Equilibrium of System of Charges

For any charge placed within a system of other charges, equilibrium is achieved only when the vector sum of all forces acting on it equals zero.

Since electrostatic forces between point charges always act along the line joining them, the net torque is automatically zero whenever the net force is zero. This means students only need to verify one condition — the force condition — rather than two separate conditions (force and torque).

Analogy: Think of a tug-of-war — a charge stays "in place" only when the pulls (or pushes) from both sides are exactly equal and opposite, just like two equally strong teams keeping the rope steady.

A charge is said to be in equilibrium when the net electrostatic force acting on it due to all other charges in the system is zero.

A group of two or more point charges arranged in space such that they exert forces on one another.

A force that acts to bring a displaced charge back toward its original equilibrium position.

For a charge q under the influence of charges q₁, q₂, …, q_n:

For the simplest two-charge case:

\[\vec F_1\] = −\[\vec F_2\]

Case 1 — Same-Sign Charges

When q₁​ and q₂ carry charges of the same sign, the equilibrium (test) charge q₀​ must be placed between them on the line joining q₁ and q₂​.

• The exact position x (measured from q_1​) depends on the ratio of the magnitudes of q₁​ and q₂.
• The charge q₀​ sits closer to the smaller-magnitude charge.

Case 2 — Opposite-Sign Charges

When q₁​ and q₂​ carry charges of opposite signs, the equilibrium charge q₀​ cannot lie between them — it must lie outside the line segment AB, on the side of the smaller-magnitude charge.

• Position x is again calculated using a ratio-based formula involving the magnitudes of q₁​ and q₂.

Setup: Two equal charges +q are suspended by two strings, each of length l, from a common point, hanging apart due to mutual repulsion.

Derivation Steps:

1. Each charge experiences three forces: gravity (mg), tension (T), and electrostatic repulsion (F_e​).
2. Resolving forces at equilibrium: T sin⁡ θ = F_e and T cos ⁡θ = mg.
3. Dividing the two equations: tan⁡θ = \[\frac {F_e}{mg}\]​​.
4. Using the small-angle approximation for small separations: sin⁡ θ ≈ θ ≈ tan ⁡θ ≈ \[\frac {x}{2l}\]​, where x is the separation between the charges.