Question

A metallic rod of length $l$ is hinged at the point $M$ and is rotating about an axis perpendicular to the plane of paper with a constant angular velocity $\omega$. A uniform magnetic field of intensity $B$ is acting in the region (as shown in the figure) parallel to the plane of paper. The potential difference between the points $M$ and $N$.

• A. is always zero
• B. varies between $\frac{1}{2}B\omega l^2$ to $0$
• C. is always $\frac{1}{2}B\omega l^2$
• D. is always $B\omega l^2$

Answer 1

The correct option is A is always zero

Let an elementary part of width $dx$.

Total induced Emf across the rod is given by $E={\int }_0^l(\overrightarrow{v}\times \overrightarrow{B}).\overrightarrow{dx}$

As according to the figure, $\overrightarrow{B}=B\hat{x}$ and $\overrightarrow{v}=-v\hat{y}=-wx\hat{y}$

$\therefore$ $\overrightarrow{v}\times \overrightarrow{B}=wxB\hat{k}$

$⟹(\overrightarrow{v}\times \overrightarrow{B}).\overrightarrow{dx}=\left(wxB\hat{k}\right).dx$ $\hat{x}=0$

Note: At every instant, $(\overrightarrow{v}\times \overrightarrow{B})\perp$ $\overrightarrow{dx}$

Thus total Emf induced is always $Zero.$