Question

A frame $ABCD$ is rotating with an angular velocity $\omega$ about an axis passing through point $O$ perpendicular to the plane of paper as shown in the figure. A uniform magnetic field $\overrightarrow{B}$ is applied into the plane of the paper in the region as in the figure. Match the following.

Answer 1

Potential difference between the ends of a rod of length $l$ rotating about one of its end with angular speed $w$ is given by $E=\frac{Bw{l}^2}{2}$ .............(1)

(A) : Join A and O

The rod AO rotates with $w$ about O

$\therefore$ $E_{AO}=\frac{Bw(AO{)}^2}{2}=\frac{Bw(\sqrt{2}L{)}^2}{2}$ $⟹E_{AO}=Bw{L}^2=constant$

(B) : Join O and D

Similarly $E_{DO}=Bw{L}^2=constant$

(C) : The rod $OC$ rotates with $w$ about O.

$\therefore$ $E_{CO}=\frac{Bw(CO{)}^2}{2}$ $⟹E_{CO}=\frac{Bw{L}^2}{2}$

Potential between C and D $E_{DC}=E_{DO}-E_{CO}$

$\therefore E_{DC}=Bw{L}^2-\frac{Bw{L}^2}{2}=\frac{Bw{L}^2}{2}=constant$

(D) : Potential between A and D $E_{DA}=E_{DO}-E_{AO}$

$\therefore E_{DA}=Bw{L}^2-Bw{L}^2=0$