Question

A conducting rod $\text{PQ}$ of length $5\text{m}$ oriented as shown in figure is moving with velocity $2\hat{i}{\text{ms}}^{-1}$ without any rotation in a uniform magnetic field $(3\hat{j}+4\hat{k})\text{T}.$ The emf induced in the rod is

• A. $36\text{V}$
• B. $34\text{V}$
• C. $32\text{V}$
• D. $38\text{V}$

Answer 1

The correct option is C $32\text{V}$

Given,
$\overrightarrow{v}=2{\text{ms}}^{-1}\hat{i};l=5\text{m}\overrightarrow{B}=3\hat{j}+4\hat{k}\text{T}$

$\overrightarrow{l}=l\cos 53\hat{i}+l\sin 53\hat{j}$

$\overrightarrow{l}=(5\times \frac{3}{5}\hat{i})+(5\times \frac{4}{5}\hat{j})$

$\overrightarrow{l}=3\hat{i}+4\hat{j}\text{m}$

The induced e.m.f in the rod $\text{PQ}$ is,

$\left|\begin{array}{c}\epsilon \end{array}\right|=|\overrightarrow{l}.(\overrightarrow{v}\times \overrightarrow{B})|$

$\epsilon =|(3\hat{i}+4\hat{j}).[2\hat{i}\times (3\hat{j}+4\hat{k}\left)\right]|$

$\epsilon =|(3\hat{i}+4\hat{j}).(6\hat{k}-8\hat{j})|$

$\epsilon =32\text{V}$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(C\right)$ is the correct answer.