# 14.2.4 Forces between Two Straight Wires

For electric fields, like charges repel, and unlike charges attract. We have something similar for magnetic fields: like currents attract, and unlike currents repel.

Consider two long parallel wires W1 and W2 spaced r apart with currents I1 and I2 in the same direction.

I1 is going to produce a magnetic field of strength $\displaystyle {{B}_{1}}=\frac{{{{\mu }_{0}}{{I}_{1}}}}{{2\pi r}}$ at W2. W2 is carrying a current I2, and sitting in B1. It thus experiences a force per unit length of $\displaystyle \frac{{{{F}_{b}}}}{L}={{B}_{1}}{{I}_{2}}=\frac{{{{\mu }_{0}}{{I}_{1}}}}{{2\pi r}}{{I}_{2}}=\frac{{{{\mu }_{0}}{{I}_{1}}{{I}_{2}}}}{{2\pi r}}$.

What about W1? Similarly, W1 is carrying I1 sitting in B2. So it experiences a force per unit length of $\displaystyle \frac{{{{F}_{b}}}}{L}={{B}_{2}}{{I}_{1}}=\frac{{{{\mu }_{0}}{{I}_{2}}}}{{2\pi r}}{{I}_{1}}=\frac{{{{\mu }_{0}}{{I}_{1}}{{I}_{2}}}}{{2\pi r}}$.

Using the FLHR, one can deduce that the two wires are mutually attracting each other if they carry currents in the same direction.  If W1 and W2 are carrying currents in opposite directions, then it is going to be a mutual repulsion.

Do realize that even if I1 and I2 are different in magnitude, the magnetic forces each wire exerts on the other are still going to be exactly equal in magnitude. This is yet another example of two objects obeying N3L.

Concept Test

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