Question

A charge ${10}^{-9}\text{C}$ is located at the origin of a coordinate system and another charge $Q$ at $(2,0,0)$. If the $x-$ component of the electric field at $(3,1,1)$ is zero, calculate the value of $Q$.

• A. $4.3\times {10}^{-10}\text{C}$
• B. $-2.3\times {10}^{-10}\text{C}$
• C. $-4.3\times {10}^{-10}\text{C}$
• D. $2.3\times {10}^{-10}\text{C}$

Answer 1

The correct option is C $-4.3\times {10}^{-10}\text{C}$


Electric field due to a point charge $q_i$ at position ${\overrightarrow{r}}_i$ in vector form is expressed as
${\overrightarrow{E}}_i=\frac{kq}{r_i^3}{\overrightarrow{r}}_i$

Here,
${\overrightarrow{r}}_1=(3-0)\hat{i}+(1-0)\hat{j}+(1-0)\hat{k}=3\hat{i}+\hat{j}+\hat{k}$
& $r_1=\sqrt{3^2+1^2+1^2}=\sqrt{11}\text{m}$

${\overrightarrow{r}}_2=(3-2)\hat{i}+(1-0)\hat{j}+(1-0)\hat{k}=1\hat{i}+1\hat{j}+1\hat{k}$
& $r_2=\sqrt{1^2+1^2+1^2}=\sqrt{3}\text{m}$

Thus, electric fields due to the charges are,
${\overrightarrow{E}}_1=\frac{k\times {10}^{-9}}{{11}^{3/2}}[3\hat{i}+\hat{j}+\hat{k}]$
${\overrightarrow{E}}_2=\frac{k\times Q}{3^{3/2}}[\hat{i}+\hat{j}+\hat{k}]$

According to the given condition, ${\overrightarrow{E}}_x=0$
$\Rightarrow {\overrightarrow{E}}_{x_1}+{\overrightarrow{E}}_{x_2}=0$
$\Rightarrow \frac{3k\times {10}^{-9}}{{11}^{3/2}}=-\frac{k\times Q}{3^{3/2}}$
$\Rightarrow Q=-4.3\times {10}^{-10}\text{C}$