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7.2: Force Between Two Current-carrying Wires

In Figure \(VII.1\), we have two parallel currents, \(I_1\) and \(I_2\), each directed away from you (i.e. into the plane of the paper) and a distance \(r\) apart. The current \(I_1\) produces a magnetic field at \(I_2\), directed downward as shown, and of magnitude \(B=\mu I_1/(2\pi r)\) where \(\mu\) is the permeability of the medium in which the two wires are immersed. Therefore, following Equation 7.1.1, \(I_2\) experiences a force per unit length towards the left \(F'=\mu I_1I_2/(2\pi r)\) You must also go through the same argument to show that the force per unit length on \(I_1\) from the magnetic field produced by \(I_2\) is of the same magnitude but directed towards the right, thus satisfying Newton’s third law of motion.

\(\text{FIGURE VII.1}\)

Thus the force of attraction per unit length between two parallel currents a distance \(r\) apart is

\[F' = \dfrac{\mu I_1I_2}{2\pi r}\]