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The ``solution'' to this problem is to look at the melting point of copper, which is $\sim$1358 K at standard pressure. The copper cable will melt long before such a temperature is reached. \end{enumerate} + + \ex{1.7} + A $20,000 \Ohm/\V$ meter read, on its $1 \V$ scale, puts an $20,000 \Ohm/\V \cdot 1 \V = 20,000 \Ohm = 20\k\Ohm$ resistor in series with an ideal ammeter (ampere meter). Also, a voltage source with an internal resistance is equivalent to an ideal voltage source with its internal resistance in series. + \begin{enumerate} + \item In the first question, we have the following circuit: + \begin{circuit}{fig:1.7.1}{A voltage source with internal resistance and a $20,000 \Ohm/\V$ meter read in its $1 \V$ scale.} + (0,0) to[V=$1 \V$,invert] (0,4) + to[R=$10 \k\Ohm$,i>^=$I$] (5,4); + \draw[blue] (5,4) to[R=$20 \k\Ohm$,*-,color=blue] (5,2) + to node[draw,circle,fill=white] {A} (5,0); + \draw[blue] node[draw,circle,fill=blue,inner sep=1pt] at(5,0) {}; + \draw (5,0) to (0,0) + \end{circuit} + Then, we have that the current in the ideal ammeter and the voltage in the meter resistance are given by\footnote{When a meter only measures currents, it puts a resistance in series to measures the current through that resistance and internally converts that current into voltage to \textit{measure voltages}.} + \[I = \frac{1 \V}{10\k\Ohm + 20\k\Ohm} = \mans{0.0333 \m\A} \quad \text{ and } \quad V = 0.0333 \m\A \cdot 20\k\Ohm = \mans{0.666 \V}\] + \item In the second question, we have the following circuit: + \begin{circuit}{fig:1.7.2}{A $10\k\Ohm-10\k\Ohm$ voltage divider and a $20,000 \Ohm/\V$ meter read in its $1 \V$ scale.} + (0,0) to[V=$1 \V$,invert] (0,4) + to[short] (3,4) + to[R=$10 \k\Ohm$] (3,2) + to[R=$10 \k\Ohm$] (3,0); + \draw[blue] (3,2) to node[draw,circle,fill=white] {A} (5,2) + to[R=$20 \k\Ohm$,color=blue] (5,0) + to[short,-*,color=blue] (3,0); + \draw[blue] node[draw,circle,fill=blue,inner sep=1pt] at(3,2) {}; + \draw (3,0) to (0,0) + \end{circuit} + Now, we can to obtain the Thévenin equivalent circuit of circuit in Figure \ref{fig:1.7.2} with + \[R_{\Th} = \frac{10\k\Ohm \cdot 10\k\Ohm}{10\k\Ohm + 10\k\Ohm} = 5\k\Ohm\] + and + \[V_{\Th} = 1\V \cdot \frac{10\k\Ohm}{10\k\Ohm + 10\k\Ohm} = 0.5\V\] + Then, we have the following equivalent circuit: + \begin{circuit}{fig:1.7.3}{Thévenin equivalent circuit of circuit in Figure \ref{fig:1.7.2}.} + (0,0) to[V=$V_{\Th}$,invert] (0,4) + to[R=$R_{\Th}$,i>^=$I$] (5,4); + \draw node[draw,circle,fill=blue,inner sep=1pt] at(5,4) {}; + \draw[blue] (5,4) + to node[draw,circle,fill=white] {A} (5,2) + to[R=$20 \k\Ohm$,-*,color=blue] (5,0); + \draw (5,0) to (0,0) + \end{circuit} + Finally, we have that the current in the ideal ammeter and the voltage in the meter resistance are given by + \[I = \frac{0.5 \V}{5\k\Ohm + 20\k\Ohm} = \mans{0.02 \m\A} \quad \text{ and } \quad V = 0.02 \m\A \cdot 20\k\Ohm = \mans{0.4 \V}\] + \end{enumerate} + + \ex{1.10} \begin{enumerate}