Lorentz Force or Simple Harmonic Motion

[mateomy]

So my friend and I were going through problems and this came up...

Consider 3 straight, infinitely long, equally spaced wires (with zero radius, separated each by a distance d), each carrying a current I in the same direction.


blah blah blah...


part c) asks us...

If the middle wire is rigidly displaced a very small distance x (x<<d) upward while the other 2 wires are held fixed, describe qualitatively the subsequent motion of the wire.


One of us is invoking the Lorentz force and saying as a the middle wire is displaced the Force between the middle wire and it's now closer neighbor will increase, thereby creating a very strong attractive force. However, the other of us is manipulating the equations to show that what will happen is in fact, simple harmonic motion. Lorentz force seems obvious but, how can we be pulling SHM out of this? Which makes the most sense?

Any input would be appreciated.

Thanks!

 

[TSny]

I assume that "displaced upward" means that the middle wire is displaced in a direction that is perpendicular to the plane that originally contained the three wires. If so, does the middle wire remain equidistant from the other two wires? Does the net force on the middle wire remain zero? If not, what is the direction of the net force?

 

[mateomy]

No, the displacement is within the wire plane. It's displaced closer to another wire and then released.

 

[TSny]

Oh, I see. Well, after the displacement, what will be the direction of the net force on the middle wire?