A pair of long, rigid metal rods, each of length L , lie...

A pair of long, rigid metal rods, each of length $L$ , lie parallel to each other on a perfectly smooth table. Their ends are connected by identical, very light conducting springs of force constant $k$ (Figure 20.79$)$ and negligible unstretched length. If a current $I$ runs through this circuit, the springs will stretch. At what separation will stretch. At remain at rest? Assume that $k$ is large enough so that the separation of the rods will be much less than $L .$

Key Concepts

Magnetic Force Between Parallel Currents

When current flows through parallel conductors, each conductor generates a magnetic field which exerts a force on the other conductor. This interaction is described by Ampère’s law and the Biot–Savart law, and the resulting force per unit length typically varies inversely with the separation distance between the wires. In configurations with currents flowing in the same direction, the force is attractive, and its magnitude is a key factor in determining the behavior of the system.

Hooke's Law

Hooke's Law governs the behavior of springs by stating that the force exerted by a spring is directly proportional to its displacement from the equilibrium position, with the proportionality constant being the spring constant k. This principle is critical in problems involving mechanical deformation and restoring forces, as it allows the calculation of spring stretch or compression when subjected to external forces.

Equilibrium of Forces

In systems where multiple forces act, equilibrium is achieved when the net force on each part of the system is zero. In this context, the equilibrium condition requires a balance between the magnetic attractive forces generated by the currents and the restoring forces of the springs as dictated by Hooke’s Law. Establishing this balance is essential for determining the steady-state separation of components.

Small Separation Approximation

The assumption that the separation between the rods is much less than their length simplifies the analysis by allowing one to neglect edge effects and assume that the magnetic forces are uniformly distributed along the rod lengths. This approximation is useful in reducing complex integrations to more manageable expressions, making it easier to relate the force per unit length to the overall force in the system.

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