Two small beads having charges q1 and q2 of the same sign...

Two small beads having charges $q_{1}$ and $q_{2}$ of the same sign are fixed at the opposite ends of a horizontal insulating rod of length $d$. The bead with charge $q_{1}$ is at the origin. As shown in Figure $\mathrm{P} 22.8,$ a third small, charged bead is free to slide on the rod. (a) At what position $x$ is the third bead in equilibrium? (b) Can the equilibrium be stable?

Key Concepts

Coulomb's Law

This law describes the force between two point charges and states that the magnitude of the force between them is directly proportional to the product of their charges and inversely proportional to the square of the distance that separates them. It is fundamental in determining the magnitude and direction of electrostatic forces acting in systems with charged particles.

Electrostatic Equilibrium

In electrostatics, equilibrium refers to the condition where the net force acting on a charge is zero, meaning that all forces balance each other out. This concept is crucial when analyzing systems where charges influence each other, as it helps in determining possible positions for a particle where it can remain at rest under the influence of electric forces.

Force Balance Analysis

Force balance analysis involves setting the vector sum of forces acting on a charge to zero in order to find the conditions or positions where the net force is null. This method is widely used in problems involving multiple interacting charges, ensuring that the electric forces provided by Coulomb’s law cancel out for a particle in equilibrium.

Stability of Equilibrium

Stability analysis examines whether a small displacement from the equilibrium position leads to forces that restore the particle back to equilibrium or move it further away. For electrostatic systems, this involves analyzing the potential energy profile around the equilibrium position, determining whether it corresponds to a minimum (indicating stable equilibrium) or a maximum or saddle point (unstable equilibrium).

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