Three long parallel wires are 3.5 cm from one another....

Three long parallel wires are $3.5 \mathrm{~cm}$ from one another. (Looking along them, they are at three corners of an equilateral triangle.) The current in each wire is $8.00 \mathrm{~A}$, but its direction in wire $\mathrm{M}$ is opposite to that in wires $\mathrm{N}$ and $\mathrm{P}$ (Fig. $28-54$ ). Determine the magnetic force per unit length on each wire due to the other two.

Key Concepts

Magnetic Field of a Current-Carrying Wire

A long, straight wire carrying an electric current generates a magnetic field that circles around the wire. The magnitude of this magnetic field at a distance is given by the formula B = ??I/(2?r), where ?? is the permeability of free space, I is the current, and r is the radial distance from the wire. This formula stems from Ampère's circuital law and is fundamental in analyzing the interactions between current-carrying wires.

Magnetic Force Between Parallel Currents

Two parallel current-carrying wires experience a magnetic force due to the interaction of the magnetic field produced by one wire with the current in the other. The force per unit length between them is given by F/L = ??I?I?/(2?d), where d is the distance between the wires. The direction of this force depends on whether the currents are in the same or opposite directions, leading to attraction or repulsion, respectively.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field around a current-carrying wire and to find the direction of the force experienced by a current in a magnetic field. By orienting the thumb in the direction of the current, the curling of the fingers illustrates the circular direction of the magnetic field, which helps in establishing how forces interact between different wires.

Superposition Principle

The superposition principle in electromagnetism states that the net magnetic field or force in a region is the vector sum of the individual fields or forces produced by each source. When multiple wires are present, the total force on any one wire is found by calculating the contributions from each of the other wires separately and then vectorially adding them.

Geometrical Symmetry in Multi-Wire Configurations

In problems involving multiple conductors arranged in a symmetric pattern, such as the vertices of an equilateral triangle, symmetry can simplify the analysis by enabling the decomposition of forces into components. This approach can lead to cancellations of certain components or the reinforcement of others, thereby easing the computation of the net force acting on each conductor.

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