Q.Two point charges, each equal to $-q$, are held fixed on a straight line, separated by a distance $2d$ (so each is a distance $d$ from the mid-point of the line). A third charge $+q$ of mass $m$ is placed at the mid-point and is then displaced by a small distance $x$ (with $x \ll d$) in the direction perpendicular to the line joining the two fixed charges. Show that the charge $+q$ executes simple harmonic motion, and that its time period is $T = \left[\dfrac{8\pi^3 \varepsilon_0 m d^3}{q^2}\right]^{1/2}$.

Coulomb Force Superposition – From Intuition to Precision

Imagine you're in a room with three friends. Each friend can push or pull you. If two friends push you from the same side, you feel a stronger push — the combined effect. If one pushes from the left and another from the right, you feel the net effect, which might be smaller or even zero if they push equally hard.

This is exactly how electric forces work. When multiple charged particles are present, each one exerts its own force on a given charge. The total force that charge feels is simply the vector sum of all the individual forces — as if each other charge were acting alone, completely ignoring the presence of the rest.

That's the core idea: forces add like arrows, not like numbers.

The Precise Statement

Where:

• $q_0$ is the charge you're calculating the force on
• $q_i$ are all other charges (excluding $q_0$ itself)
• $r_{i0}$ is the distance between $q_i$ and $q_0$
• $\hat{r}_{i0}$ is a unit vector pointing from $q_i$ to $q_0$ (or away, depending on sign convention — be consistent)

The key point: Each pair of charges interacts independently. The presence of a third charge does not alter the force between the first two. This is what "superposition" means — the forces simply layer on top of each other.

Why This Matters (and a Common Trap)

If two forces point in opposite directions, they partially cancel. If they're at right angles, the net force is found using the Pythagorean theorem, not simple addition.

Example: Three charges on a line:

• $q_1 = +2\,\mu\text{C}$ at $x=0$
• $q_2 = -1\,\mu\text{C}$ at $x=3\,\text{cm}$
• $q_0 = +1\,\mu\text{C}$ at $x=1\,\text{cm}$

Step 1: Force from $q_1$ on $q_0$ — both positive, so repulsive. $q_0$ is pushed to the right.

Step 2: Force from $q_2$ on $q_0$ — opposite signs, so attractive. $q_0$ is pulled to the right (toward $q_2$).

Step 3: Both forces point right. Now you add magnitudes: $F_{\text{net}} = F_{1\to0} + F_{2\to0}$.

If $q_2$ were also positive, the force from $q_2$ would push $q_0$ left, and you'd subtract.

The Deeper Reason

Coulomb's law is a linear law — the force is proportional to each charge individually. If you double $q_1$, the force from $q_1$ doubles, but the force from $q_2$ stays the same. This linearity is what makes superposition possible. It's not a coincidence — it's a fundamental property of electromagnetic interactions at the classical level.

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