Which of the following capacitors has the largest charge? a) a parallel plate capacitor with an area of \(10 \mathrm{~cm}^{2}\) and a plate separation of \(2 \mathrm{~mm}\) connected to a \(10-\mathrm{V}\) battery b) a parallel plate capacitor with an area of \(5 \mathrm{~cm}^{2}\) and a plate separation of \(1 \mathrm{~mm}\) connected to a \(10-\mathrm{V}\) battery c) a parallel plate capacitor with an area of \(10 \mathrm{~cm}^{2}\) and a plate separation of \(4 \mathrm{~mm}\) connected to a \(5-\mathrm{V}\) battery d) a parallel plate capacitor with an area of \(20 \mathrm{~cm}^{2}\) and a plate separation of \(2 \mathrm{~mm}\) connected to a \(20-\mathrm{V}\) battery e) All of the capacitors have the same charge.

Step by step solution

01

Recall the formula for the charge of a parallel plate capacitor

The formula for the charge of a parallel plate capacitor is given by \(Q=\epsilon _0 A \frac{V}{d}.\)

02

Convert units

Before we calculate the charges, we need to convert the given areas from \(\mathrm{cm}^2\) to \(\mathrm{m}^2,\) and the plate separations from \(\mathrm{mm}\) to \(\mathrm{m}.\) The conversion factors are: 1 cm = 0.01 m; 1 mm = 0.001 m.

03

Calculate the charge for each capacitor

The vacuum permittivity is given by \(\epsilon _0 = 8.854\times10^{-12}\mathrm{~F~m}^{-1}\). Calculate the charges for each capacitor using the formula and the converted units: a) \(Q_a = \epsilon_0 A_a \frac{V_a}{d_a} = (8.854\times10^{-12})(0.001) \frac{10}{0.002} = 44.27\times10^{-12}\mathrm{C}\) b) \(Q_b = \epsilon_0 A_b \frac{V_b}{d_b} = (8.854\times10^{-12})(0.0005) \frac{10}{0.001} = 44.27\times10^{-12}\mathrm{C}\) c) \(Q_c = \epsilon_0 A_c \frac{V_c}{d_c} = (8.854\times10^{-12})(0.001) \frac{5}{0.004} = 11.0675\times10^{-12}\mathrm{C}\) d) \(Q_d = \epsilon_0 A_d \frac{V_d}{d_d} = (8.854\times10^{-12})(0.002) \frac{20}{0.002} = 177.08\times10^{-12}\mathrm{C}\)

04

Compare the charges

Compare the charges: \(Q_a = 44.27\times10^{-12}\mathrm{C}\) \(Q_b = 44.27\times10^{-12}\mathrm{C}\) \(Q_c = 11.0675\times10^{-12}\mathrm{C}\) \(Q_d = 177.08\times10^{-12}\mathrm{C}\) The largest charge is found in option d) with a value of \(177.08\times10^{-12}\mathrm{C}\). Therefore, the parallel plate capacitor with an area of \(20\mathrm{~cm}^2\) and a plate separation of \(2\mathrm{~mm}\) connected to a \(20-\mathrm{V}\) battery has the largest charge.

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