# Solving Laplace equation in Cylindrical coordinates with azimuthal symmetry?

I am having some trouble solving what the potential field looks like for a finite cylinder along the $$z$$ direction.

Physical Scenario

Consider a cylindrical electrode with radius $$R_0$$ that extends along $$z$$ from $$z = 0$$ to $$z= z_0$$. This electrode is at a fixed potential $$\phi_0$$. The electrode is an empty shell.

What have I tried

I want to solve Laplace's equation: $$\nabla ^2\phi(r,z) = 0$$

with boundary conditions: $$\phi(R_0, z) = \phi_0, z \in [0,z_0]$$

and $$\lim_{z \to \pm\infty} \phi(r,z) = 0.$$

Given the azimuthal symmetry in my configuration then Laplace equation is: $$\frac{\partial ^2 \phi}{\partial r^2} + \frac{1}{r}\frac{\partial \phi}{\partial r} + \frac{\partial ^2 \phi}{\partial z^2} = 0$$

And after separation of variables the problem reduces to: $$\frac{d^2Z}{dz^2} - k^2Z = 0$$ $$\frac{d^2R}{dr^2} + \frac{1}{r}\frac{dR}{dr} + k^2R = 0$$ where $$k$$ is just a constant.

The solutions would then be of the form: $$Z(z) = A\exp(kz)+B\exp(-kz)$$ $$R(r) = J_0(kr)$$

where $$J_0$$ is a Bessel function of first kind.

What is my Problem?

As $$\phi$$ vanishes when $$z$$ tends to infinity and minus infinity, wouldn't these conditions set both $$A$$ and $$B$$ in $$Z(z)$$ to zero?

I am also very confused as to how the value of $$k$$ is dealt with. Is the solution $$Z \times R$$ summed over values of $$k$$? How do the set of boundary conditions come into place, would they restrict $$k$$?

My end goal is to take advantage of the in-built C++ Bessel functions to get the solution for the potential.

EDIT

The potential $$\phi_0$$ is a constant on the surface of the electrode. My end goal is to simulate the electrostatic potential caused in a Penning-Malmberg Trap. As of this question I am only interested in a single electrode.

• It is not so easy. Try to decompose $\phi_0$ into a complete basis, for example using Fourier series. Oct 24 '19 at 13:08
• To be frank, the real problem here is that the physical situation isn't quite clear enough, and until that is settled, worrying about the analytical details is pretty pointless. What happens at the edges of the electrode? What happens at the electrode itself ─ are you looking for solutions with constant potential at the electrode wall? Are you looking for a specific solution, or for all possible solutions consistent with that electrode? Oct 24 '19 at 15:54
• Related Math.SE post by OP: math.stackexchange.com/q/3390194/11127 Oct 24 '19 at 17:07
• @EmilioPisanty The electrode is set at a fixed potential i.e. $\phi (R,z) = V$ where $V$ is a constant. My end goal is to simulate the Electric field caused by a Penning-Malmberg Trap (alpha.web.cern.ch/penningtrap). But for now, before worrying at multiple electrodes I am even struggling with a single one. Oct 24 '19 at 18:32
• This PhD-thesis shows an approximate solution to this problem on pages 54ff. It also provides links to theses, where the same problem is solved with different approaches. Disclaimer: The author was a colleague of mine, many years ago. Jan 1 '20 at 22:30

You are partly correct. If $$\phi(z,r)$$ vanishes at $$z\pm \infty$$ then $$A=B=0$$ and $$\phi$$ is identically zero. Furthermore, this is in conflict with the value of $$\phi(z,R_0)=\phi_0$$ unless $$\phi_0=0$$. The problem is that you have chosen the wrong set of solutions to Laplace's equation.
All solutions of Laplace oscillate in one direction and decay expoentially in the other. You want ones that oscillate along the $$z$$ direction (so you can match the $$\phi(z,R_0) = \phi_0$$ when $$0 boundary conditions) and decay or grow along the radial direction. In other words you need solutions like $$e^{ikz}I_0(kr)$$ or $$e^{ikz} K_0(kr)$$ where $$I$$ and $$K$$ are the modified Bessel functions. You will need to use $$I_0(kr)$$ for $$r and $$K_0(kr)$$ for $$r>R_0$$.
The problem is that although the physical situation is clear, the BC's are not. You have a nice boundadry condition at $$r=R_0$$ for $$z\in[0,z_0]$$ there is no simple BC for $$z\notin[0,z_0]$$ Even if there were it would still be a hard problem. For example think of a case when the electrode is strip of width $$w$$ lying parallel in the $$x,y$$ plane and parallel to the x axis. On the strip you have the BC $$\phi(x,y,0)=\phi_0$$ for $$0 and by symmetry you have vanishing electric field off the strip so $$\partial_z \phi(x,y,z)|_{z=0}=0$$ for $$y\notin [0,w]$$. This is a mixed boundary problem (Dirichlet on the strip and Neumann offf the strip) and is hard. You, however, do not even know the analogue of the Neumann BC off your cylinder.
• If I choose my solution to decrease exponentially along $r$, and I want them to tend to zero as $r$ goes to infinity, wouldn't that imply that they are higher at $r=0$?(so that they can decrease as they move far away from 0). Isn't this opposite to what the solution should actually be like? Shouldn't the potential be lower at $r=0$ compared to $r = R_0$? Oct 24 '19 at 13:23