FOUR CAPACITORS CONNECTED IN PARALLEL GOAL Analyze a circuit with several capacitors in parallel. PROBLEM (a) Determine the capacitance of the single capacitor that is equivalent to the parallel combination of capacitors shown in Figure 16.19. Find (b) the charge on the 12.0-μF capacitor and (c)

Question

FOUR CAPACITORS CONNECTED IN PARALLEL

GOAL Analyze a circuit with several capacitors in parallel.

PROBLEM (a) Determine the capacitance of the single capacitor that is equivalent to the parallel combination of capacitors shown in Figure 16.19. Find (b) the charge on the 12.0-[latex]\mu \mathrm{F}[/latex] capacitor and (c) the total charge contained in the configuration. (d) Derive a sym bolic expression for the fraction of the total charge contained on one of the capacitors.

STRATEGY For part (a), add the individual capacitances. For part (b), apply the formula [latex]C=Q / \Delta V[/latex] to the [latex]12.0-\mu \mathrm{F}[/latex] capacitor. The volt age difference is the same as the difference across the battery. To find the total charge contained in all four capacitors, use the equiva lent capacitance in the same formula.

Accepted Answer

(a) Find the equivalent capacitance.

Apply Equation 16.12:

[latex]C_{eq} = C_1 + C_2 + C_3 + \cdots \left( {}_{ ext{combination}}^{ ext{parallel}} ight)[/latex] [16.12]

[latex]\begin{aligned}C_{ ext {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}[/latex]

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

Solve the capacitance equation for [latex]Q[/latex] and substitute:

[latex]\begin{aligned}Q &=C_{3} \Delta V=\left(12.0 imes 10^{-6} \mathrm{~F} ight)(18.0 \mathrm{~V})=216 imes 10^{-6} \mathrm{C} \&=216 \mu \mathrm{C}\end{aligned}[/latex]

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

Use the equivalent capacitance:

[latex]C_{\mathrm{eq}}=\frac{Q}{\Delta V} \quad ightarrow \quad Q=C_{\mathrm{eq}} \Delta \mathrm{V}=(45.0 \mu \mathrm{F})(18.0 \mathrm{~V})=8.10 imes 10^{2} \mu \mathrm{C}[/latex]

(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 [latex]i[/latex] th capacitor and use the capacitor definition:

[latex]\frac{Q_{i}}{Q_{ ext {tot }}}=\frac{C_{i} \Delta V}{C_{ ext {eq }} \Delta V}=\frac{C_{i}}{C_{ ext {eq }}}[/latex]

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.

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

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