- #1
rmiller70015
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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.