 Transverse oscillation of the middle of three parallel currents

Two fixed outer wires at $x=\pm d$ attract a middle wire carrying the same electric current $I$. If the middle wire is displaced in the perpendicular direction $y$, the net magnetic force per unit length is $-{\mu }_0{I}^{2}y/[\pi (d^2+y^2)]$. For $|y|\ll d$, the wire's linear mass density ${\mu }_m$ gives the displayed simple harmonic motion frequency. Motion towards either outer wire in the original plane is unstable. Gravity must be absent from or balanced in the oscillation direction.