Force Between Two Parallel Currents — Question 10

Question 10

A triangular arrangement requires vector addition even when all pair forces are attractive

Two fixed wires at $(-a,0)$ and $(+a,0)$ carry I toward +z. A third wire at $(0,h)$ carries i toward +z. All are very long. Compute the force per length on the upper wire as a function of positive h. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Find the vertical force; show that the horizontal components cancel.

2. Find the value of h that maximizes the force magnitude for fixed a, I and i, and compare the limits h approaching zero and infinity.

Question 10 - Solution

A triangular arrangement requires vector addition even when all pair forces are attractive

(a) Method and calculation

$$r=\sqrt{a^2+h^2},\quad\vec f_\pm=\frac{\mu_0Ii}{2\pi r^2}(\pm a\hat x-h\hat y).$$$$\boxed{\vec f=-\frac{\mu_0Ii\,h}{\pi(a^2+h^2)}\hat y}.$$

(b) Interpretation and further calculation

$$\frac d{dh}\frac h{a^2+h^2}=\frac{a^2-h^2}{(a^2+h^2)^2}=0\Rightarrow\boxed{h=a}.$$$$|f|_{\max}=\frac{\mu_0Ii}{2\pi a},\quad |f|\to0\text{ as }h\to0\text{ or }h\to\infty.$$

Physical check

Near the base the two attractions are almost horizontal; far away both are weak. Their vertical sum therefore has an intermediate maximum.