Question

Identical charges -q each are placed at 8 corners of a cube of each side b. Electrostatic potential energy of a charge +q which is placed at the centre of cube will be :

• A. $\frac{-4\sqrt{2}{q}^2}{\pi {ϵ}_{0}b}$
• B. $\frac{-8\sqrt{3}{q}^2}{\pi {ϵ}_{0}b}$
• C. $\frac{-4q^2}{\sqrt{3}\pi {ϵ}_{0}b}$
• D. $\frac{-8\sqrt{2}{q}^2}{\pi {ϵ}_{0}b}$

Answer 1

Answer: (C)

$\frac{-4q^2}{\sqrt{3}\pi {ϵ}_{0}b}$

Here, $AD=CD=OC=b/2$ and $A{C}^2=A{D}^2+C{D}^2=\frac{b^2}{4}+\frac{b^2}{4}=\frac{b^2}{2}$

Thus distance of each charge from centre is $OA=r=\sqrt{A{C}^2+O{C}^2}=\sqrt{\frac{b^2}{2}+\frac{b^2}{4}}=\frac{\sqrt{3}b}{2}$

Potential energy at the centre is $W=\frac{8(-q)q}{4\pi {ϵ}_{0}r}=\frac{8(-q)q}{4\pi {ϵ}_0\left(\frac{\sqrt{3}b}{2}\right)}=\frac{-4q^2}{\sqrt{3}\pi {ϵ}_{0}b}$