Magnetic Moment of a spinning shell

[Ataman]

The problem reads:

Find the magnetic moment of a spinning shell of charge Q, radius R, and angular velocity $$\vec{\omega}$$.

My solution:

I split the sphere into infinitely small loops of current (from top to bottom) and add them up. I set my origin on the center of the sphere and integrate with respect to y from 0 to R times two to get the entire sphere.

The start of the problem:

$$\vec{\mu} \,= A\vec{I}$$
$$d\vec{\mu} \,= A(d\vec{I})$$

Solving for $$d\vec{I}$$:

$$d\vec{I} = \frac{dQ}{t}$$
$$d\vec{I} = \frac{\sigma dA}{t}$$
$$d\vec{I} = \frac{\sigma dA}{(\frac{2\pi}{\omega})}$$
$$d\vec{I} = \frac{\sigma \omega dA}{2\pi}$$

Plugging $$d\vec{I}$$ back:

$$d\vec{\mu} \,= \frac{\sigma \omega A dA}{2\pi}$$

Changing A and dA, so I could integrate in terms of dy:

$$A \,= \pi r^2$$
$$A \,= \pi (R^2-y^2)$$
$$dA \,= -2 \pi y dy$$

Then setting up the integral:
$$d\vec{\mu} \,= - \frac{2 \sigma \omega \pi^2}{2\pi} y (R^2-y^2) dy$$
$$d\vec{\mu} \,= - \sigma \omega \pi y (R^2-y^2) dy$$
$$\vec{\mu} \,= - 2 \sigma \omega \pi \int_{0}^{R} y (R^2-y^2) dy$$

Without sigma:
$$\vec{\mu} \,= - \frac {Q \omega}{2R^2} \int_{0}^{R} y (R^2-y^2) dy$$

This is incorrect, and I have a suspicion it has to do with the "dA" part.

If you don't mind his handwriting, http://nebula.deanza.fhda.edu/physics/Newton/4B/MagneticMomentSphere.jpg is the correct way to do the problem. I am rather uncomfortable integrating over angles, so I opted for a different way.

I would like to know why my way does not work.

-Ataman

 

[BigJDubs]

The problem with your solution is that you are not taking into account the fact that the angular velocity \vec{\omega} is changing over the surface of the sphere. This means that the amount of current in each loop (and thus the magnetic moment) is not constant. In order to calculate the magnetic moment, you need to take into account the angular velocity at each point on the surface of the sphere. The solution given in the link you provided does this by integrating the magnetic moment over the entire surface of the sphere.