Question

Two charges of equal magnitude ‘$q$’ are placed in air at a distance ‘$2a$’ apart and third charge ‘$-2q$’ is placed at midpoint. 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

Answer: (D)

Potential energy of the system

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