Q: A parallel plate capacitor is constructed using two plates of area $$A = 0.02\ \mathrm{m}^2$$ separated by a distance $$d = 1\ \mathrm{mm}$$ and filled with air ($$\$$varepsilon_0$$ = 8.85 \times 10$$^{-12}$$\ \mathrm{F/m}). $$What is the capacitance of this capacitor?

$$C = \dfrac{\$$varepsilon_0$$ A}{d} = \dfrac{8.85 \times 10$$^{-12}$$ \times 0.02}{0.001} = 1.77 \times 10$$^{-10}$$\ \mathrm{F}$$

Q: If the voltage across a $$5\ \mathrm{\mu F}$$ capacitor is increased from $$2\ \mathrm{V} to$$ $$6\ \mathrm{V}$$, what is the change in charge stored on the capacitor?

$$\Delta Q = C \Delta V = 5 \times 10$$^{-6}$$ \times (6 - 2) = 2 \times 10$$^{-5}$$\ \mathrm{C}$$

Q: A $$10\ \mathrm{\mu F}$$ capacitor is charged to $$12\ \mathrm{V}. $$How much energy is stored in the capacitor?

$$U = \dfrac{1}{2} C V^2$ = \dfrac{1}{2} \times 10 \times 10$$^{-6}$$ \times 12^2$ = 0.00072\ \mathrm{J}$$

Q: Two capacitors, $$C_1 = 4$$\ \mathrm{\mu F}$$ and C_2$ = 6\ \mathrm{\mu F}$$, are connected in series. What is their equivalent capacitance?

$$C_{eq} = \dfrac{C_1$ C_2$}{C_1$ + C_2$} = \dfrac{4 \times 6}{4 + 6} = 2.4\ \mathrm{\mu F}$$

Q: In a circuit, three capacitors $$C_1 = 2$$\ \mathrm{\mu F}$$, C_2$ = 4\ \mathrm{\mu F}$$, and $$C_3 = 6$$\ \mathrm{\mu F}$$ are connected in parallel. What is the total charge stored if the combination is connected to a 12$$\ \mathrm{V}$$ battery?

$$Q_{total} = (C_1$ + C_2$ + C_3$) V = (2 + 4 + 6) \times 12 = 144\ \mathrm{\mu C}$$

Question 1

A parallel plate capacitor is constructed using two plates of area $$A = 0.02\ \mathrm{m}^2$$ separated by a distance $$d = 1\ \mathrm{mm}$$ and filled with air ($$\$$varepsilon_0$$ = 8.85 	imes 10$$^{-12}$$\ \mathrm{F/m}). $$What is the capacitance of this capacitor?

Accepted Answer

$$C = \dfrac{\$$varepsilon_0$$ A}{d} = \dfrac{8.85 	imes 10$$^{-12}$$ 	imes 0.02}{0.001} = 1.77 	imes 10$$^{-10}$$\ \mathrm{F}$$

Question 2

If the voltage across a $$5\ \mathrm{\mu F}$$ capacitor is increased from $$2\ \mathrm{V} to$$ $$6\ \mathrm{V}$$, what is the change in charge stored on the capacitor?

Accepted Answer

$$\Delta Q = C \Delta V = 5 	imes 10$$^{-6}$$ 	imes (6 - 2) = 2 	imes 10$$^{-5}$$\ \mathrm{C}$$

Question 3

A $$10\ \mathrm{\mu F}$$ capacitor is charged to $$12\ \mathrm{V}. $$How much energy is stored in the capacitor?

Accepted Answer

$$U = \dfrac{1}{2} C V^2$ = \dfrac{1}{2} 	imes 10 	imes 10$$^{-6}$$ 	imes 12^2$ = 0.00072\ \mathrm{J}$$

Question 4

Two capacitors, $$C_1 = 4$$\ \mathrm{\mu F}$$ and C_2$ = 6\ \mathrm{\mu F}$$, are connected in series. What is their equivalent capacitance?

Accepted Answer

$$C_{eq} = \dfrac{C_1$ C_2$}{C_1$ + C_2$} = \dfrac{4 	imes 6}{4 + 6} = 2.4\ \mathrm{\mu F}$$

Question 5

A $$3\ \mathrm{\mu F}$$ and a $$6\ \mathrm{\mu F}$$ capacitor are connected in parallel. What is the equivalent capacitance?

Accepted Answer

$$C_{eq} = C_1$ + C_2$ = 3 + 6 = 9\ \mathrm{\mu F}$$

Question 6

A parallel plate capacitor has plates of area $$A$$ and separation $$d. If a $$dielectric with dielectric constant $$K is $$inserted fully between the plates, how does the capacitance change?

Accepted Answer

$$C_{new} = K \dfrac{\$$varepsilon_0$$ A}{d}$$

Question 7

A $$2\ \mathrm{\mu F}$$ capacitor is connected to a $$9\ \mathrm{V}$$ battery. If the capacitor is then disconnected from the battery and a dielectric with $$K = 4 is $$inserted, what is the new voltage across the capacitor?

Accepted Answer

$$V_{new} = \dfrac{V_{old}}{K} = \dfrac{9}{4} = 2.25\ \mathrm{V}$$

Question 8

Which of the following best describes the effect of increasing the distance $$d$$ between the plates of a parallel plate capacitor (with no dielectric)?

Accepted Answer

The capacitance decreases, since $$C = \dfrac{\$$varepsilon_0$$ A}{d}$$

Question 9

A $$5\ \mathrm{\mu F}$$ capacitor is charged to $$10\ \mathrm{V}$$ and then disconnected from the battery. If the plate separation is doubled, what happens to the energy stored in the capacitor?

Accepted Answer

The energy increases by a factor of 2, since $$U = \dfrac{Q^2$}{2C}$$ and $$C$$ halves

Question 10

In a circuit, three capacitors $$C_1 = 2$$\ \mathrm{\mu F}$$, C_2$ = 4\ \mathrm{\mu F}$$, and $$C_3 = 6$$\ \mathrm{\mu F}$$ are connected in parallel. What is the total charge stored if the combination is connected to a 12$$\ \mathrm{V}$$ battery?

Accepted Answer

$$Q_{total} = (C_1$ + C_2$ + C_3$) V = (2 + 4 + 6) 	imes 12 = 144\ \mathrm{\mu C}$$

Question 11

A capacitor is being charged in a circuit. Which of the following statements is true about the current and voltage across the capacitor during charging?

Accepted Answer

The current decreases and the voltage increases as the capacitor charges

Question 12

A $$1\ \mathrm{\mu F}$$ capacitor and a $$2\ \mathrm{\mu F}$$ capacitor are connected in series across a $$12\ \mathrm{V}$$ battery. What is the voltage across the $$2\ \mathrm{\mu F}$$ capacitor?

Accepted Answer

$$V_2 = Q$$ / $$C_2$$, where $$Q = C_{eq} V = \dfrac{1 	imes 2}{1 + 2} 	imes 12 = 8\ \mathrm{\mu C}$$, so $$V_2 = 8 / 2 = 4$$\ \mathrm{V}$$

Question 13

Which of the following best explains why inserting a dielectric between the plates of a capacitor increases its capacitance?

Accepted Answer

The dielectric reduces the electric field for a given charge, allowing more charge to be stored for the same voltage

Question 14

A capacitor breakdown occurs when the electric field between the plates exceeds a certain value. What is the physical consequence of this breakdown?

Accepted Answer

The dielectric becomes conductive, allowing charge to flow and destroying the capacitor's ability to store energy

Question 15

A $$10\ \mathrm{\mu F}$$ capacitor is connected in series with a $$20\ \mathrm{\mu F}$$ capacitor and a $$24\ \mathrm{V}$$ battery. What is the charge on each capacitor?

Accepted Answer

$$Q = C_{eq} V = \dfrac{10 	imes 20}{10 + 20} 	imes 24 = 6.67 	imes 24 = 160\ \mathrm{\mu C}$$

Question 16

A parallel plate capacitor is filled halfway with a dielectric of $$K = 3$$ and halfway with air. What is the equivalent capacitance?

Accepted Answer

$$C_{eq} = \dfrac{2 \$$varepsilon_0$$ A}{d} \left( \dfrac{K}{K+1} \right)$$

Question 17

Which of the following is the correct expression for the energy stored in a capacitor in terms of charge $$Q$$ and capacitance $$C$$?

Accepted Answer

$$U = \dfrac{Q^2$}{2C}$$

Question 18

A $$4\ \mathrm{\mu F}$$ capacitor is connected to a $$16\ \mathrm{V}$$ battery. If the battery is disconnected and the plate separation is halved, what happens to the voltage across the capacitor?

Accepted Answer

The voltage doubles to $$32\ \mathrm{V}$$

Capacitors: Principles, Calculations, and Applications

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