Multi Resistor/Capacitor Circuit

Electricity and Magnetism Level 2

In hopes of creating an RC circuit , you have a $9V$ battery arranged in a circuit of five capacitors ( $C$ ) and five resistors ( $R$ ). This circuit has resistors with the following resistances: $R_1=5Ω$ , $R_2=3Ω$ , $R_3=7Ω$ , $R_4=6Ω$ , $R_5=3Ω$ . The capacitors have the following capacitance values: $C_1=1F$ , $C_2=5F$ , $C_3=2F$ , $C_4=4F$ , and $C_5=8F$ . Determine the circuit's time constant ( $\tau$ ) in seconds.

Note : $\tau =RC$ .

David's Electricity Set

The answer is 34.

1 solution

David Hontz
May 24, 2016

The key is to simplify the resistors and capacitors by noticing which are in series and which are in parallel. $C_{parallel} = C_1+C_2+C_3=1+5+2=\underline{C_{parallel}=8F} \\ \frac{1}{C_{circuit}} = \frac{1}{C_4}+\frac{1}{C_5}+\frac{1}{C_{parallel}} = \frac{1}{8}+\frac{1}{4}+\frac{1}{8}=\frac{1}{2} \Rightarrow \boxed{C_{circuit}=2F} \\ \frac{1}{R_{parallel}}=\frac{1}{R_4}+\frac{1}{R_5}=\frac{1}{3}+\frac{1}{6}=\frac{1}{2} \Rightarrow \underline{R_{parallel}=2Ω} \\ R_{circuit}=R_1+R_2+R_3+R_{parallel}=5+3+7+2=\boxed{R_{circuit}=17Ω} \\ \\ \tau = R_{circuit}C_{circuit} = (17Ω)(2F)=34 \frac{Ωs}{Ω} = \boxed{34s}$

Multi Resistor/Capacitor Circuit

David's Electricity Set

The answer is 34.

1 solution

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