Two equal negative charge –q are fixed at point (0, –a) and (0, a) on…

The correct answer is Option 4: execute oscillatory but not simple harmonic motion

1. Net force on the charge

At any point $x$ on the x-axis, forces due to the two charges are symmetric:

• Vertical components cancel
• Horizontal components add toward origin

So, net force is along x-axis and given by:

$F = -\frac{2kQqx}{(x^2 + a^2)^{3/2}}$

Negative sign shows restoring force toward origin

2. Nature of motion

• At $x = 2a$, charge moves toward origin
• At origin $\rightarrow$ force $= 0$ but velocity is maximum
• Due to inertia, it crosses origin
• After crossing $\rightarrow$ force reverses direction and pulls it back

Hence, motion is to and fro (oscillatory)

3. Why not SHM?

For SHM:

$F = -kx$

But here:

$F = -\frac{2kQqx}{(x^2 + a^2)^{3/2}}$

• Force is not directly proportional to $x$
• Denominator depends on $x \rightarrow$ non-linear

So, motion is not simple harmonic