Magnetic field of a uniformly magnetized sphere.

[rmiller70015]

Homework Statement

Find the magnetic field of a uniformly magnetized sphere.
(This is an example in my book, I have underlined what I am having trouble understanding down below.)

Homework Equations

$$\vec{J}_b = \nabla \times \vec{M}$$
$$\vec{K}_b = \vec{M}\times \hat{n}$$
$$\vec{A}(\vec{r}) = \frac{\mu _0}{4\pi}\int_v \frac{\vec{J_b(\textbf{r'})}}{\eta}d\tau + \frac{\mu _0}{4\pi}\oint_S \frac{\vec{K_b (\textbf{r'})}}{\eta}da'$$
$\eta$ is the script r vector that Griffith's uses in his books because I couldn't figure out how to do it in mathjax.

The Attempt at a Solution

This is Example 6.1 from Griffith's Introduction to Electrodynamics 4th edition. He says that the $\vec{M}$ vector should be aligned with the z-axis and then $\vec{J_b} = \nabla \times \vec{M} = 0$ and $\vec{K_b} = \vec{M} \times \hat{n} = Msin\theta \hat{\phi}$

This tells us that the rotating volume is equivalent to a shell with a uniform surface charge density of $\sigma$, when this shell rotates with angular velocity, $\omega$, it can be thought of as a surface current density of:
$$\vec{K} = \sigma \vec{v} = \sigma \omega Rsin\theta$$

This is where I get lost, the book says that "with the identification that $\underline{\sigma R\omega \rightarrow M}$. Conclude that:"
$$\vec{B} = \frac{2}{3}\mu _0 \vec{M}$$

I have no idea where the author is getting this from, I think he is using Ampere's law, but I can't seem to find out where the relationship between B and M is that allows the author to get here.

 

[kuruman]

I don't have the book, but it looks like Griffiths is drawing an analogy between the magnetized sphere and a charged rotating conducting shell. K_b in the case of the magnetized sphere is compared to the current generated by the rotation of the shell. If there is an example of the rotating shell in the book, be sure to study it.

 

[TSny]

I second kuruman's post. The example of the rotating spherical shell of charge was worked out in chapter 5. (Example 5.11 in the 3rd edition).