Help with E/M Potential using Laplace Solution

[Fjolvar]

I'm struggling to learn how to find the potential using laplace solution. I know that X(x) can be rewritten in terms of C1e^kx + D1e-kx OR C1 cosh kx + D1 sinh kx OR cos kx + sin kx... but when do you know how to use which form. I understand it partially but not fully. And then how do you consider Y(y)? Given its a 2-dim problem in cartesian. Any information or references would be immensely appreciated!

 

[Fjolvar]

C1e^kx + D1e-kx can be rewritten as C1 cosh kx + D1 sinh kx, correct? Depending on the geometry of the problem, but cos kx + sin kx cannot be rewritten right? You need one sinusoidal and one hyperbolic function in the 2dim case, is that right?

 

[Born2bwire]

Cosine and sine can be related to cosh and sinh by scaling the argument by the imaginary number i. So there really is no difference between either one of the three formulations because in the end, all it will change is the phase of your solved modes k. But as long as you properly solve for the coefficients and k then you should come up with equivalent solutions for either of the three cases because all three form equivalent bases for the solutions to Laplace's equation in terms of the modes k. Although, some cases can be easier than others if you know more about the conditions of the problem. For example, if you know that the boundary conditions at infinity are Dirichlet with values of zero then using the exponential form is desirable because we can quickly see that the coefficient on the exp(kx) term must be zero otherwise the solution would blow up. This is not so readily seen if we took the sinusoidal or even hyperbolic sinusoidal cases.

 

[Nabeshin]

If it's truly a 2-dimensional problem you just suppress the 3rd component (you can equally well do the formalism of separation of variables in only 2D, it turns out to be the same just with only two functions).

As far as choosing sin/cos or sinh/cosh, this kind of thing is revealed only by the boundary conditions. Let's say you know that X(x) is sins and cosines. Then the other component (assuming 2D) must be sinh/cosh (or the specific case of exponentials). This follows directly from the laplace equation, where you know that not all of the coefficients can be real/imaginary, but there must be some mixing.

Laplace equation is just all about the boundary conditions.