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: Pair list
There are three pairs: A-B, A-C and B-C.

Step 2: Pair energies in units of $k q^2/x$
A-B: $+\frac{1}{3}$. A-C: $-3$. B-C: $-\frac{3}{2}$.

Step 3: Add
$0.333 - 3 - 1.5 = -4.17$ in units of $\dfrac{kq^2}{x}$ with $k = \dfrac{1}{4\pi\varepsilon_0}$.

Step 4: Choose
The closest listed value is $-4$, option (D). The attractive pairs dominate, so the total is negative, ruling out (A) and (C).

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