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?

 

[paranormal]

I would say that both interpretations have valid points and it ultimately depends on the specific scenario and parameters involved. Let's break down the two approaches and see how they apply in this situation.

First, let's consider the Lorentz force approach. According to the Lorentz force law, a charged particle moving in a magnetic field experiences a force perpendicular to both its velocity and the magnetic field. In this case, the middle wire is carrying a current and is therefore acting as a charged particle. As it is displaced upward, it will experience a force from the neighboring wires due to the magnetic field created by their currents. This force will indeed be attractive, and since the displacement is small, it can be approximated as a simple harmonic motion.

On the other hand, the simple harmonic motion approach is based on the idea that any system with a restoring force proportional to its displacement will exhibit a periodic motion. In this case, the restoring force comes from the tension in the wires, which increases as the middle wire is displaced upward. This tension can be calculated using the equations for the forces between two parallel wires carrying currents. By manipulating these equations, it can be shown that the resulting motion will indeed be simple harmonic.

So, which approach makes the most sense? Both can be valid, and in this case, it may be helpful to consider the specific values of the parameters involved. If the displacement is very small, then the Lorentz force approach may be more accurate. However, if the displacement is larger, the simple harmonic motion approach may provide a better approximation. Ultimately, both approaches are based on fundamental principles and can provide valuable insights into the behavior of the system. It may be helpful to compare the results from both approaches and see which one better fits the observed data.