Solution: CP CALC A region in space contains a total

QUESTION:

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

\(\begin{array}{l}\rho(r)=3 \alpha r /(2 R) \quad \text {for } r \leq R / 2\end{array}\)

\(\rho(r)=\alpha\left[1-(r / R)^{2}\right] \quad \text { for } R / 2 \leq r \leq R\)

\(\begin{array}{l}\rho(r)=0 \quad \text {for } r \geq R\end{array}\)

Here \(\alpha\) is a positive constant having units of \(\mathrm{C} / \mathrm{m}^{3}\). (a) Determine \(\alpha\) 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 the total charge Q. (c) What fraction of the total charge is contained within the region \(R / 2 \leq r \leq R\)? (d) What is the magnitude of \(\overrightarrow{\boldsymbol{E}}\) at r = R/2? (e) If an electron with charge \(q^{\prime}=-e\) is released from rest at any point in any of the three regions, the resulting motion will be oscillatory but not simple harmonic. Why? (See Challenge Problem 22.66.)