Chapter 22, Problem 22.62CP

CP CALC A region in space contains a total positive charge Q that is distributed spherically such that the volume charge density ρ(r) is given by

$\begin{array}{cc}\rho (\text{r})=3\text{αr}/2\text{R}& for\text{ r }\le \text{ R}/2\\ \rho (\text{r})=\text{ α}[1-{(\text{r}/\text{R})}^2]& for\text{ R}/2\le \text{ r }\le \text{ R}\\ \rho (\text{r})=0& for\text{ r }\ge \text{ R}\end{array}$

Here α is a positive constant having units of C/m³, (a) Determine α in terms of Q and R. (b) Using Gauss’s law, derive an expression for the magnitude of the electric field as a function of r. Do this separately for all three regions. Express your answers in terms of Q. (c) What fraction of the total charge is contained within the region R/2 ≤ r ≤ R? (d) What is the magnitude of $\stackrel{\to }{\text{E}}$ at r = R/2? (e) If an    electron with charge q' = −e is released from rest at any point in any of the three regions, the resulting     motion will be oscillatory but not simple harmonic. Why?