Question

A wire of length l carries a current i along the X-axis. A magnetic field exists which is given as $B=B_0(\hat{i}+\hat{j}+\hat{k})$ T. Find the magnitude of the magnetic force acting on the wire

• A. $B_{0}il$
  
• B. $B_{0}il\times \sqrt{2}$
  
• C. $2B_{0}il$
  
• D. $\frac{1}{\sqrt{2}}\times B_{0}il$

Answer 1

The correct option is B $B_{0}il\times \sqrt{2}$


By using $\overrightarrow{F}=i(\overrightarrow{l}\times \overrightarrow{B})\Rightarrow \overrightarrow{F}=i[l\hat{i}\times B_0(\hat{i}+\hat{j}+\hat{k}]=B_{0}il[\hat{i}\times (\hat{i}+\hat{j}+\hat{k}\left)\right]$
$\Rightarrow$ $\overrightarrow{F}=B_{0}il[\hat{i}\times \hat{i}+\hat{i}\times \hat{j}+\hat{i}\times \hat{k}]=B_{0}il[\hat{k}-\hat{j}]$
$\{\hat{i}\times \hat{i}=0,\hat{i}\times \hat{j}=\hat{k},\hat{i}\times \hat{k}=-\hat{j}\}$
It's magnitude $F=\sqrt{2}{B}_{0}il$