\begin{problem*}{28.57} % photoelectric effect
The following table shows data obtained in a photoelectric
experiment.  \Part{a} Using these data, make a graph similar to Active
Figure 28.9 that plots as a straight line.  From the graph,
determine \Part{b} an experimental value for Planck's constant (in
joule-seconds) and \Part{c} the work function (in electron volts) for
the surface.  (Two significant figures for each answer are
sufficient.)
\begin{center}
\begin{tabular}{r r}
Wavelength (nm) & Maximum Kinetic Energy of Photoelectrons (eV) \\
$588$ & $0.67$ \\
$505$ & $0.98$ \\
$445$ & $1.35$ \\
$399$ & $1.63$
\end{tabular}
\end{center}
\end{problem*} % problem 28.57

\begin{solution}
\Part{a}
We can convert the wavelengths to frequencies and graph them
\begin{center}
\begin{tabular}{r r}
$\lambda$ (nm) & $f=c/\lambda$ (THz) \\
$588$ & $510$ \\
$505$ & $594$ \\
$445$ & $674$ \\
$399$ & $751$
\end{tabular} \\
\begin{asy}
import graph;
import stats;
size(8cm, 5cm, IgnoreAspect);

real c = 299792458.0;

real fmin = 0;
real fmax = 800;

real[] x = {588,  505,  445,  399};  // nm
real[] y = {0.67, 0.98, 1.35, 1.63}; // eV

// convert wavelength in nm to freq in THz
int i;
for (i=0; i<x.length; ++i) {
  x[i] = c/x[i]/1e3;
}

linefit L=leastsquares(x, y);
write("work function");
write(L.b);
write("h/e");
write(L.m);

real fit(real f)
{
  return L.b + L.m*f;
}

draw(graph(fit, fmin, fmax), red);
draw(graph(x, y), p=invisible, marker=marker(scale(1pt)*unitcircle, blue));

pen thin=linewidth(0.5*linewidth())+grey;
xaxis("$f$ (THz)",BottomTop,
      LeftTicks(begin=false,end=false,extend=true,ptick=thin));
yaxis("$K_{max}$ (V)",LeftRight,
      RightTicks(begin=false,end=false,extend=true,ptick=thin));
\end{asy}
\end{center}

The least-squares fit yields parameters \Part{c}
$\phi=\ans{1.4\U{eV}}$ and $h/e = 4.04\U{mV/THz}$.  So \Part{b}
$h=\ans{6.5\E{-34}\U{J$\cdot$s}}$.
\end{solution}
