. The magnetic field is increasing at a rate $\frac{dB}{dt}$. The induced electric field at a distance $r$ from the centre for $r<R$ is

• A. $\frac{dB}{dt}\frac{r}{2}$
• B. Zero
• C. $\frac{dB}{dt}$
• D. $\frac{dB}{dt}\frac{R^2}{{2r}^2}$

Answer 1

The correct option is A $\frac{dB}{dt}\frac{r}{2}$

A time varying magnetic field always produces induced electric field in all space surrounding it.

Now,

$E=+\oint \overrightarrow{E}.\overrightarrow{dl}$

And, $E=-\frac{d\varphi }{dt}$

$\oint E_{in}.\overrightarrow{dl}=-\frac{d\varphi }{dt}$

$\oint E_{in}.\overrightarrow{dl}=A\frac{dB}{dt}$

$E\int dl=\pi r^2\frac{dB}{dt}$

As induced electric field $E$ at any point on the circle will be same.

$\therefore E\times 2\pi r=\pi r^2\left(\frac{dB}{dt}\right)$

$E=\frac{dB}{dt}\times \frac{r}{2}$