The long, straight wire A B shown in Fig. P28.60 carries a...

The long, straight wire $A B$ shown in Fig. P28.60 carries a current of $14.0 \mathrm{~A}$. The rectangular loop whose long edges are parallel to the wire carries a current of $5.00 \mathrm{~A}$. Find the magnitude and direction of the net force exerted on the loop by the magnetic field of the wire.

Key Concepts

Superposition of Magnetic Forces

When analyzing systems with multiple current-carrying conductors or loops in a magnetic field, the net force on the system is obtained by vectorially adding the forces acting on each segment. Since different segments may experience different magnitudes and directions of force due to variations in the magnetic field, the overall force is the cumulative result of these differential contributions.

Net Force on a Current Loop

In a current loop placed in a spatially varying magnetic field, each segment of the loop experiences a force according to the Lorentz force law. Due to the geometry of the loop and the non-uniform magnetic field, forces on different segments may not completely cancel each other out, resulting in a net force. Calculating this net force involves considering both the magnitude and direction of the forces on the various segments of the loop.

Magnetic Field of a Current-Carrying Conductor

A long, straight wire generates a magnetic field that encircles the wire with a direction given by the right-hand rule. The magnitude of this field decreases inversely with the distance from the wire according to the formula B = ??I/(2?r), where ?? is the permeability of free space, I is the current through the wire, and r is the distance from the wire. This relationship is derived from the Biot–Savart law and Ampère's law and is fundamental in understanding how currents produce magnetic fields.

Lorentz Force on a Current-Carrying Conductor

A segment of a current-carrying conductor placed in a magnetic field experiences a force given by F = I (l × B), where F is the force vector, I is the current, l is the length vector indicating the direction and magnitude of the current element, and B is the magnetic field. The direction of this force is determined by the right-hand rule, and its magnitude depends on the sine of the angle between the current direction and the magnetic field.

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