Question 16.7: FOUR CAPACITORS CONNECTED IN PARALLEL GOAL Analyze a circuit...

(a) Find the equivalent capacitance.

Apply Equation 16.12:

C_{eq} = C_1 + C_2 + C_3 + \cdots \left( {}_{\text{combination}}^{\text{parallel}} \right)        [16.12]

\begin{aligned}C_{\text {eq }} &=C_{1}+C_{2}+C_{3}+C_{4} \\&=3.00  \mu \mathrm{F}+6.00  \mu \mathrm{F}+12.0  \mu \mathrm{F}+24.0  \mu \mathrm{F} \\&=45.0  \mu \mathrm{F}\end{aligned}

(b) Find the charge on the 12-\mu \mathrm{F} capacitor (designated C_{3} ).

Solve the capacitance equation for Q and substitute:

\begin{aligned}Q &=C_{3} \Delta V=\left(12.0 \times 10^{-6} \mathrm{~F}\right)(18.0 \mathrm{~V})=216 \times 10^{-6}  \mathrm{C} \\&=216  \mu \mathrm{C}\end{aligned}

(c) Find the total charge contained in the configuration.

Use the equivalent capacitance:

C_{\mathrm{eq}}=\frac{Q}{\Delta V}  \quad \rightarrow \quad Q=C_{\mathrm{eq}} \Delta \mathrm{V}=(45.0  \mu \mathrm{F})(18.0 \mathrm{~V})=8.10 \times 10^{2}  \mu \mathrm{C}

(d) Derive a symbolic expression for the fraction of the total charge contained in one of the capacitors.

Write a symbolic expression for the fractional charge in the i th capacitor and use the capacitor definition:

\frac{Q_{i}}{Q_{\text {tot }}}=\frac{C_{i} \Delta V}{C_{\text {eq }} \Delta V}=\frac{C_{i}}{C_{\text {eq }}}

REMARKS The charge on any one of the parallel capacitors can be found as in part (b) because the potential difference is the same. Notice that finding the total charge does not require finding the charge on each individual capacitor and adding. It’s easier to use the equivalent capacitance in the capacitance definition.