\begin{problem*}{Q19.13} % internal resistance
Explain in detail how you could measure the internal resistance of a
battery.
\end{problem*}

\begin{solution}
  \begin{center}
\begin{asy}
  import Circ;
  real u = 1cm;
  TwoTerminal B = source((0,0), DC, 0, "", "$V$");
  TwoTerminal r = resistor(B.end, normal, 0, "", "$r$");
  TwoTerminal R = resistor((0,u), normal, 0, "$R$", "");
  TwoTerminal I = current((r.end.x, R.end.y), 180, "", "$I$");
  wire(B.beg, R.beg, udsq);
  wire(r.end, I.beg, udsq);
  wire(I.end, R.end, nsq);
\end{asy}
  \end{center}
Make a circuit using a known resistance $R$ to connect the two
terminals of the battery, and measure the current $I$.

From Kirchhoff's loop rule
\begin{align*}
  V - Ir - IR &= 0 \\
  Ir &= V - IR \\
  r &= \ans{\frac{V}{I} - R}
\end{align*}
\end{solution}
