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Section 9.2 Densities with Step Functions

Typically, charge and mass densities do not extend throughout all of space, rather they are limited to the inside of some physical object. In some instances, it is useful to have an algebraic way of describing such a charge density that shows explicitly where it turns on and turns off. This is an example of the use of Step Functions; see Section 8.1.

Activity 9.2.1. Charge density inside a spherical shell.

Use step functions to write the charge density inside an insulating spherical shell (with finite thickness).

Hint.

  1. Make sure to name the things you don't know, like the inside and outside radii of the shell.

  2. There are several equivalent ways of writing step functions that “turn off”. Choose any of them.

  3. Don't assume that the charge density is constant inside the shell, or that it has any particular symmetry, unless you are explicitly told so.

Answer.

If the inside radius is \(a\) and the outside radius is \(b\text{,}\) then:

\begin{equation*} \rho(\vec{r})_{\textrm{everywhere}} = \rho(r,\theta,\phi)_{\textrm{inside}} \left(\theta(r-a)-\theta(r-b)\right) . \end{equation*}