Part 3 - Capacitors in Parallel
3.1 Charge capacitor \( C_{1} \) up to some potential difference \( \Delta V_{1 i} \) (Write down the value below), disconnect it from the supply, and connect it in parallel with the second capacitor \( \mathrm{C}_{2} \), which is initially uncharged.
\[
\begin{array}{l}
c_{1}=3.17 \mathrm{mF} \quad \mathrm{C}_{2}=\mathrm{1}-49 \mathrm{mF} \\
\Delta \mathrm{V}_{1 \mathrm{i}}=9.987 \quad \Delta \mathrm{V}_{2 \mathrm{i}}=0 \text { volts } \\
\end{array}
\]
a) Predictions
When the capacitors are again separated, what will the final potential differences across each of the capacitors be?
Calculation or reasoning for prediction:
\[
\begin{array}{crl}
Q_{1 i}=C_{1} \Delta V_{1 i} & Q_{2}=C_{2} \Delta V_{2 i}=0 \mathrm{C} \\
Q_{1 i}=\left(3.17 \times 10^{-3} \mathrm{~F}\right)(9.987 \mathrm{~V}) & C_{p}=C_{1}+C_{2} \\
Q_{\text {total }}=Q_{1 i}=0.031659 \mathrm{C} & C_{p}=3.17 \mathrm{mF}+1.99 \mathrm{mF} \\
\Delta V_{F}=\frac{Q_{\text {total }}}{C_{p}} & C_{p}=4.65 \mathrm{mF} \\
\Delta V_{F}=\frac{0.021459 \mathrm{C}}{4.66 \times 10^{-3} \mathrm{~F}}=6.79378 \mathrm{~V}
\end{array}
\]
7 Capacitors in parallel have the some voltage across their plates.
\( \Delta \mathrm{V}_{\mathrm{If}}= \) 6. \( 794 \vee \)
\[
\Delta \mathrm{V}_{2 \mathrm{f}}=
\]
\[
6.794
\]
b) Measurements
\[
\Delta v_{1 f}=6.740 \mathrm{~V}
\]
\[
\Delta \mathrm{V}_{2 \mathrm{f}}=
\]
\( 740 \mathrm{~V} \)
Comment on prediction vs measurement:
\[
\begin{aligned}
\% \text { diff } & =\frac{6.794-6.740}{6.740} \times 10 \% \\
& =0.8 \%
\end{aligned}
\]
\( > \) The measured value is relatively rimilar to the predicted value with only \( 0.9 \% \) percent dipperence.
3.2 Energy stored in the capacitors
a) Use the results of the previous section to calculate the initial energy stored in capacitor \( \mathrm{C}_{1} \) after it has been charged up to a potential difference \( \Delta V_{1} \)
b) Now calculate the total energy stored by \( \mathrm{C}_{1} \) and \( \mathrm{C}_{2} \) after they have been connected together, and then separated.
c) Is the energy stored initially the same as the energy stored finally? If not, why?
