Question

Two equal charges \(q\) are kept fixed at \(-a\) and \(+a\) along the \(x\)-axis. A particle of mass \(m\) and charge \(\dfrac{q}{2}\) is brought to the origin and given a small displacement along the \(x\)-axis, then

• A. the particle executes oscillatory motion
• B. the particle remains stationary
• C. the particle executes, SHM along \(x\)-axis
• D. the particle executes SHM along \(y\)-axis

Hint:

For two like charges, the midpoint along the axis is unstable equilibrium, while the perpendicular bisector is stable equilibrium -- giving SHM for transverse displacements.

Answer 1

The Correct Option is D

Step 1: Basic Principle
At the midpoint (origin), the forces from both charges cancel. For displacement along \(x\)-axis, it is unstable equilibrium; for displacement along \(y\)-axis, restoring force acts.
Step 2: Solution Procedure:
If displaced along \(y\)-axis by small \(y\), the net force is directed back towards origin (restoring), producing SHM. Along \(x\)-axis, displacement leads to a repulsive force away from origin (unstable).
Step 3: Required Answer:
The particle executes SHM along the \(y\)-axis.