Question

Equal charges $Q$ are placed at the four corners $A,B,C,D$ of a square of length $a$. The magnitude of the force on the charge at $B$ will be

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

Answer 1

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



Net force on the charge at $B$ is shown in the following figure.

$F_{net}=F_{AC}+F_D=\sqrt{(F_A^2+F_C^2}+F_D$
Since $F_A=F_C=\frac{k{Q}^2}{a^2}$
and $F_D=\frac{k{Q}^2}{(a\sqrt{2}{)}^2}$
$F_{net}=\frac{\sqrt{2}k{Q}^2}{a^2}+\frac{\left(k{Q}^2\right)}{\left(2a^2\right)}=\frac{k{Q}^2}{a^2}(\sqrt{2}+\frac{1}{2})$
$=\frac{Q^2}{4\pi {ϵ}_0{a}^2}\left(\frac{1+2\sqrt{2}}{2}\right)$

Hence, option (C) is correct.