Force Between Two Parallel Currents — Question 3

Question 3

Three parallel currents reveal a force balance that is unstable along their line

Two fixed wires at x=-a and x=+a carry current I toward +z. A third wire of mass per length lambda carries i toward +z and can slide along x between them. Let its displacement from the midpoint be x; neglect gravity along x. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Find the net force per unit length and its leading term for small x.

2. Classify the midpoint stability. Repeat the linear stability analysis if only i is reversed.

Question 3 - Solution

Three parallel currents reveal a force balance that is unstable along their line

(a) Method and calculation

$$f_x=\frac{\mu_0Ii}{2\pi}\left(\frac1{a-x}-\frac1{a+x}\right).$$$$f_x=\frac{\mu_0Ii\,x}{\pi(a^2-x^2)}\simeq\frac{\mu_0Ii}{\pi a^2}x.$$

(b) Interpretation and further calculation

$$\lambda\ddot x=+\frac{\mu_0Ii}{\pi a^2}x\Rightarrow x=0\text{ is unstable}.$$$$i\to-i:\quad\ddot x=-\Omega^2x,\quad\boxed{\Omega=\sqrt{\frac{\mu_0Ii}{\pi\lambda a^2}}}.$$

Physical check

An attractive displacement favors the nearer fixed wire and grows; repulsion reverses that linear response when motion is confined to x.