\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;
  MultiTerminal B = source(type=DC, label="$V$");
  MultiTerminal r = resistor(B.terminal[1], type=normal, label="$r$");
  MultiTerminal R = resistor((0,u), type=normal, label="$R$");
  MultiTerminal I = current((r.terminal[1].x, R.terminal[1].y), dir=180,
      label=Label("$I$", align=N));
  wire(B.terminal[0], R.terminal[0], udsq);
  wire(r.terminal[1], I.terminal[0], udsq);
  wire(I.terminal[1], R.terminal[1], 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}
