Force between two parallel wires carrying currents in the same direction

Problem

Two parallel wires are $d = 5$ cm apart and carry equal currents $i = 10$ A in the same direction. Describe (without calculating) the direction of the force that each wire exerts on the other and explain why, using the Laplace force and the right-hand rule.

Solution

Wire 1 generates, at the position of wire 2, a magnetic field $\vec B_1$ whose field lines are concentric circles around wire 1; using the right-hand rule (thumb in the direction of the current, fingers wrapping around the wire), we find that, at the position of wire 2, $\vec B_1$ is perpendicular to the plane of the two wires.

Wire 2, carrying the current $i$ in the same direction, experiences the Laplace force $\vec F = i\,\vec\ell\times\vec B_1$ Applying the right-hand rule to the cross product, this force turns out to be directed towards wire 1. By symmetry, wire 1 is attracted towards wire 2.

So currents in the same direction attract each other (currents in opposite directions would repel). $\ev{\text{attractive force}}$

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Topics: Magnetism Concepts: Laplace force · Magnetic field Skills: Right-hand rule