A classic electrostatics problem is finding the position where a charge stays in equilibrium, i.e. where the net force exerted on it by all other charges cancels out. It is the electric analogue of finding equilibrium points in mechanics, and it is solved by requiring that the magnitudes of the opposing forces be equal.
The simplest case is that of a free charge placed on a line together with two fixed charges. For to be in equilibrium, the two Coulomb forces acting on it must have the same magnitude and opposite direction. Equating the magnitudes gives an equation in the unknown position :
The test charge cancels out: the equilibrium position does not depend on the value or sign of the charge placed there, only on the two fixed charges. This is an important conceptual point.
Watch the signs and the physical solutions
The equation of magnitudes, once squared, often yields two algebraic solutions. You must always discuss which one is physically acceptable:
- if the two fixed charges have the same sign, the equilibrium point lies between them;
- if they have opposite signs, the equilibrium point is outside the segment, on the side of the smaller-magnitude charge (where its contribution, being closer, can match that of the larger, more distant charge).
In the detailed solution (see the exercise “Equilibrium on a line” and the challenge “Equilibrium of three aligned charges”) you can see how to set up the condition, solve the equation, and discard the unphysical root by reasoning about the geometry of the problem.
Links
Topics: Electrostatics Concepts: Coulomb’s law Skills: Superposition principle · Symbolic set-up
Related exercises: Worked exercise — Three charges at the vertices of a triangle · Problem — Force between two 1-microcoulomb charges · Problem — Force between two spheres with the same charge