A long, horizontal wire A B rests on the surface of a table...

A long, horizontal wire $A B$ rests on the surface of a table and carries a current $I .$ Horizontal wire $C D$ is vertically above wire $A B$ and is free to slide up and down on the two vertical metal guides $C$ and $D$ (Fig. E28.27). Wire $C D$ is connected through the sliding contacts to another wire that also carries a current $I$, opposite in direction to the current in wire $A B$. The mass per unit length of the wire $C D$ is $\lambda$. To what equilibrium height $h$ will the wire $C D$ rise, assuming that the magnetic force on it is due entirely to the current in the wire $A B ?$

Key Concepts

Magnetic Field of a Long Straight Conductor

This concept involves understanding that a long straight current-carrying wire produces a magnetic field that circulates around it. The magnetic field strength decreases with distance from the wire according to a 1/r dependence, which can be derived using Ampère's Law or the Biot–Savart Law. This concept is critical in determining the strength and direction of the field experienced by nearby conductors.

Lorentz Force on a Current-Carrying Conductor

This principle states that a conductor carrying an electric current in the presence of a magnetic field experiences a force. The direction and magnitude of this force are given by the cross product of the current element and the magnetic field, which is succinctly stated by the formula F = I (L × B). This force is pivotal in understanding how current-carrying wires interact with magnetic fields and subsequently move or exert pressure.

Force Equilibrium in a Gravitational Field

Equilibrium involves balancing forces so that an object remains at rest or moves uniformly. In this context, the upward magnetic force acting on the wire must be balanced by the gravitational force (weight) acting downward on the wire. Establishing this equilibrium condition is essential in determining the stable position or height where the forces cancel out.

Interaction between Parallel Currents

When two current-carrying wires are positioned in proximity, the magnetic field created by one wire exerts a force on the other. The direction and magnitude of the force depend on the directions of the currents. In particular, currents flowing in opposite directions will repel each other, a principle that is central to analyzing the forces in the scenario described.

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