\begin{problem}{19.40}
An insulating solid sphere of radius $a$ has a uniform volume charge
density $\rho$ and carries a total positive charge $Q$.  A spherical
gaussian surface of radius $r$, which shares a common center with the
insulating sphere, is inflated starting from $r=0$.
\Part{a} Find an expression for the electric flux $\Phi_E$ passing through the
surface of the gaussian sphere as a function of $r$ for $r < a$.
\Part{b} Find an expression for the electric flux $\Phi_E$ for $r > a$.
\Part{c} Plot $\Phi_E$(r).
\end{problem*} % problem 19.40

\begin{solution}
\Part{a}
\begin{equation}
 \Phi_E = \frac{q_{in}}{\epsilon_0}
        = \frac{Q \cdot 4/3 \cdot \pi r^3}{\epsilon_0 \cdot 4/3 \cdot \pi R^3}
        = \ans{\frac{Q r^3}{\epsilon_0 R^3}}
\end{equation}

\Part{b}
\begin{equation}
 \Phi_E = \frac{q_{in}}{\epsilon_0}
        = \ans{\frac{Q}{\epsilon_0}}
\end{equation}

\Part{c}
\empaddtoprelude{input graph}
\begin{center}
\begin{empfile}[3]
\begin{empgraph}(5cm, 3cm)
  % scaled so x=r/R, y=Phi/(Q/e0)
  glabel.bot(btex $r/R$ etex, OUT);
  glabel.lft(btex $\displaystyle \frac{\Phi_E \epsilon_0}{Q}$ etex, OUT);
  gdraw (0,0){right}..(.2,.008)..(.5, .125)..(.8,.512)..(1,1)--(2,1) withcolor green;
\end{empgraph}
\end{empfile}
\end{center}
\end{solution}
