Force Between Two Parallel Currents — Question 4

Question 4

Four corner currents exert a diagonal force on each wire

Four infinite parallel wires lie at the corners of a square of side a in the xy plane. All carry the same current I toward +z. Find the force per unit length on the wire at the lower-left corner, with +x rightward and +y upward. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Add the force vectors from the two adjacent wires and the opposite corner.

2. Evaluate the magnitude for I=10.0 A and a=.0200 m.

Question 4 - Solution

Four corner currents exert a diagonal force on each wire

(a) Method and calculation

$$C=\frac{\mu_0I^2}{2\pi a},\quad\vec f_{\mathrm{adjacent}}=C(\hat x+\hat y).$$$$\vec f_{\mathrm{diagonal}}=\frac C{\sqrt2}\frac{\hat x+\hat y}{\sqrt2}=\frac C2(\hat x+\hat y),\quad\vec f=\frac{3C}{2}(\hat x+\hat y).$$

(b) Interpretation and further calculation

$$C=1.00\times10^{-3}\,\mathrm{N/m},\quad|f|=\frac{3\sqrt2}{2}C.$$$$\boxed{|f|=2.12132\times10^{-3}\,\mathrm{N/m}}\quad\text{toward the square center}.$$

Physical check

The diagonal wire is farther away and its force must also be resolved into components; it does not contribute a full adjacent-wire force to either axis.