Magnetic Force Between Two Parallel Currents

Two long, straight, and parallel current-carrying conductors exert a force of equal magnitude on one another. The direction of the force depends on the current direction in the conductors.

The force exerted by the magnetic field due to the first conductor over a finite length of the second conductor is given as the product of the current in the second conductor and  the vector product of the length vector along the current element and the field due to the first conductor. According to the right-hand rule for the cross-product, the force on the second conductor is directed towards the first conductor. Similarly, the field due to the second conductor exerts an equal magnitude of force on the first conductor and is directed towards the second conductor. Thus the forces are attractive when the current in the both the conductors flow in the same direction. Reversing the direction of current in any one of the conductors makes the force repulsive.

Since the wires are very long, the force is often expressed in terms of the force per unit length, which forms the basis for the definition of the unit 'Ampere' for the current. Quantitatively, one Ampere is the amount of current present in each of the two parallel conductors of an infinite length separated by one meter in empty space, which causes each conductor to experience a force of exactly 1 N/m.

This force is responsible for the pinch effect in electric arcs and other plasmas. The force is apparent if the overall charge density is zero; otherwise, the Coulomb repulsion overwhelms the magnetic attraction. An attractive force squeezes currents into a smaller tube in an electric arc, where charges are moving parallel to one another. In large circuit breakers, such as those used in neighbourhood power distribution systems, the pinch effect can concentrate an arc between plates of a switch trying to break a large current, burn holes, and even ignite the equipment. Another example of the pinch effect is found in solar plasma, where jets of ionized material, such as solar flares, are shaped by magnetic forces.