Laplace's Equation in Cylindrical Coordinates

(392) |

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

where 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

(396) |

Let us try a separable solution of the form

Proceeding in the usual manner, we obtain

Note that we have selected exponential, rather than oscillating, solutions in the -direction [by writing , instead of , in Equation (399)]. As will become clear, this implies that the radial solutions oscillate, which is the appropriate choice for the particular set of boundary conditions under consideration. The solution to Equation (399), subject to the constraint that [which follows from the first boundary condition, (394)] is

(401) |

The most general solution to Equation (400) is

(402) |

Note that, to ensure that the potential is single-valued in , the constant is constrained to be an integer. Finally, if we write then Equation (401) becomes

(403) |

This equation is known as

(404) |

This solution, which is known as a

(405) | ||

(406) |

In other words, at small arguments the function has a power-law behavior, whereas at large arguments it takes the form of an oscillation of slowly decaying amplitude. It follows that

(407) |

Let denote the th zero of the Bessel function . In other words, is the th root (in order, as increases from zero) of . The values of the can be looked up in standard reference books.

Thus, our separable solution becomes

It is convenient to express the specified function in the form of a Fourier series: that is,

(410) |

Our final boundary condition, (396), then yields

(411) | ||

(412) |

It remains to invert the previous two expressions to obtain the coefficients and . In fact, it is possible to demonstrate that if

(413) |

then

(414) |

Hence,

(415) | ||

(416) |

and our solution is fully determined.

Consider the limit that . In this case, according to Equation (409), the allowed values of become more and more closely spaced. Consequently, the sum over discrete -values in (410) morphs into an integral over a continuous range of -values. For instance, suppose that we wish to solve Laplace's equation in the region , subject to the boundary condition that as and , with , where is specified. In this case, we would choose in order to satisfy the boundary condition at large . The choice ensures that the potential is well behaved at , and automatically satisfies the boundary condition at large . Hence, our general solution becomes

(417) |

If we write

(418) |

then the final boundary condition implies that

(419) | ||

(420) |

We can invert the previous two expressions by means of the identity

(421) |

Hence, we obtain

(422) | ||

(423) |

and our solution is fully defined.

Suppose that we wish to solve Laplace's equation in a cylindrical volume of radius and height , subject to the boundary conditions

where is specified. In other words, the potential is zero on the two flat portions of the bounding surface, and given on the curved portion. We can again look for a separable solution of the form (398). Proceeding in the usual manner, we obtain

Note that we have selected oscillating, rather than exponential solutions in the -direction [by writing , instead of , in Equation (428)]. This is the appropriate choice for the particular set of boundary conditions under consideration. The solution to Equation (428), subject to the constraints that [which follow from the boundary conditions (425) and (426)] is

(430) |

where

(431) |

Here, is a positive integer. The single-valued solution to Equation (429) is again

(432) |

Finally, writing , Equation (430) takes the form

(433) |

This equation is known as the

This solution, which is known as a

(435) | ||

(436) |

In other words, at small arguments the function has a power-law behavior, whereas at large arguments it grows exponentially. It follows that

(437) |

Thus, our separable solution becomes

(438) |

If we express the function as a Fourier series in and , so that

(439) |

then the boundary condition (427) yields

(440) | ||

(441) |

Hence, our solution is fully specified.