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Physics · Ch 3 — Magnetism and Magnetic Effects of Electric Current

Force Between Two Long Parallel Current Carrying Conductors

3.10.6

Force Between Two Long Parallel Current Carrying Conductors

Two long straight parallel wires A and B, separated by distance rr, carry currents I1I_1 and I2I_2. Wire A produces, at the location of wire B, a field B1=μ0I12πrB_1 = \dfrac{\mu_0 I_1}{2\pi r} (§3.8.2); the force this exerts on a length dldl of wire B (from §3.10.5, dF=I2 dl B1dF=I_2\,dl\,B_1) works out, per unit length, to

Fl=μ0I1I22πr\frac{F}{l} = \frac{\mu_0 I_1 I_2}{2\pi r}

Working through the direction carefully (using the right-hand rule for B1B_1 and Fleming's left-hand rule for the resulting force) shows that when the two currents flow in the same direction, the force is attractive; when they flow in opposite directions, the force is repulsive -- the same magnitude of force acts symmetrically back on wire A due to wire B's field, by Newton's third law.

This force is used to define the ampere itself: …

Figure 3.53Two long straight parallel wires

What this figure shows. Two long straight wires A and B run parallel to each other, separated by distance r, carrying currents I1 and I2 respectively along the same (z) direction. The field B1 produced by wire A at the location of wire B is drawn, directed into the page there by the right- …

Figure 3.55Two parallel conductors carrying current in the same direction

What this figure shows. Two parallel wires, both carrying current in the same direction, are drawn with a force arrow on each pointing toward the other wire -- illustrating that like-directed parallel currents attract, exactly the way each wire's own circular field lines pull the neighbouring wire's current element in toward it. …

Figure 3.56Two parallel conductors carrying current in opposite directions

What this figure shows. Two parallel wires, carrying current in opposite directions, are drawn with a force arrow on each pointing away from the other wire -- illustrating that oppositely-directed parallel currents repel, the mirror-image case of Figure 3.55 obtained simply by reversing the direction of one of the two currents. …