Oscillations in a magnetic field

[w3390]

Homework Statement

A long, narrow bar magnet that has magnetic moment $$\vec{\mu}$$ parallel to its long axis is suspended at its center as a frictionless compass needle. When placed in a region with a horizontal magnetic field $$\vec{B}$$, the needle lines up with the field. If it is displaced by a small angle theta, show that the needle will oscillate about its equilibrium position with frequency f= (1/2pi)*sqrt(uB/I), where I is the moment of inertia of the needle about the point of suspension.

Homework Equations

No specific equations

The Attempt at a Solution

I remember from my mechanics physics class that I need to figure out what the restoring force is. However, that is where I run into my first problem. I do not know how to model an equation to show how the magnetic field will restore the magnet to equilibrium.

 

[w3390]

This equation looks the exact same as the equation for the oscillation frequency of a spring f=(1/2pi)*sqrt(k/m). I know it is a harmonic oscillator, but can anyone get me started on how to change from sqrt(k/m) to sqrt(uB/I). Any help on how to start would be greatly appreciated.

 

[nlightner]

Oscillations in a magnetic field occur when a magnetic object, such as a bar magnet with a magnetic moment parallel to its long axis, is placed in a region with a horizontal magnetic field. When the object is displaced from its equilibrium position, it will experience a restoring force due to the interaction between its magnetic moment and the external magnetic field.

To calculate the frequency of these oscillations, we need to consider the equation for the restoring force. Since we are dealing with a magnetic field, we can use the equation F = \vec{\mu} x \vec{B}, where \vec{\mu} is the magnetic moment and \vec{B} is the external magnetic field. We also know that the restoring force is directly proportional to the displacement, so we can write F = -k\theta, where k is the spring constant and \theta is the displacement.

By setting these two equations equal to each other, we can solve for the spring constant k. This will give us an equation for the restoring force in terms of the magnetic moment, magnetic field, and displacement. We can then use this equation to calculate the frequency of the oscillations using the formula f = 1/2\pi * \sqrt{k/m}, where m is the mass of the object.

In this case, the moment of inertia I can be substituted for the mass m since the object is rotating about a fixed point. This gives us the final equation for the frequency of oscillations as f = 1/2\pi * \sqrt{\mu B/I}. This equation shows that the frequency is directly proportional to the magnetic moment and the strength of the external magnetic field, and inversely proportional to the moment of inertia of the object.

I hope this explanation helps you understand the physics behind oscillations in a magnetic field. Please let me know if you have any further questions.