# Calculating magnetic moment given magnetization

Tags:
1. Nov 8, 2014

### dcnairb

1. The problem statement, all variables and given/known data
Consider a square slab of magnetized material with sides $2a$, and
thickness $d$, as shown. The magnetization is not uniform: $$M = M_o cos(\frac{2πx}{ a}) \hat{z}$$

b) Calculate the total magnetic moment of this object. How would
you guess the magnetic field decreases with distance away from the object?

2. Relevant equations

$M \equiv$ magnetic moment density

3. The attempt at a solution

Because it's a density, my first thought to find the overall $\bar{m}$ was to integrate $\bar{M}$over the volume $dx dy dz$. This of course ends up as 0 because it's over two full periods of a cosine in x, so I got $\bar{m} = 0.$ Of course with 0 magnetic moment a magnetic field far away will be 0, which jives with me because oh how the magnetization is 'cancelled' over the whole thing. However I then became a little unsure as the question asks "how" it would decrease which I would imagine would be like"as $\frac{1}{r^{5}}$" or something like that. It occurred to me that maybe I should just multiply $\bar{M}$ by the volume, to get a variable $\bar{m}$ but... that just seems incorrect to me, especially since $\bar{M}$ is not constant. Can anyone confirm that my method is correct to find $\bar{m}$ and if so am I interpreting the question correctly?

2. Nov 13, 2014