Question

General solution of laplace equation in cylindrical coordinates

Answer 1

Laplace's Equation in Cylindrical Coordinates

Suppose that we wish to solve Laplace's equation,

$\displaystyle \nabla^{\,2}\phi = 0,$ (392)

within a cylindrical volume of radius $ a$ and height $ L$ . Let us adopt the standard cylindrical coordinates, $ r$ , $ \theta$ , $ z$ . Suppose that the curved portion of the bounding surface corresponds to $ r=a$ , while the two flat portions correspond to $ z=0$ and $ z=L$ , respectively. Suppose, finally, that the boundary conditions that are imposed at the bounding surface are

$\displaystyle \phi(r,\theta,0)$ $\displaystyle =0,$ (393)

$\displaystyle \phi(a,\theta,z)$ $\displaystyle =0,$ (394)

$\displaystyle \phi(r,\theta,L)$ $\displaystyle = {\mit\Phi}(r,\theta),$ (395)

where $ {\mit\Phi}(r,\theta)$ is a given function. In other words, the potential is zero on the curved and bottom surfaces of the cylinder, and specified on the top surface.

In cylindrical coordinates, Laplace's equation is written

$\displaystyle \frac{1}{r}\,\frac{\partial}{\partial r}\left(r\,\frac{\partial\p...

...{\,2}\phi}{\partial\theta^{\,2}}+\frac{\partial^{\,2}\phi}{\partial z^{\,2}}=0.$ (396)