Question

A conducting rod $\text{PQ}$ of length $5\text{m}$ oriented as shown in figure is moving with velocity $2\hat{i}\text{m/s}$, without rotation, in a uniform magnetic field of $(3\hat{j}+4\hat{k})\text{Tesla}$. The induced emf $(\text{in V}$) in the rod is

Answer 1

Given,

Velocity of rod $\text{PQ},\overrightarrow{v}=\left(2\hat{i}\right)\text{m/s}$

Magnetic field, $\overrightarrow{B}=(3\hat{j}+4\hat{k})\text{T}$


From the figure, we can see that,

Length vector for rod $\text{PQ}$,

$\overrightarrow{l}=(5\cos {53}^{\circ }\hat{i}+5\sin {53}^{\circ }\hat{j})\text{m/s}$

$\overrightarrow{l}=(3\hat{i}+4\hat{j})\text{m/s}$

$\therefore$ The induced emf is given by,

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

Here,
$\overrightarrow{v}\times \overrightarrow{B}=\hat{i}(0-0)-\hat{j}(8-0)+\hat{k}(6-0)$

$=6\hat{k}-8\hat{j}$

$\therefore E=(3\hat{i}+4\hat{j}).(6\hat{k}-8\hat{j})=-32\text{V}$

Considering the magnitude only, $|E|=32\text{V}$

Accepted answers: $32,32.0,32.00$