Can Green's Function Solve Poisson's Equation in Cylindrical Geometry?

[pfknapp]

Hey everyone,
first time poster. I am looking for a general solution to poisson's equation in a cylindrical geometry. By general, I am thinking of two concentric hollow cylinders of finite height, and the solution for the field using the green's function in between the two cylinders. With this kind of geometry one can take the limit as either the inner radius goes to zero or the outer radius goes to infinity and therby cover all your bases and have the solution everywhere. Let me know if you can help. I have general solutions for spherical and cartesian, but I can't seem to find one for cylindrical. Thanks everyone.

 

[AiRAVATA]

What about polar coordinates?

 

[tacman]

Hi there,

Thank you for sharing your question with us. It sounds like you are looking for a general solution to Poisson's equation in cylindrical geometry using Green's function. This is a very interesting problem and I would be happy to provide some guidance.

To start, let's briefly review what Green's function is. Green's function is a mathematical tool used to solve differential equations by decomposing the problem into simpler parts. In the case of Poisson's equation, Green's function represents the potential at a point due to a point source located at a different point. In other words, it describes the response of the system to a localized disturbance.

Now, for a cylindrical geometry, the Green's function can be derived by considering the problem in cylindrical coordinates and using separation of variables. This will result in a Bessel function solution for the potential. However, for the specific case you mentioned, with two concentric hollow cylinders, the problem becomes more complicated and the solution cannot be obtained through simple separation of variables.

One approach you can take is to use a numerical method, such as finite element analysis, to solve for the potential in between the two cylinders. This will give you a general solution for any given inner and outer radius.

Another approach is to use a series expansion method, where you express the potential in terms of a series of Bessel functions and solve for the coefficients using boundary conditions at the two cylinders.

I hope this helps guide you in the right direction. If you have any further questions, please don't hesitate to ask. Good luck with your research!