In Fig. 29-64, five long parallel wires in an x y plane are...

In Fig. $29-64,$ five long parallel wires in an $x y$ plane are separated by distance $d=8.00 \mathrm{~cm},$ have lengths of $10.0 \mathrm{~m},$ and carry identical currents of 3.00 A out of the page. Each wire experiences a magnetic force due to the currents in the other wires. In unit-vector notation, what is the net magnetic force on (a) wire 1 , (b) wire $2,$ (c) wire 3 , (d) wire $4,$ and (e) wire $5 ?$

Key Concepts

Superposition Principle in Electromagnetic Interactions

The superposition principle states that the net magnetic field at any point due to multiple sources can be found by vectorially adding the individual magnetic fields from each source. This principle is essential when dealing with several wires whose individual magnetic fields interact to produce a combined effect on each wire.

Right-Hand Rule

The right-hand rule is a mnemonic device used to determine the direction of the magnetic field around a current-carrying conductor and the direction of the magnetic force on a current. By pointing the thumb in the direction of conventional current, the curled fingers then indicate the direction of the magnetic field, and a subsequent application helps infer the force direction when the magnetic field is crossed with the current direction.

Magnetic Field of a Long Straight Conductor

This concept describes the magnetic field produced by a long, straight current?carrying wire. According to Ampere’s law, the magnetic field at a distance r from the wire is given by B = (??I)/(2?r). The magnetic field forms concentric circles around the wire and its magnitude decreases with increasing distance. This field is fundamental when determining the interactions between multiple current?carrying conductors.

Magnetic Force on a Current-Carrying Wire

A current-carrying wire placed in an external magnetic field experiences a force given by the Lorentz force law F = I(L × B), where I is the current, L is the length vector in the direction of the current, and B is the magnetic field. This concept is crucial for calculating the net force on a wire subject to magnetic fields produced by nearby currents.

Symmetry in Magnetic Arrangements

When multiple conductors are arranged in symmetric configurations, symmetry considerations can simplify the calculation of net forces. Similar distances and current magnitudes often lead to cancellation of forces in certain directions, so exploiting symmetry helps in deducing the net effect without computing each interaction independently.

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