Question

Three point charges \(4q\), \(Q\), and \(q\) are placed in a straight line of length \(L\) at points 0, \(L/2\), and \(L\) respectively. The net force on charge \(q\) is zero. The value of \(Q\) is:

• A. \(4q\)
• B. \(-q\)
• C. \(-0.5q\)
• D. \(-2q\)
• E. \(3q\)

Hint:

When forces are in opposite directions, set the magnitudes equal to each other to find the value of the unknown charge.

Answer 1

The Correct Option is B

Step 1: Understanding the problem.}
We are given three charges: \( 4q \) at position 0, \( Q \) at position \( L/2 \), and \( q \) at position \( L \). The net force on charge \( q \) is zero, so we need to find the value of \( Q \) that satisfies this condition.
Step 2: Forces on charge \( q \).}
The force on charge \( q \) due to charge \( 4q \) (placed at the origin) and charge \( Q \) (placed at \( L \)) must balance each other. The force due to \( 4q \) is repulsive, and the force due to \( Q \) is attractive if \( Q \) is negative.
Step 3: Apply Coulomb's Law.}
The force between \( 4q \) and \( q \) is: \[ F_{4q \to q} = k \frac{4q^2}{L^2} \] The force between \( Q \) and \( q \) is: \[ F_{Q \to q} = k \frac{4|Q|q}{L^2} \] For the forces to balance, we set these equal: \[ k \frac{4q^2}{L^2} = k \frac{4|Q|q}{L^2} \] Simplifying: \[ q = |Q| \] Thus, the value of \( Q \) is \( -q \).