Calculating magnetic moment given magnetization

[dcnairb]

Homework Statement

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?

Homework Equations

$M \equiv$ magnetic moment density

The Attempt at a Solution

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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?

 

[mark726]

Your method is correct for finding the total magnetic moment of the object. However, the question is asking for your guess on how the magnetic field decreases with distance away from the object. This means that you should consider the behavior of the magnetization and how it affects the magnetic field. In this case, the magnetization is not uniform and varies with position, so you should think about how this would affect the magnetic field at different distances from the object. For example, at a closer distance, the magnetic field may be stronger due to the higher magnetization, while at a farther distance, the magnetic field may be weaker due to the lower magnetization. Your guess on how the magnetic field would decrease with distance away from the object should take into account these factors.