\begin{problem*}{9}
In the Bohr theory of the hydrogen atom, an electron moves in a circular orbit about a proton, where the radius of the orbit is $r = 0.529\E{-10}\U{m}$.
\Part{a} Find the magnitude of the electric force each exerts on the other.
\Part{b} If this force causes the centripetal acceleration of the electron, what is the speed of the electron?
\end{problem*}

\empaddtoprelude{
  pair A, B, C;
  numeric r;
  r := 1cm;
  A := origin;
  B := (r, 0);
  C := (-r, 0);
  draw A--B;
  def drawCbot = 
    draw A--C;
    label.bot("r", (A+C)/2);
  enddef;
  def drawCtop = 
    draw_pcharge(A, 6pt);
    label.rt("v", draw_velocity(B-(0,1), B, .7*r));
    draw fullcircle scaled (2*r) shifted A;
    draw_ncharge(B, 3pt);
  enddef;
}

\begin{nosolution}
\begin{center}
\begin{empfile}[3ns]
\begin{emp}(0cm,0cm)
  drawCbot;
  drawCtop;
\end{emp}
\end{empfile}
\end{center}
\end{nosolution}

\begin{solution}
\begin{center}
\begin{empfile}[3]
\begin{emp}(0cm,0cm)
  drawCbot;
  label.top("F", draw_force(A, B, -r/2));
  drawCtop;
\end{emp}
\end{empfile}
\end{center}

\Part{a}
\begin{equation}
 F = k_e \frac{q^2}{r^2}
   = 8.99\E{9}\U{N$\cdot$m$^2$/C$^2$} \left(\frac{1.60\E{-19}\U{C}}{0.529\E{-10}\U{m}}\right)^2
   = \ans{8.22\E{-8}\U{N}}
\end{equation}

\Part{b}
Using $F_c = m a_c = m v^2/r$
\begin{equation}
 v = \sqrt{\frac{F r}{m}}
   = \sqrt{\frac{8.24\E{-8}\U{N} \cdot 0.529\E{-10}\U{m}}{9.11\E{-31}\U{kg}}}
   = \ans{2.19\E{6}\U{m/s}}
\end{equation}
\end{solution}
