Question

Identical charges each of value q are placed at the four corners A, B, C and D of a square of side a. The magnitude of the force on the charge at any one corner of the square is

• A. $\frac{3q^2}{4\pi {ϵ}_0{a}^5}$
• B. $\frac{q^2}{4\pi {ϵ}_0{a}^2}$
• C. $\left(\frac{1+2\sqrt{2}}{2}\right)\frac{q^2}{4\pi {ϵ}_0{a}^2}$
• D. $\left(\frac{1+2\sqrt{2}}{\sqrt{2}}\right)\frac{q^2}{4\pi {ϵ}_0{a}^2}$

Answer 1

The correct option is C $\left(\frac{1+2\sqrt{2}}{2}\right)\frac{q^2}{4\pi {ϵ}_0{a}^2}$


$F_{12}=F_{14}=\frac{q^2}{4\pi {ϵ}_0{a}^2}$ and $F_{12}\perp F_{14}$
Resultant of $F_{12}$ and $F_{14}$ is
$\sqrt{2}\frac{q^2}{4\pi {ϵ}_0{a}^2}$ along $F_{13}$
Further $F_{13}=\frac{q^2}{4\pi {ϵ}_0(\sqrt{2}a{)}^2}$
So total force on 1 is $F_1=F_{12}+F_{13}$
$\Rightarrow F_1=\frac{q^2}{4\pi {ϵ}_0{a}^2}(\frac{1}{2}+\sqrt{2})\Rightarrow F_1=\left(\frac{1+2\sqrt{2}}{2}\right)\frac{q^2}{4\pi {ϵ}_0{a}^2}$