Three charges are arranged at the corner of a square ABCD of side 'a'. At A '(+q)', at B (-q), at C (+q) in coulomb. Find the work needed to bring +q to point D of the square when given other charges at A, B and C are fixed.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{q^{2}}{4\pi\varepsilon_0 a}\left(2-\frac{1}{\sqrt{2}}\right) \)
The work done in bringing a charge from infinity to a point is the charge times the potential at that point, \( W = qV_D \). The potential at D is the sum of the potentials of the three fixed charges (superposition).
In the square ABCD, the corner D is adjacent to A and C, each at distance a, and diagonally opposite to B, at distance \( a\sqrt{2} \).
\( \displaystyle V_D = \frac{1}{4\pi\varepsilon_0}\left[\frac{q}{a} + \frac{q}{a} + \frac{(-q)}{a\sqrt{2}}\right] = \frac{q}{4\pi\varepsilon_0 a}\left(2 - \frac{1}{\sqrt{2}}\right) \)
\( \displaystyle W = qV_D = \frac{q^{2}}{4\pi\varepsilon_0 a}\left(2 - \frac{1}{\sqrt{2}}\right) \)
Shortcut: Work has the dimension of \( q^{2}/(\varepsilon_0 a) \), so Options (3) and (4), which carry \( a^{2} \) in the denominator, are ruled out at once. Of the other two, the net potential at D is positive (two nearby \( +q \) charges outweigh the distant \( -q \)), so the work is positive, which leaves Option (2).