- #1
adrian52
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Homework Statement
A cylindrical volume of length L, radius a, is given a uniform magnetization M along its axis (call it the z axis). The question is to first find the bound currents, then use them to find the magnetic field B, as well as "the auxiliary field" H - but only along the axis of the cylinder.
Homework Equations
The bound volume current is [itex] \vec J_b = \nabla \times \vec M [/itex]
The bound surface current is [itex] \vec K_b = \vec M \times \hat n [/itex]
The magnetization is [itex] \vec M = M_o \hat z [/itex]
The auxiliary field is defined as [itex] \vec H = \frac{1}{\mu_o}\vec B - \vec M [/itex]
Possibly using Amperian loops: [itex] \oint B \cdot dl = \mu_o I_e [/itex] ([itex] I_e [/itex] is the enclosed current, both bound and free)
Also possibly using similar loops for H: [itex] \oint \vec H \cdot \vec dl = I_f [/itex] ([itex] I_f [/itex] is the free, enclosed charge...which is zero in this case, as far as I can tell)
The magnetic vector potential is: [itex] \vec A = \frac{u_o}{4\pi}\int_V \frac{\vec J_b\vec (r')}{r_s}\ dV' + \frac{u_o}{4\pi}\oint_S \frac{\vec K_b\vec (r')}{r_s}\ da' [/itex]
And from magnetic vector potential, the magnetic field [itex] \vec B = \nabla \times \vec A [/itex] or the curl of [itex] \vec A [/itex].
Here [itex] r_s [/itex] is the magnitude of the separation vector. So [itex] r_s = r - r' [/itex], where [itex] r' [/itex] is the distance from the origin to the source charge and r is the distance from the origin to the field. Similarly all primed coordinates represent the distances or vectors to the source charges/currents in the system.
The Attempt at a Solution
[/B]
Using boldface as vector notation, since M is uniform, I found Jb to be zero, while Kb was zero at the end faces of the cylinder, but not along the side face, where the right-hand-rule determines that Kb = [itex] M \hat \phi [/itex] (using cylindrical coordinates).
After a horribly long period of confusion trying to apply Amperian loops across infinitesimal distances, I gave up and tried to use the formula for the magnetic vector potential directly. Only the closed surface integral remains since J is 0. In a nutshell, I chose my origin to be the center of the cylinder. I wrote [itex] r = z \hat z [/itex] while the primed, source r' was: [itex] r' = a \hat s + z' \hat z[/itex].
Then the magnitude, [itex] r_s [/itex] was found to be [itex] \sqrt{a^2 + (z-z')^2} [/itex]
The primed area, da', was written (in cylindrical) to be: [itex] ad\phi' dz' [/itex]
So here's all of it put together:
[itex] \vec A = \frac{\mu_o M}{4\pi}\int_0^{2\pi} \int_\frac{-L}{2}^\frac{L}{2} \frac{1}{\sqrt{a^2 + (z-z')^2}}ad\phi'dz' \hat \phi [/itex]
Anyway in the end I used the fact that [itex] \int \frac{1}{\sqrt{a^2 + x^2}}dx = ln(x + \sqrt{a^2 + x^2}) [/itex] to compute the integral, and it was purely in terms of z, L/2, a, and some other constants.
Then I computed the curl (in cylindrical coordinates), but here was the big problem. In my textbook (Griffiths, electrodynamics), the one of the two non-zero terms that I had to compute was [itex] \frac{1}{s}[\frac{\partial}{\partial s}(sA_\phi)] \hat z [/itex].
My magnetic vector potential has no "s" term, because I am only considering points along the axis. That's why I have r = [itex] z \hat z [/itex] and not r = [itex] s \hat s + z \hat z [/itex].
So basically this requires me to multiply by s, take the partial of s, then divide by s. In the end I have a magnetic field with a [itex] \frac{1}{s} [/itex] term! But I want to find the field at s = 0...along the axis. So my magnetic field blows up everywhere along z!
I am really stuck on this one, hopefully someone can respond soon so that I have time to study for my other two midterms, before I try to finish the other four problems :(.
Also note I'm pretty sure you can't just use the amperian loops as before because its a *finite* cylinder so the old symmetrical tricks don't work anymore. Unless you consider them in a really small area, but I have no idea how that would work. i.e. there *is* a radial component now, it doesn't get canceled out by some infinite length. And anyway even if there was a way, I am deeply confused as to why the equation they give me in the book isn't working. Where am I going wrong?
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