Question

Two charges of $-4\mu \text{C}$ and $+4\mu \text{C}$ are placed at points $\text{A}(1,0,4)$ and $\text{B}(2,-1,5)$ located in an electric field $\overrightarrow{E}=0.20\hat{i}{\text{Vcm}}^{-1}$. The torque acting on the dipole is

• A. $8\times {10}^{-5}\text{Nm}$
• B. $\frac{8}{\sqrt{2}}\times {10}^{-5}\text{Nm}$
• C. $8\sqrt{2}\times {10}^{-5}\text{Nm}$
• D. $2\sqrt{2}\times {10}^{-5}\text{Nm}$

Answer 1

The correct option is C $8\sqrt{2}\times {10}^{-5}\text{Nm}$

Given,
magnitude of charges, $q=4\mu \text{C}$

electric field, $\overrightarrow{E}=0.20\hat{i}{\text{Vcm}}^{-1}=20\hat{i}{\text{Vm}}^{-1}$

position vector of point $\text{A}$,
$r_A=1\hat{i}+0\hat{j}+4\hat{k}$

position vector of point $\text{B}$,
$r_B=2\hat{i}-1\hat{j}+5\hat{k}$

We know that torque due to equal and opposite charges,
$\overrightarrow{\tau }=\overrightarrow{p}\times \overrightarrow{E}=q\left(2\overrightarrow{a}\right)\times \overrightarrow{E}...\left(1\right)$

Here,$\overrightarrow{p}=q\left(2\overrightarrow{a}\right)$ is dipole moment.

Since, the direction of dipole is from negative to positive.
Therefore,
$2\overrightarrow{a}={\overrightarrow{r}}_B-{\overrightarrow{r}}_A$

$\Rightarrow 2\overrightarrow{a}=(2-1)\hat{i}+(-1-0)\hat{j}+(5-4)\hat{k}$

$\Rightarrow 2\overrightarrow{a}=\hat{i}-\hat{j}+\hat{k}$

Substituting the values in the above equation,

$\Rightarrow \overrightarrow{\tau }=(4\times {10}^{-6})\left[\right(\hat{i}-\hat{j}+\hat{k})\times 20\hat{i}]$

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

Magnitude of torque is

$\tau =\sqrt{(8\times {10}^{-5}{)}^2+(8\times {10}^{-5}{)}^2}$

$\therefore \tau =8\sqrt{2}\times {10}^{-5}\text{Nm}$

Hence, option (c) is the correct answer.