Question

Four electric charges $+q,+q,-q$ and $-q$ are placed at the corners of a square of side $2L$ (see figure). The electric potential at point A, midway between the two charges $+q$ and $+q$, is

• A. $\frac{1}{4\pi {ϵ}_0}\frac{2q}{L}(1+\sqrt{5})$
• B. $\frac{1}{4\pi {ϵ}_0}\frac{2q}{L}(1+\frac{1}{\sqrt{5}})$
• C. $\frac{1}{4\pi {ϵ}_0}\frac{2q}{L}(1-\frac{1}{\sqrt{5}})$
• D. zero

Answer 1

The correct option is C

$\frac{1}{4\pi {ϵ}_0}\frac{2q}{L}(1-\frac{1}{\sqrt{5}})$

Here, $AB=AE=L$ and $AC=AD=\sqrt{A{B}^2+B{C}^2}=\sqrt{L^2+(2L{)}^2}=\sqrt{5}L$

Potential at A is $V_A=\frac{1}{4\pi {ϵ}_0}[\frac{q}{AB}+\frac{-q}{AC}+\frac{-q}{AD}+\frac{q}{AE}]$

or $V_A=\frac{1}{4\pi {ϵ}_0}[\frac{q}{L}+\frac{-q}{\sqrt{5}L}+\frac{-q}{\sqrt{5}L}+\frac{q}{L}]=\frac{1}{4\pi {ϵ}_0}\frac{2q}{L}[1-\frac{1}{\sqrt{5}}]$