An electric charge $10^{-3} \mu \mathrm{C}$ is placed at the origin $(0,0)$ of $(x-y)$ co-ordinate system. Two points $A$ and $B$ are situated at $(\sqrt{2}, \sqrt{2})$ and $(2,0)$ respectively. The potential difference between the points $A$ and $B$ will be

• A

  4.5 volt
• B

  9 volt
• Correct answer
  C

  zero
• D

  2 volt

Solution

$$ \begin{aligned} & \text { } \overrightarrow{r_1}=\sqrt{2} \hat{i}+\sqrt{2} \hat{j} \\ & \left|\vec{r}_1\right|=r_1=\sqrt{(\sqrt{2})^2+(\sqrt{2})^2}=2 \end{aligned} $$ $$ \begin{aligned} & \vec{r}_2=2 \hat{i}+0 \hat{j} \\ & \text { or }\left|\vec{r}_2\right|=r_2=2 \end{aligned} $$ Potential at point $A$ is $$ \begin{aligned} & V_A=\frac{1 q}{4 \pi \varepsilon_0 r_1} \\ & =\frac{1}{4 \pi \varepsilon_0} \frac{10^{-3} \times 10^{-6}}{2} \end{aligned} $$ Potential at point $B$ is
$$ \begin{aligned} & \quad V_B=\frac{1}{4 \pi \varepsilon_0} \frac{q}{r_2}=\frac{1}{4 \pi \varepsilon_0} \frac{10^{-3} \times 10^{-6}}{2} \\ & \therefore \quad V_A-V_B=0 . \end{aligned} $$

Hence, the correct answer is option: 'C'.