Calculating magnetic moment given magnetization

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?

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Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

"Calculating magnetic moment given magnetization"

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