Advanced Applications: Coulomb's Law & Simple Harmonic Motion

While the fundamental statement of Coulomb's Law dictates the force between two stationary charges, JEE Advanced problems rarely stop there. A common archetype involves placing a system of charges in equilibrium and analyzing what happens when that equilibrium is slightly disturbed.

The Two-Charge System: Axial and Equatorial Displacement

Consider two identical positive point charges, $+Q$, fixed at positions $(d, 0)$ and $(-d, 0)$. A third test charge, $q$, is placed at the origin $(0,0)$. By symmetry, the net electrostatic force on $q$ is zero. The system is in equilibrium.

However, the nature of this equilibrium depends entirely on the sign of $q$ and the axis of displacement.

Case 1: Displacing a Positive Charge $+q$ along the X-axis (Axial)

Let's displace $+q$ by a small distance $x$ ($x \ll d$) along the positive x-axis. The force from the right charge increases (distance decreases to $d-x$), and the force from the left charge decreases (distance increases to $d+x$).

The net restoring force $\vec{F}_{net}$ directed towards the origin is:

$F_{net} = \frac{1}{4\pi\epsilon_0} \frac{Qq}{(d-x)^2} - \frac{1}{4\pi\epsilon_0} \frac{Qq}{(d+x)^2}$

Factoring out the constants and $d^2$:

$F_{net} = \frac{Qq}{4\pi\epsilon_0 d^2} \left[ \left(1 - \frac{x}{d}\right)^{-2} - \left(1 + \frac{x}{d}\right)^{-2} \right]$

Since $x \ll d$, we use the binomial approximation $(1+z)^n \approx 1 + nz$:

$F_{net} \approx \frac{Qq}{4\pi\epsilon_0 d^2} \left[ \left(1 + 2\frac{x}{d}\right) - \left(1 - 2\frac{x}{d}\right) \right] = \frac{Qq}{4\pi\epsilon_0 d^2} \left( \frac{4x}{d} \right)$

$F_{net} = -\left( \frac{Qq}{\pi\epsilon_0 d^3} \right) x$

Because $F_{net} \propto -x$, the charge executes Simple Harmonic Motion (SHM). The effective spring constant is $k_{eff} = \frac{Qq}{\pi\epsilon_0 d^3}$, giving a time period of:

$T = 2\pi \sqrt{\frac{m}{k_{eff}}} = 2\pi \sqrt{\frac{m \pi \epsilon_0 d^3}{Qq}}$

Equilibrium Summary Matrix

JEE Pro-Tip: Earnshaw's Theorem dictates that a charged particle cannot be in a stable equilibrium in all three spatial dimensions relying solely on electrostatic forces. If it is stable along the x-axis, it must be unstable along the y or z-axis.