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# Capacitors Circuits

## Circuits containing capacitors are easy to understand, take a closer look at them!

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Capacitors Circuits

When a capacitor is placed in a circuit, current does not actually travel across it. Rather, equal and opposite charge begins to build up on opposite sides of the capacitor --- mimicking a current --- until the electric field in the capacitor creates a potential difference across it that balances the voltage drop across any parallel resistors or the voltage source itself (if there are no resistors in parallel with the capacitor). The ratio of charge on a capacitor to potential difference across it is called capacitance.

It is important to break down a complicated circuit into the equivalent capacitance using the rules for capacitors in series and capacitors in parallel. Also remember that capacitors in parallel have the same voltage while capacitors in series have the same charge.

Key Equations
C=QVDefinition of Capacitance\begin{align*} C = \frac{Q}{V} && \text{Definition of Capacitance} \end{align*}

Cparallel1Cseries=C1+C2+C3+=1C1+1C2+1C3+Capacitors in parallel add like resistors in seriesCapacitors in series add like resistors in parallel\begin{align*} C_{\text{parallel}} &= C_1 + C_2 + C_3 + \dots && \text{Capacitors in parallel add like resistors in series}\\ \frac{1}{C_{\text{series}}} &= \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3}+ \dots && \text{Capacitors in series add like resistors in parallel} \end{align*}

#### Example

In the circuit shown below, determine (a) the total capacitance and (b) the total charge stored.

(a): In solving this problem, we'll call the 20μF\begin{align*}20\;\mu\text{F}\end{align*} capacitor C1\begin{align*}C_1\end{align*}, the 60μF\begin{align*}60\;\mu\text{F}\end{align*} capacitor C2\begin{align*}C_2\end{align*}, the 40μF\begin{align*}40\;\mu\text{F}\end{align*} capacitor C3\begin{align*}C_3\end{align*}, and the 100μF\begin{align*}100\;\mu\text{F}\end{align*} capacitor C4\begin{align*}C_4\end{align*}.

Our first step will be to find the equivalent capacitance of C2\begin{align*}C_2\end{align*} and C3\begin{align*}C_3\end{align*}.

1C2,31C2,31C2,3C2,3=1C2+1C3=160μF+140μF=5120μF=24F\begin{align*} \frac{1}{C_{2,3}}&=\frac{1}{C_2}+\frac{1}{C_3}\\ \frac{1}{C_{2,3}}&=\frac{1}{60\;\mu\text{F}} + \frac{1}{40\;\mu\text{F}}\\ \frac{1}{C_{2,3}}&=\frac{5}{120\;\mu\text{F}}\\ C_{2,3}&=24\;\text{F}\\ \end{align*}

Next, we'll combine the capacitance of C2,3\begin{align*}C_{2,3}\end{align*} and C4\begin{align*}C_4\end{align*}.

C2,3,4C2,3,4C2,3,4=C2,3+C4=24μF+100μF=124μF\begin{align*} C_{2,3,4}&=C_{2,3} + C_4\\ C_{2,3,4}&=24\;\mu\text{F} + 100\;\mu\text{F}\\ C_{2,3,4}&=124\;\mu\text{F}\\ \end{align*}

Finally, we can combine C2,3,4\begin{align*}C_{2,3,4}\end{align*} with C1\begin{align*}C_1\end{align*} to find the total capacitance.

1Ceq1Ceq1CeqCeq=1C1+1C2,3,4=120μF+1124μF=.058μF1=17.22μF\begin{align*} \frac{1}{C_{eq}}&=\frac{1}{C_1} + \frac{1}{C_{2,3,4}}\\ \frac{1}{C_{eq}}&=\frac{1}{20\;\mu\text{F}} + \frac{1}{124\;\mu\text{F}}\\ \frac{1}{C_{eq}}&=.058\;\mu\text{F}^{-1}\\ C_{eq}&=17.22\;\mu\text{F}\\ \end{align*}

(b): Now we can use this value to find the total charge stored on all the capacitors by also using the voltage provided on the diagram.

QQQ=CV=17.22μF10V=172.2μC\begin{align*} Q&=CV\\ Q&=17.22\;\mu\text{F} * 10\;\text{V}\\ Q&=172.2\mu\text{C} \end{align*}

### Simulation

Circuit Constructoin Kit (AC+DC) (PhET Simlation)

### Review

1. Consider the figure above with switch, S\begin{align*}S\end{align*}, initially open and the power supply set to 24 V:
1. What is the voltage drop across the 20Ω\begin{align*}20 \Omega \end{align*} resistor?
2. What current flows thru the 60Ω\begin{align*}60 \Omega\end{align*} resistor?
3. What is the voltage drop across the 20\begin{align*}20\end{align*} microfarad capacitor?
4. What is the charge on the capacitor?
5. How much energy is stored in that capacitor?
6. Find the capacitance of capacitors B\begin{align*}B\end{align*}, C\begin{align*}C\end{align*}, and D\begin{align*}D\end{align*} if compared to the 20μF\begin{align*}20 \mu \mathrm{F}\end{align*}capacitor where...
1. B\begin{align*}B\end{align*} has twice the plate area and half the plate separation
2. C\begin{align*}C\end{align*} has twice the plate area and the same plate separation
3. D\begin{align*}D\end{align*} has three times the plate area and half the plate separation
2. Now the switch in the previous problem is closed.
1. What is the total capacitance of branch with B and C?
2. What is the total capacitance of the circuit?
3. What is the voltage drop across capacitor B\begin{align*}B\end{align*}?

1. a. 6V\begin{align*}6\mathrm{V}\end{align*} b. 0.3A\begin{align*}0.3\mathrm{A}\end{align*} c. 18V\begin{align*}18\mathrm{V}\end{align*} d. 3.6×104C\begin{align*}3.6 \times 10^{-4}\mathrm{C}\end{align*} e. 3.2×103J\begin{align*}3.2 \times 10^{-3}\mathrm{J}\end{align*} f. i) 80μF\begin{align*}80 \mu \mathrm{F}\end{align*} ii) 40μF\begin{align*}40\mu \mathrm{F}\end{align*} iii) 120μF\begin{align*}120 \mu \mathrm{F}\end{align*}
2. a. 26.7μF\begin{align*}26.7\mu \mathrm{F}\end{align*} b. 166.7μF\begin{align*}166.7\mu \mathrm{F}\end{align*}

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