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This is overwhelmingly more of a maths problem than a physics problem, because it's all theoretical. I'll give some background to modle it incase the math's isn't enough.

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 m_{o}k.cos(kx) the vector field?, not like m_{x}+m_{y}or something instead? It looks like they've just differentiated m_{x}and that's the vector function, maybe just a coincidence?

I also have no idea how that is thegeneralsolution? Specifically the homogenious part.

I get that for the part of theparticularyou 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 del^{2}:

(Asin(kx) + Bcos(kx))''= m_{o}k.Cos(kx)

therefore: -A.k^{2}sin(kx)-Bk^{2}cos(kx) = m_{o}k.Cos(kx)

therefore: -A.ksin(kx)-Bkcos(kx) = m_{o}.Cos(kx) and equating coefficients yields:

B = - m_{o}/k

A = 0

so Yp = - m_{o}/k * Cos(kx)

But why does thehomogenouspart 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.

THANKS!!!

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

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# General Solution of a Poisson Equation (maybe difficult)

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