A long, horizontal wire Figure E28.27 A B rests on the...

A long, horizontal wire Figure E28.27 $A B$ rests on the surface of a table and carries a current $I$. Horizontal $C D$ is vertically above wire is free to slide up and down two vertical metal guides $C$ (Fig. E28.27). Wire $C D$ nected 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 Wire

This concept involves understanding that a long, straight current-carrying wire produces a magnetic field that circulates around it. The magnitude of this magnetic field at any point a distance r from the wire is given by the expression B = ??I/(2?r). This field distribution is derived using Ampère’s law and is fundamental when analyzing the interaction of current-carrying wires in space.

Lorentz Force on a Current-Carrying Conductor

The Lorentz force law describes how a current-carrying conductor in the presence of a magnetic field experiences a force. The force per unit length on such a wire is calculated by the cross product F = I L × B, where I is the current, L is the length vector of the wire, and B is the magnetic field. This principle is key to analyzing how magnetic interactions can produce mechanical motion, as seen when balancing forces in electromagnetic systems.

Mechanical Equilibrium and Force Balance

When a system reaches mechanical equilibrium, the net force acting on it is zero. In the context of current-carrying wires in a magnetic field, the upward magnetic force must balance the downward gravitational force, which is a product of the mass per unit length (?) and the acceleration due to gravity (g). This balance determines the equilibrium position of a movable wire in magnetic and gravitational fields.

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