Two small beads having positive charges q1=3 q and q2=q are...

Two small beads having positive charges $q_{1}=3 q$ and $q_{2}=q$ are fixed at the opposite ends of a horizontal insulating rod of length $d=1.50 \mathrm{~m}$. The bead with charge $q_{1}$ is at the origin. As shown in Figure $\mathrm{P} 22.7$, 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

Stability of Equilibrium

Stability of equilibrium refers to whether a system at equilibrium will return to that state after a small perturbation. In the context of electrostatics, this involves examining how the net force or potential energy changes with a slight displacement of the charge. A stable equilibrium is characterized by a restoring force that pushes the system back towards equilibrium, whereas an unstable equilibrium lacks this restoring influence.

Coulomb's Law

Coulomb's Law describes the force between two point charges. It states that the magnitude of the electrostatic force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This law is fundamental for calculating forces in systems involving multiple charged objects, allowing the determination of the interactions between fixed and moving charges.

Net Force Equilibrium

Net force equilibrium in an electrostatic system occurs when the sum of all forces acting on a charged particle is zero. This condition is critical for finding the equilibrium position, as it ensures that the particle remains at rest if undisturbed. Setting the algebraic sum of the Coulomb forces to zero is the mathematical approach used to determine where equilibrium is achieved along the rod.

Electrostatic Potential Energy

Electrostatic potential energy is the energy that a charged particle possesses due to its position in an electric field created by other charges. Analyzing the potential energy function helps in understanding the stability of an equilibrium point. An equilibrium corresponding to a local minimum in potential energy is stable, while one at a maximum or saddle point is unstable.

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