Skip to main content Accessibility help
×
Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-10-31T23:27:10.188Z Has data issue: false hasContentIssue false

7 - Spherical Systems

Published online by Cambridge University Press:  05 June 2012

Andrea Prosperetti
Affiliation:
The Johns Hopkins University and University of Twente
Get access

Summary

In many ways the sphere is the prototypical three-dimensional body and the consideration of fields in the presence of spherical boundaries sheds light on several features of more general three-dimensional cases.

In all the examples of this chapter extensive use is made of expansions in series of Legendre polynomials, for axi-symmetric problems, or spherical harmonics, for the general three-dimensional case. After a review of the polar coordinate system, we begin with a summary of the properies of these functions which are dealt with in greater detail in Chapters 13 and 14, respectively. While the axi-symmetric situation is somewhat simpler, it is also contained as a special case in the general three-dimensional one and it is therefore expedient to treat it as a special case of the latter.

We start with the general solution of the Laplace and Poisson equations (§7.3) and apply it to several axisymmetric (§§7.4 and 7.5) and non-axisymmetric situations. In all these cases the radial part of the solution consists of powers of r. The examples in the second part of the chapter (§7.13 and §7.14) deal with the scalar Helmholtz equation, for which the radial dependence is expressed in terms of spherical Bessel functions, the fundamental properties of which are summarized in §7.12. The last four sections deal with problems involving vector fields and vector harmonics.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×