Question

The magnetic field at all points within the cylindrical region whose cross-section is indicated in the accompanying figure starts increasing at a constant rate $\beta$. The induced electric field $E$ satisfies


$E\propto r^p(r<R)$ and $E\propto r^{-q}(r>R)$
where, $r$ is the distance from the axis of the region. Find the value of $p+q$.(Integer only)

Answer 1

$\text{Case 1 :}$ For $r<R$


Using, $E\times \left(2\pi r\right)=A\left|\begin{array}{c}\frac{dB}{dt}\end{array}\right|$

$\Rightarrow E\times 2\pi r=\left(\pi r^2\right)\times \beta [\because \frac{dB}{dt}=\beta \text{given}]$

Or, $E=\frac{1}{2}\beta r$

$\therefore E\propto r^1$

$\Rightarrow p=1$

$\text{Case 2 :}$ For $r>R$


Using, $E\times \left(2\pi r\right)=A\left|\begin{array}{c}\frac{dB}{dt}\end{array}\right|$

$\Rightarrow E\times 2\pi r=\pi R^2\times \beta$

[ $\because$ magnetic field is present up to $R$ only]

$\Rightarrow E=\frac{\beta R^2}{2r}$

$\Rightarrow E\propto \frac{1}{r}$ Or, $E\propto r^{-1}$

$\Rightarrow q=1$

$\therefore p+q=1+1=2$

Hence, Correct answer $=2$