Four point charges $-Q,-q, 2 q$ and $2 Q$ are placed, one at each corner of the square. The relation between $Q$ and $q$ for which the potential at the centre of the square is zero is

• Correct answer
  A

  $Q=-q$
• B

  $Q=-1 / q$
• C

  $Q=q$
• D

  $Q=1 / q$

Solution

Let $a$ be the side length of the square $A B C D$. $\begin{aligned} & A C=B D=\sqrt{a^2+a^2}=a \sqrt{2} \\ & O A=O B=O C=O D=\frac{a \sqrt{2}}{2}=\frac{a}{\sqrt{2}}\end{aligned}$ Potential is a scalar quantity. Potential at the centre $O$ due to given charge configuration is $\begin{aligned} & V=\frac{1}{4 \pi \varepsilon_0}\left[\frac{(-Q)}{\left(\frac{a}{\sqrt{2}}\right)}+\frac{(-q)}{\left(\frac{a}{\sqrt{2}}\right)}+\frac{(2 q)}{\left(\frac{a}{\sqrt{2}}\right)}+\frac{2 Q}{\left(\frac{a}{\sqrt{2}}\right)}\right]=0 \\ & \Rightarrow-Q-q+2 q+2 Q=0 \\ & \text { or } Q+q=0 \text { or } Q=-q\end{aligned}$

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