Question

Three point charges, each of charge \( q \), are placed on vertices of a triangle \( ABC \), with \( AB = AC = 5L \), \( BC = 6L \). The electrostatic potential at the midpoint of side \( BC \) will be:

• A. \( \frac{11q}{48\pi \epsilon_0 L} \)
• B. \( \frac{8q}{36\pi \epsilon_0 L} \)
• C. \( \frac{5q}{24\pi \epsilon_0 L} \)
• D. \( \frac{q}{16\pi \epsilon_0 L} \)

Hint:

The electrostatic potential at a point due to multiple charges is the algebraic sum of the potentials due to individual charges.

Answer 1

The Correct Option is A

Electrostatic Potential at the Midpoint of \( BC \)

1: Formula for Electrostatic Potential
- The electrostatic potential at a point due to a charge \( q \) at distance \( r \) is: \[ V = \frac{1}{4\pi \epsilon_0} \frac{q}{r} \]
2: Contribution of Charges at \( B \) and \( C \) - The midpoint of \( BC \) is equidistant from charges at \( B \) and \( C \).
- Distance of midpoint from \( B \) or \( C \): \[ r_{B} = r_C = \frac{BC}{2} = \frac{6L}{2} = 3L \] - Potential due to charge at \( B \): \[ V_B = \frac{1}{4\pi \epsilon_0} \frac{q}{3L} \] - Potential due to charge at \( C \): \[ V_C = \frac{1}{4\pi \epsilon_0} \frac{q}{3L} \] - Total contribution from \( B \) and \( C \): \[ V_{BC} = 2 \times \frac{q}{12\pi \epsilon_0 L} = \frac{2q}{12\pi \epsilon_0 L} = \frac{q}{6\pi \epsilon_0 L} \]
3: Contribution of Charge at \( A \)
- Distance from \( A \) to the midpoint of \( BC \) (Using Apollonius theorem): \[ r_A = \sqrt{(5L)^2 - (3L)^2} = \sqrt{25L^2 - 9L^2} = \sqrt{16L^2} = 4L \] - Potential due to charge at \( A \): \[ V_A = \frac{1}{4\pi \epsilon_0} \frac{q}{4L} = \frac{q}{16\pi \epsilon_0 L} \]
4: Total Electrostatic Potential at Midpoint \[ V = V_{BC} + V_A \] \[ V = \frac{q}{6\pi \epsilon_0 L} + \frac{q}{16\pi \epsilon_0 L} \] \[ V = \frac{16q + 6q}{96\pi \epsilon_0 L} \] \[ V = \frac{22q}{96\pi \epsilon_0 L} = \frac{11q}{48\pi \epsilon_0 L} \] Thus, the electrostatic potential at the midpoint of \( BC \) is: \[ \frac{11q}{48\pi \epsilon_0 L} \] which matches option (A).