Question

Two charges $-4\mu \text{C}$ and $+4\mu \text{C}$ are placed at the points $\text{A}(1,0,4)$ and $\text{B}(2,-1,5)$ located in an uniform electric field $\overrightarrow{E}=20\hat{i}\text{N/C}$. The torque acting on the dipole has a magnitude of

• A. $0.1\times {10}^{-4}\text{Nm}$
• B. $1.13\times {10}^{-4}\text{Nm}$
• C. $2.5\times {10}^{-4}\text{Nm}$
• D. $3.7\times {10}^{-4}\text{Nm}$

Answer 1

The correct option is B $1.13\times {10}^{-4}\text{Nm}$

Given:
$q=4\mu \text{C};\overrightarrow{E}=20\hat{i}\text{N/C}$

End points of dipole are $\text{A}(1,0,4)$ and $\text{B}(2,-1,5)$.

So, the distance between charges of the dipole
$2\overrightarrow{l}=\overrightarrow{\text{AB}}$

$\Rightarrow 2\overrightarrow{l}=\left[\right(2-1)\hat{i}+(-1-0)\hat{j}+(5-4\left)\hat{k}\right]$

$\Rightarrow 2\overrightarrow{l}=(\hat{i}-\hat{j}+\hat{k})$

Torque acting on dipole placed in electric field is given by

$\overrightarrow{\tau }=\overrightarrow{p}\times \overrightarrow{E}=\left[q\right(2\overrightarrow{l}\left)\right]\times \overrightarrow{E}$

$\Rightarrow \overrightarrow{\tau }=\left[q\right(\hat{i}-\hat{j}+\hat{k}\left)\right]\times 20\hat{i}$

As, $\hat{i}\times \hat{i}=0;\hat{j}\times \hat{i}=-\hat{k};\hat{k}\times \hat{i}=\hat{j}$

$\Rightarrow \overrightarrow{\tau }=4\times {10}^{-6}\times 20(\hat{k}+\hat{j})=8\times {10}^{-5}(\hat{k}+\hat{j})$

So, for magnitude,

$|\overrightarrow{\tau }|=8\times {10}^{-5}\sqrt{(1{)}^2+(1{)}^2}=8\times {10}^{-5}\times \sqrt{2}=1.13\times {10}^{-4}\text{Nm}$

So, option $\left(b\right)$ is the correct answer.