Question

Two charges of equal magnitude $q$ are placed in air at a distance $2a$ apart and third charge $-2q$ is placed at mid-point. The potential energy of the system is (${\epsilon }_0=$ permittivity of free space) :

• A. $-\frac{q^2}{8\pi {\epsilon }_{0}a}$
• B. $-\frac{3q^2}{8\pi {\epsilon }_{0}a}$
• C. $-\frac{5q^2}{8\pi {\epsilon }_{0}a}$
• D. $-\frac{7q^2}{8\pi {\epsilon }_{0}a}$

Answer 1

The correct option is D $-\frac{7q^2}{8\pi {\epsilon }_{0}a}$

Potential energy of the system
$U=\frac{1}{4\pi {\epsilon }_0}[\frac{q_1{q}_2}{r_{12}}+\frac{q_1{q}_3}{r_{13}}+\frac{q_2{q}_3}{r_{23}}]$
$=\frac{1}{4\pi {\epsilon }_0}[\frac{q(-2q)}{a}+\frac{q(-2q)}{a}+\frac{qq}{2a}]$
$=\frac{1}{4{\pi \epsilon }_0}[-\frac{2q^2}{a}-\frac{2q^2}{a}+\frac{q^2}{2a}]$
$=\frac{1}{4\pi {\epsilon }_0}[\frac{-4q^2}{a}+\frac{q^2}{2a}]=\frac{-7q^2}{8\pi {\epsilon }_{0}a}$