Time Varying Magnetic Field - Practice Questions & MCQ

As we learn Induced electric field is given by

$
\varepsilon=\oint \overrightarrow{E_{i n}} \cdot \overrightarrow{d l}=\frac{-d \phi}{d t}
$

But using $\phi=B . A_{\text {so we can also write }}$

$
\varepsilon=\oint \overrightarrow{E_{i n}} \cdot \overrightarrow{d l}=\frac{-d \phi}{d t}=-A \frac{d B}{d t}
$

Where
$\mathrm{A} \rightarrow$ constant Area
$B \rightarrow$ Varying Magnetic field
For example-
A uniform but time-varying magnetic field $\mathrm{B}(\mathrm{t})$ exists in a circular region of radius ' a ' and is directed into the plane of the paper as shown in the below figure, the magnitude of the induced electric field ( $E_{\text {in }}$ ) at point P lies at a distance $r$ from the centre of the circular region is calculated as follows.

As due to the time-varying magnetic field induced electric field will be produced whose electric field lines are concentric circular closed curves of radius $r$.

$
\text { if } r \leq a
$

then $E_{\text {in }}(2 \pi r)=\pi r^2\left|\frac{d B}{d t}\right|$

$
\Rightarrow E_{\text {in }}=\frac{r}{2}\left|\frac{d B}{d t}\right|
$
 

For $r>R$,

$
\begin{aligned}
& E_{\text {in }} * 2 \pi r=\pi a^2\left|\frac{d B}{d t}\right| \\
& \Rightarrow E_{\text {in }}=\frac{a^2}{2 r}\left|\frac{d B}{d t}\right|
\end{aligned}
$

- The graph of E vs r

where E=induced electric field