Force Between Two Parallel Currents — Question 1

Question 1

The force on each of two unequal currents follows from the other wire field

Two very long parallel wires carry I1=2.00 A and I2=5.00 A toward +z, separated by d=.0400 m. Consider a length ell=.300 m far from their ends; neglect return paths. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Calculate the magnetic field each wire produces at the other wire.

2. Find the force on each length ell, including direction, and explain why the forces have equal magnitudes.

Question 1 - Solution

The force on each of two unequal currents follows from the other wire field

(a) Method and calculation

$$B_{1\to2}=\frac{\mu_0I_1}{2\pi d}=10.0\,\mu\mathrm T.$$$$B_{2\to1}=\frac{\mu_0I_2}{2\pi d}=25.0\,\mu\mathrm T.$$

(b) Interpretation and further calculation

$$F_2=I_2\ell B_{1\to2}=F_1=I_1\ell B_{2\to1}=\frac{\mu_0I_1I_2\ell}{2\pi d}.$$$$\boxed{F_1=F_2=15.0\,\mu\mathrm N};\quad\vec F_1=-\vec F_2\text{ and both point toward the other wire}.$$

Physical check

Although the two source fields differ, each force multiplies that field by the other current, giving the same product I1 I2.