Q: A parallel-plate capacitor has plates of area $$A = 0.1\ \mathrm{m}^2$$ separated by $$d = 1\ \mathrm{mm} of $$air. What is the capacitance of this capacitor? ($$\$$epsilon_0$$ = 8.85 \times 10$$^{-12}$$\ \mathrm{F/m})$$

$$C = \dfrac{\$$epsilon_0$$ A}{d} = 8.85 \times 10$$^{-10}$$\ \mathrm{F}$$

Q: A $$100\ \mu\mathrm{F}$$ capacitor is charged to $$12\ \mathrm{V}. $$How much energy is stored in the capacitor?

$$U = \dfrac{1}{2} C V^2$ = 7.2 \times 10$$^{-2}$$\ \mathrm{J}$$

Q: A $$10\ \mu\mathrm{F}$$ capacitor is connected to a $$20\ \mathrm{V}$$ battery. If a dielectric with $$\kappa = 2 is $$inserted while the battery remains connected, what is the new charge on the capacitor?

$$Q = \kappa C_0$ V = 2 \times 10 \times 10$$^{-6}$$ \times 20 = 4.0 \times 10$$^{-4}$$\ \mathrm{C}$$

Q: A $$2\ \mu\mathrm{F}$$ and a $$4\ \mu\mathrm{F}$$ capacitor are connected in parallel across a $$12\ \mathrm{V}$$ battery. What is the total charge stored in the system?

$$Q_{total} = (C_1$ + C_2$)V = (2 + 4) \times 12 = 72\ \mu\mathrm{C}$$

Q: A defibrillator stores $$400\ \mathrm{J} of $$energy in a capacitor. If the voltage across the capacitor is $$2000\ \mathrm{V}$$, what is the capacitance?

$$C = \dfrac{2U}{V^2$} = \dfrac{2 \times 400}{(2000)^2$} = 0.2\ \mathrm{mF}$$

Question 1

A parallel-plate capacitor has plates of area $$A = 0.1\ \mathrm{m}^2$$ separated by $$d = 1\ \mathrm{mm} of $$air. What is the capacitance of this capacitor? ($$\$$epsilon_0$$ = 8.85 	imes 10$$^{-12}$$\ \mathrm{F/m})$$

Accepted Answer

$$C = \dfrac{\$$epsilon_0$$ A}{d} = 8.85 	imes 10$$^{-10}$$\ \mathrm{F}$$

Question 2

A $$100\ \mu\mathrm{F}$$ capacitor is charged to $$12\ \mathrm{V}. $$How much energy is stored in the capacitor?

Accepted Answer

$$U = \dfrac{1}{2} C V^2$ = 7.2 	imes 10$$^{-2}$$\ \mathrm{J}$$

Question 3

If the distance between the plates of a parallel-plate capacitor is doubled while keeping the charge constant, what happens to the electric field $$E$$, voltage $$V$$, and capacitance $$C$$?

Accepted Answer

$$E$$ remains the same, $$V$$ doubles, $$C$$ halves

Question 4

Two capacitors, $$C_1 = 2$$\ \mu\mathrm{F}$$ and C_2$ = 4\ \mu\mathrm{F}$$, are connected in series across a $$12\ \mathrm{V}$$ battery. What is the equivalent capacitance?

Accepted Answer

$$C_{eq} = \dfrac{1}{\dfrac{1}{C_1$} + \dfrac{1}{C_2$}} = 1.33\ \mu\mathrm{F}$$

Question 5

A $$10\ \mu\mathrm{F}$$ capacitor is connected to a $$20\ \mathrm{V}$$ battery. If a dielectric with $$\kappa = 2 is $$inserted while the battery remains connected, what is the new charge on the capacitor?

Accepted Answer

$$Q = \kappa C_0$ V = 2 	imes 10 	imes 10$$^{-6}$$ 	imes 20 = 4.0 	imes 10$$^{-4}$$\ \mathrm{C}$$

Question 6

Which of the following statements is true for capacitors connected in parallel?

Accepted Answer

The equivalent capacitance is the sum of the individual capacitances: $$C_{eq} = C_1$ + C_2$ + \ldots$$

Question 7

A cylindrical capacitor consists of an inner cylinder of radius $$a = 1\ \mathrm{cm}$$ and an outer cylinder of radius $$b = 2\ \mathrm{cm}$$, both of length $$L = 50\ \mathrm{cm}. $$What is its capacitance? ($$\$$epsilon_0$$ = 8.85 	imes 10$$^{-12}$$\ \mathrm{F/m})$$

Accepted Answer

$$C = \dfrac{2 \pi \$$epsilon_0$$ L}{\ln(b/a)}$$

Question 8

A $$150\ \mu\mathrm{F}$$ capacitor is charged to $$200\ \mathrm{V}$$ and then disconnected from the battery. If the plate separation is doubled, what happens to the voltage across the plates?

Accepted Answer

The voltage doubles.

Question 9

A spherical capacitor consists of two concentric spheres with radii $$a$$ and $$b$$ ($$b \u003e a). $$What is the formula for its capacitance?

Accepted Answer

$$C = 4 \pi \$$epsilon_0$$ \dfrac{ab}{b - a}$$

Question 10

A parallel-plate capacitor is connected to a battery and then a dielectric is inserted between the plates. Which of the following quantities increases?

Accepted Answer

The charge on the plates

Question 11

A $$2\ \mu\mathrm{F}$$ and a $$4\ \mu\mathrm{F}$$ capacitor are connected in parallel across a $$12\ \mathrm{V}$$ battery. What is the total charge stored in the system?

Accepted Answer

$$Q_{total} = (C_1$ + C_2$)V = (2 + 4) 	imes 12 = 72\ \mu\mathrm{C}$$

Question 12

A defibrillator stores $$400\ \mathrm{J} of $$energy in a capacitor. If the voltage across the capacitor is $$2000\ \mathrm{V}$$, what is the capacitance?

Accepted Answer

$$C = \dfrac{2U}{V^2$} = \dfrac{2 	imes 400}{(2000)^2$} = 0.2\ \mathrm{mF}$$

Question 13

A parallel-plate capacitor has an energy density $$u_E = \dfrac{1}{2} \$$epsilon_0$$ E^2$ between its plates. If the electric field is doubled, what happens to the energy density?

Accepted Answer

It increases by a factor of 4.

Question 14

Three capacitors, each of $$2\ \mu\mathrm{F}$$, are connected so that two are in parallel and the combination is in series with the third. What is the equivalent capacitance?

Accepted Answer

$$C_{eq} = \dfrac{2C}{3} = 1.33\ \mu\mathrm{F}$$

Question 15

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

Accepted Answer

$$V_1$$ = $$Q/C_1 = 8$$\ \mu\mathrm{C} / 1\ \mu\mathrm{F} = 8\ \mathrm{V}$$

Question 16

A parallel-plate capacitor is charged and then disconnected from the battery. If a dielectric is inserted, what happens to the energy stored in the capacitor?

Accepted Answer

The energy decreases by a factor of $$\kappa.$$

Question 17

A parallel-plate capacitor with area $$A$$ and separation $$d is $$filled with two dielectrics of constants $$\$$kappa_1$$ and $$\$$kappa_2$$ stacked in parallel, each covering half the area. What is the equivalent capacitance?

Accepted Answer

$$C_{eq} = \dfrac{1}{2}(\$$kappa_1$$ + \$$kappa_2$$)\$$epsilon_0$$ \dfrac{A}{d}$$

Question 18

A camera flash uses a $$150\ \mu\mathrm{F}$$ capacitor charged to $$200\ \mathrm{V}. $$How much charge is stored on the capacitor?

Accepted Answer

$$Q = C V = 150 	imes 10$$^{-6}$$ 	imes 200 = 0.03\ \mathrm{C}$$

Capacitors and Capacitance: Concepts, Calculations, and Applications

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