Question

A conducting rod of length $l$ is moved with a constant velocity $\overrightarrow{v}$ in a magnetic field $\overrightarrow{B}$. A potential difference appears across the two ends

• A. If $\overrightarrow{v}||\overrightarrow{l}$
• B. If $\overrightarrow{v}||\overrightarrow{B}$
• C. If $\overrightarrow{l}||\overrightarrow{B}$
• D. None of these

Answer 1

The correct option is D None of these

If $\overrightarrow{l}\parallel \overrightarrow{v}$, the area swept by the rod will be zero i.e, $A=0$.

So, the change in the flux will also be zero, i.e, $d\varphi =0$

Hence, emf is not induced in this case.


The emf is induced in the rod moving with constant velocity $\overrightarrow{v}$ in a magnetic field $\overrightarrow{B}$ is given by

$E=(\overrightarrow{v}\times \overrightarrow{B})\cdot \overrightarrow{l}$

$E=\left(vB\sin \theta \right)l\cos \varphi$

here,
$\theta =$ angle between $\overrightarrow{v}$ and $\overrightarrow{B}$
$\varphi =$ angle between $(\overrightarrow{v}\times \overrightarrow{B})$ and $\overrightarrow{l}$

So, when $\overrightarrow{v}||\overrightarrow{l}\&\overrightarrow{l}||\overrightarrow{B}\Rightarrow \varphi ={90}^{\circ }$

$\therefore E=0$
Similarly, when $\overrightarrow{v}||\overrightarrow{B}\Rightarrow \theta =0^{\circ }$

$\therefore E=0$

Hence, $\left(\text{D}\right)$ is the correct answer.