Question

A frame $ABCD$ is rotating with angular velocity $\omega$ about an axis which passes through the point $O$ perpendicular to the plane of paper as shown in the figure. A uniform magnetic field $B$ is applied into the plane of the paper in the region as shown . Choose the correct statement(s):

• A.
  Potential difference between $A$ and $O$ is $B\omega L^2$
• B. Potential difference between $O$ and $D$ is $B\omega L^2$
• C. Potential difference between $C$ and $D$ is $\frac{B\omega L^2}{2}$
• D. Potential difference between $A$ and $D$ is zero

Answer 1

Answer: (A), (B), (C), (D)

Potential difference between $A$ and $D$ is zero

For a rod rotating in a perpendicular uniform magnetic field of intensity $B$ with angular velocity $\omega$,
emf induced between any point on the rod and axis of rotation is $\frac{B\omega l^2}{2}$
where $l$ is perpendicular distance between point and axis of rotation.

Emf induced between $AO$ is
$V_{AO}=\frac{1}{2}B\omega (AO{)}^2$
$V_{AO}=\frac{1}{2}B\omega \left(2L^2\right)=B\omega L^2....\left(a\right)$
Emf induced between $DO$ is
$V_{DO}=\frac{1}{2}B\omega \left(O{D}^2\right)=B\omega L^2....\left(b\right)$
Emf induced between $CO$ is
$V_{CO}=\frac{1}{2}B\omega L^2$
Emf induced between $DC$ is
$\to V_{DC}=V_{DO}-V_{CO}=B\omega L^2-\frac{1}{2}B\omega L^2=\frac{1}{2}B\omega L^2....\left(c\right)$
Similarly
$\to V_A=V_D$
So, $V_A-V_D=0....\left(d\right)$