Question

A uniform but time varying magnetic field $B\left(t\right)$ exists in a cylindrical region of radius $a$ and is directed into the plane of the paper, as shown in figure. The magnetic field decreases at a constant rate inside the region. If $r$ is the distance from the axis of the cylindrical region, then match column I with column II.

Answer 1

Electric field at a point lie inside the cylinder $E\left(r\right)=\frac{r}{2}\frac{dB}{dt}$ for $r<a$

$⟹$ Electric field at B increases as $r$

Electric field at a point lie outside the cylinder $E\left(r\right)=\frac{a^2}{2r}\frac{dB}{dt}$ for $r>a$

$⟹$ Electric field at A decreases as $\frac{1}{r}$

Both the electric fields are in clockwise direction.

$\therefore$ Electric field at A as well as at B is directed along 1

As force on the electron $\overrightarrow{F_e}=-q\overrightarrow{E}$ i.e $F_e$ acts in opposite direction to that of $E\left(r\right)$

$\therefore$ Force on electron placed at A as well as as at $B$ acts along 3

Also $F_e$ decreases as $\frac{1}{r}$ for A whereas increases as $r$ for B.