Three long wires all lie in an x y plane parallel to the x...

Three long wires all lie in an $x y$ plane parallel to the $x$ axis. They are spaced equally, $10 \mathrm{~cm}$ apart. The two outer wires each carry a current of $5.0 \mathrm{~A}$ in the positive $x$ direction. What is the magnitude of the force on a $3.0 \mathrm{~m}$ section of either of the outer wires if the current in the center wire is $3.2 \mathrm{~A}$ (a) in the positive $x$ direction and (b) in the negative $x$ direction?

Key Concepts

Interaction Between Parallel Currents

When two current-carrying wires are placed near each other, they exert magnetic forces on one another. Wires carrying currents in the same direction attract, while wires with currents in opposite directions repel. This concept of mutual interaction between parallel currents is crucial for understanding how the configuration and direction of current in the center wire affect the force experienced by the outer wires.

Magnetic Field of a Long Straight Wire

A long current-carrying conductor produces a magnetic field in its surrounding space. The magnitude of this field at a distance r is given by B = ??I/(2?r), where ?? is the permeability of free space and I is the current in the wire. The direction of the field is determined by the right-hand rule. This concept is essential for calculating the interactions between the wires in the problem.

Magnetic Force on a Current-Carrying Conductor

A current-carrying conductor placed in a magnetic field experiences a force given by F = IL × B, where I is the current, L is the length of the segment, and B is the magnetic field. This force is perpendicular to both the direction of current and the magnetic field, as determined by the right-hand rule. This principle is key to determining the force on a section of the wire due to the magnetic field produced by another wire.

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