# General Solution of a Poisson Equation of a magnetic array

Hi,
I'll give some background, say you've got a planar structure of thickness 'd', lying on the z plane. Also say the upper and lower surfaces are y = 0 and y = -d, respectively.
The structure has scalar potentials inside it as so:

As you can see the vector fields cancel out on one side, As it says below, there is a Poisson equation of:

BUT I HAVE NO IDEA WHY that is the poission equation, I get that Fi inside is a scalar potential, but why is mok.cos(kx) the vector field?, not like mx+my or something instead? It looks like they've just differentiated mx and that's the vector function, maybe just a coincidence?
I also have no idea how that is the general solution? Specifically the homogenious part.
I get that for the part of the particular you can solve the Poisson equation of using method of undetermined coefficients with a guess of
(Asin(kx) + Bcos(kx)) and just differentiate that twice for del2:
(Asin(kx) + Bcos(kx))'' = mok.Cos(kx)
therefore: -A.k2sin(kx)-B.k2cos(kx) = mok.Cos(kx)
therefore: -A.ksin(kx)-B.kcos(kx) = mo.Cos(kx) and equating coefficients yields:
B = - mo/k
A = 0
so Yp = - mo/k * Cos(kx)

But why does the homogenous part have exponentials and y in them? I thought they'd just be zero.
If someone could explain that or even just why the Poisson equation is what they say I'd be greatful.
By the way, I understand that a Poisson equation that = 0 is a Laplace equation, but as far as I know The General solution = homogenious + particular, solutions. As you can see in (3b) the particular solution I worked out is in there, but so is the 'homogenous' aspect which I can't figure out. As you can see in (2a) the Poisson is non-homogenous (not laplace) but it must use a homogenous part in the method of undetermined coefficients, I expect, to find (3a) and (3c) as well as the afformentioned aspect of (3b).

The boundary conditions are given later (below) so I didn't think they were used to find the general solution. It would seem weird that I can find half of the solution and not the rest. And that doesn't explain how they came to that value of the Poisson equation.

P.S I also wonder, which point is chosen as x = 0 on the diagram...?
THANKS!!!

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