Question

Two equal positive charges each of value 'q' are placed at points A and B, where \(AB = 3x\). A third charge \(-3q\) is placed at point C at a distance \(x\) from A on AB. The potential energy of the system is nearly (\(ε_0\) = permittivity of free space)

• A. \(\frac{q^2}{4πε_0x}\)
• B. \(\frac{-2q^2}{4πε_0x}\)
• C. \(\frac{3q^2}{4πε_0x}\)
• D. \(\frac{-4q^2}{4πε_0x}\)

Hint:

Add the pair energies kq1q2/r for all three pairs.

Answer 1

The Correct Option is D

Step 1: Understand the concept
The potential energy of a system of point charges is the sum of \(\dfrac{q_iq_j}{4\pi\varepsilon_0 r_{ij}}\) over every pair of charges.

Step 2: Find the distances
\(AB = 3x\) and \(AC = x\), so \(CB = 3x - x = 2x\). The charges are \(q\) at A, \(q\) at B and \(-3q\) at C.

Step 3: Add the three pair energies
\[ U = \frac{1}{4\pi\varepsilon_0}\left[\frac{q\cdot q}{3x} + \frac{q(-3q)}{x} + \frac{q(-3q)}{2x}\right] = \frac{q^2}{4\pi\varepsilon_0x}\left[\frac{1}{3} - 3 - \frac{3}{2}\right] \]

Step 4: Evaluate
The bracket is \(\dfrac{2 - 18 - 9}{6} = -\dfrac{25}{6} \approx -4.17\). So \(U \approx \dfrac{-4q^2}{4\pi\varepsilon_0x}\), which is nearly option (D). The question says "nearly", so rounding \(-4.17\) to \(-4\) is intended.

Final Answer:
The potential energy is nearly -4 q^2 / (4 pi epsilon_0 x). This is option (D). \[ \boxed{\text{(D) }\frac{-4q^2}{4\pi\varepsilon_0x}} \]