Chapter 25 Sketch a schematic diagram of a circuit with a parallel

Chapter: Chapter 25

Learning Objectives

LO 25.1.0 Solve problems related to capacitance.

LO 25.1.1 Sketch a schematic diagram of a circuit with a parallel-plate capacitor, a battery, and

an open or closed switch.

LO 25.1.2 In a circuit with a battery, an open switch, and an uncharged capacitor, explain what

happens to the conduction electrons when the switch is closed.

LO 25.1.3 For a capacitor, apply the relationship between the magnitude of charge q on either

plate (“the charge on the capacitor”), the potential difference V between the plates (“the potential

across the capacitor”), and the capacitance C of the capacitor.

LO 25.2.0 Solve problems related to calculating the capacitance.

LO 25.2.1 Explain how Gauss’ law is used to find the capacitance of a parallel-plate capacitor.

LO 25.2.2 For a parallel-plate capacitor, a cylindrical capacitor, a spherical capacitor, and an

isolated sphere, calculate the capacitance.

LO 25.3.0 Solve problems related to capacitors in parallel and in series.

LO 25.3.1 Sketch schematic diagrams for a battery and (a) three capacitors in parallel and (b)

three capacitors in series.

LO 25.3.2 Identify that capacitors in parallel have the same potential difference, which is the

same value that their equivalent capacitor has.

LO 25.3.3 Calculate the equivalent capacitance of several capacitors in parallel.

LO 25.3.4 Identify that the total charge stored on parallel capacitors is the sum of the charges

stored on the individual capacitors.

LO 25.3.5 Identify that capacitors in series have the same charge, which is the same value that

their equivalent capacitor has.

LO 25.3.6 Calculate the equivalent capacitance of several capacitors in series.

LO 25.3.7 Identify that the potential applied to capacitors in series is equal to the sum of the

potentials across the individual capacitors.

LO 25.3.8 For a circuit with a battery and some capacitors in parallel and some in series,

simplify the circuit in steps by finding equivalent capacitors, until the charge and potential on the

final equivalent capacitor can be determined, and then reverse the steps to find the charge and

potential on the individual capacitors.

LO 25.3.9 For a circuit with a battery, an open switch, and one or more uncharged capacitors,

determine the amount of charge that moves through a point in the circuit when the switch is

closed.

LO 25.3.10 When a charged capacitor is connected in parallel to one or more uncharged

capacitors, determine the charge and potential difference on each capacitor when equilibrium is

reached.

LO 25.4.0 Solve problems related to energy stored in an electric field.

LO 25.4.1 Explain how the work required to charge a capacitor results in the potential energy of

the capacitor.

LO 25.4.2 For a capacitor, apply the relationship between the potential energy U, the

capacitance C, and the potential difference V.

LO 25.4.3 For a capacitor, apply the relationship between the potential energy, the internal

volume, and the internal energy density.

LO 25.4.4 For any electric field, apply the relationship between the potential energy density u

in the field and the field’s magnitude E.

LO 25.4.5 Explain the danger of sparks in airborne dust.

LO 25.5.0 Solve problems related to capacitor with a dielectric.

LO 25.5.1 Identify that capacitance is increased if the space between the plates is filled with a

dielectric material.

LO 25.5.2 For a capacitor, calculate the capacitance with and without a dielectric.

LO 25.5.3 For a region filled with a dielectric material with a given dielectric constant κ, identify

that all electrostatic equations containing the permittivity constant ε0 are modified by multiplying

that constant by the dielectric constant to get κε0.

LO 25.5.4 Name some of the common dielectrics.

LO 25.5.5 In adding a dielectric to a charged capacitor, distinguish the results for a capacitor (a)

connected to a battery and (b) not connected to a battery.

LO 25.5.6 Distinguish polar dielectrics from nonpolar dielectrics.

LO 25.5.7 In adding a dielectric to a charged capacitor, explain what happens to the electric field

between the plates in terms of what happens to the atoms in the dielectric.

LO 25.6.0 Solve problems related to dielectrics and Gauss’ law.

LO 25.6.1 In a capacitor with a dielectric, distinguish free charge from induced charge.

LO 25.6.2 When a dielectric partially or fully fills the space in a capacitor, find the free charge,

the induced charge, the electric field between the plates (if there is a gap, there is more than one

field value), and the potential between the plates.

Multiple Choice

1. The units of capacitance are equivalent to:

A) J/C

B) V/C

C) J2/C

D) C/J

E) C2/J

2. A farad is the same as a:

A) J/V

B) V/J

C) C/V

D) V/C

E) N/C

3. A parallel-plate capacitor C has a charge Q. The actual charges on its plates are:

A) Q, Q

B) Q/2, Q/2

C) Q, –Q

D) Q/2, –Q/2

E) Q, 0

4. Each plate of a capacitor stores a charge of magnitude 1 mC when a 100-V potential

difference is applied. The capacitance is:

A) 5



F

B) 10



F

C) 50



F

D) 100



F

E) none of these

5. To charge a 1-F capacitor with 2 C requires a potential difference of:

A) 0.2 V

B) 0.5 V

C) 2 V

D) 5 V

E) none of these

6. If the charge on a parallel-plate capacitor is doubled:

A) the capacitance is halved

B) the capacitance is doubled

C) the electric field is halved

D) the electric field is doubled

E) the surface charge density is not changed on either plate

7. The capacitance of a parallel-plate capacitor with plate area A and plate separation d is given

by:

A) 0d/A

B) 0d/2A

C) 0A/d

D) 0A/2d

E) d/0

8. The capacitance of a parallel-plate capacitor is:

A) proportional to the plate area

B) proportional to the charge stored

C) independent of any material inserted between the plates

D) proportional to the potential difference of the plates

E) proportional to the plate separation

9. The plate areas and plate separations of five parallel plate capacitors are

capacitor 1: area A0, separation d0

capacitor 2: area 2A0, separation 2d0

capacitor 3: area 2A0, separation d0/2

capacitor 4: area A0/2, separation 2d0

capacitor 5: area A0, separation d0/2

Rank these according to their capacitances, least to greatest.

A) 1, 2, 3, 4, 5

B) 5, 4, 3, 2, 1

C) 5, then 3 and 4 tie, then 1, then 2

D) 4, then 1 and 2 tie, then 5, then 3

E) 3, then 5, then 1 and 2 tie, then 4

10. The capacitance of a parallel-plate capacitor can be increased by:

A) increasing the charge

B) decreasing the charge

C) increasing the plate separation

D) decreasing the plate separation

E) decreasing the plate area

11. If both the plate area and the plate separation of a parallel-plate capacitor are doubled, the

capacitance is:

A) doubled

B) halved

C) unchanged

D) one-fourth the original

E) quadrupled

12. If the plate area of an isolated charged parallel-plate capacitor is doubled:

A) the electric field is doubled

B) the potential difference is halved

C) the charge on each plate is halved

D) the surface charge density on each plate is doubled

E) none of the above

13. If the plate separation of an isolated charged parallel-plate capacitor is doubled:

A) the electric field is doubled

B) the potential difference is halved

C) the charge on each plate is halved

D) the surface charge density on each plate is doubled

E) none of the above

14. Pulling the plates of an isolated charged capacitor apart:

A) increases the capacitance

B) increases the potential difference

C) does not affect the potential difference

D) decreases the potential difference

E) does not affect the capacitance

15. A parallel-plate capacitor has a plate area of 0.2 m2 and a plate separation of 0.1 mm. To

obtain an electric field of 2.0  106 V/m between the plates, the magnitude of the charge on each

plate should be:

A) 3.5  10–6 C

B) 7.1  10–6 C

C) 1.4  10–5 C

D) 1.8  10–5 C

E) 8.9  10–5 C

16. A parallel-plate capacitor has a plate area of 0.2 m2 and a plate separation of 0.1 mm. If the

charge on each plate has a magnitude of 4  10–6 C the potential difference across the plates is

approximately:

A) 0 V

B) 4  10–2 V

C) 2  102 V

D) 2  105 V

E) 4  108 V

17. The capacitance of a spherical capacitor with inner radius a and outer radius b is

proportional to:

A) a/b

B) b – a

C) b2 – a2

D) ab/(b – a)

E) ab/(b2 – a2)

18. The capacitance of a single isolated spherical conductor with radius R is proportional to:

A) R

B) R2

C) 1/R

D) 1/R2

E) none of these

19. The capacitance of a cylindrical capacitor can be increased by:

A) decreasing both the radius of the inner cylinder and the length

B) increasing both the radius of the inner cylinder and the length

C) increasing the radius of the outer cylindrical shell and decreasing the length

D) decreasing the radius of the inner cylinder and increasing the radius of the outer cylindrical

shell

E) only by decreasing the length

20. Two conducting spheres have radii of R1 and R2 with R1 greater than R2. If they are far

apart the capacitance is proportional to:

A) R1R2/(R1 – R2)

B) 𝑅2

2− 𝑅1

2

C) (R1 – R2)/R1R2

D) 𝑅2

2+ 𝑅1

2

E) none of these

21. A parallel-plate capacitor has a plate area of 0.3 m2 and a plate separation of 0.1 mm. If the

charge on each plate has a magnitude of 5  10–6 C then the force exerted by one plate on the

other has a magnitude of about:

A) 5 N

B) 9 N

C) 1  104 N

D) 9  105 N

E) 2  107 N

22. Each of the four capacitors shown is 500



F. The voltmeter reads 1000V. The

magnitude of the charge on each capacitor plate is:

A) 0.2 C

B) 0.5 C

C) 20 C

D) 50 C

E) none of these

23. Two parallel-plate capacitors with the same plate area but different capacitance are

connected in parallel to a battery. Both capacitors are filled with air. The quantity that is the

same for both capacitors when they are fully charged is:

A) potential difference

B) energy density

C) electric field between the plates

D) charge on the positive plate

E) plate separation

24. Two parallel-plate capacitors with different plate separation but the same capacitance are

connected in series to a battery. Both capacitors are filled with air. The quantity that is NOT the

same for both capacitors when they are fully charged is:

A) potential difference

B) stored energy

C) electric field between the plates

D) charge on the positive plate

E) dielectric constant

25. Two parallel-plate capacitors with different capacitance but the same plate separation are

connected in series to a battery. Both capacitors are filled with air. The quantity that is the same

for both capacitors when they are fully charged is:

A) potential difference

B) stored energy

C) energy density

D) electric field between the plates

E) charge on the positive plate

26. Capacitor C1 and C2 are connected in parallel. The equivalent capacitance is given by:

A) C1C2/(C1 + C2)

B) (C1 + C2)/C1C2

C) 1/(C1 + C2)

D) C1/C2

E) C1 + C2

27. A battery is used to charge a parallel combination of two identical capacitors. If the

potential difference across the battery terminals is V and the total charge Q flows through the

battery during the charging process then the charge on the positive plate of each capacitor and

the potential difference across each capacitor are:

A) Q/2 and V/2, respectively

B) Q and V, respectively

C) Q/2 and V, respectively

D) Q and V/2, respectively

E) Q and 2V, respectively

28. A 2-



F and a 1-



F capacitor are connected in series and a potential difference is applied

across the combination. The 2-



F capacitor has:

A) twice the charge of the 1-



F capacitor

B) half the charge of the 1-



F capacitor

C) twice the potential difference of the 1-



F capacitor

D) half the potential difference of the 1-



F capacitor

E) none of the above

29. A 2-



F and a 1-



F capacitor are connected in parallel and a potential difference is applied

across the combination. The 2-



F capacitor has:

A) twice the charge of the 1-



F capacitor

B) half the charge of the 1-



F capacitor

C) twice the potential difference of the 1-



F capacitor

D) half the potential difference of the 1-



F capacitor

E) none of the above

30. Capacitors C1 and C2 are connected in series and a potential difference is applied to the

combination. If the capacitor that is equivalent to the combination has the same potential

difference, then the charge on the equivalent capacitor is the same as:

A) the charge on C1

B) the sum of the charges on C1 and C2

C) the difference of the charges on C1 and C2

D) the product of the charges on C1 and C2

E) none of the above

31. Capacitors C1 and C2 are connected in series. The equivalent capacitance is given by:

A) C1C2/(C1 + C2)

B) (C1 + C2)/C1C2

C) 1/(C1 + C2)

D) C1/C2

E) C1 + C2

32. Two identical capacitors are connected in series and two, each identical to the first, are

connected in parallel. The equivalent capacitance of the series connection is ________ the

equivalent capacitance of parallel connection.

A) twice

B) four times

C) half

D) one fourth

E) the same as

33. A 2-



F and a 1-



F capacitor are connected in series and charged from a battery. They store

charges P and Q, respectively. When disconnected and charged separately using the same

battery, they have charges R and S, respectively. Then:

A) R > S > Q = P

B) P > Q > R = S

C) R > P = Q > S

D) R = P > S = Q

E) R > P > S = Q

34. Capacitors C1 and C2 are connected in parallel and a potential difference is applied to the

combination. If the capacitor that is equivalent to the combination has the same potential

difference, then the charge on the equivalent capacitor is the same as:

A) the charge on C1

B) the sum of the charges on C1 and C2

C) the difference of the charges on C1 and C2

D) the product of the charges on C1 and C2

E) none of the above

35. A battery is used to charge a series combination of two identical capacitors. If the

potential difference across the battery terminals is V and total charge Q flows through the battery

during the charging process then the charge on the positive plate of each capacitor and the

potential difference across each capacitor are:

A) Q/2 and V/2, respectively

B) Q and V, respectively

C) Q/2 and V, respectively

D) Q and V/2, respectively

E) Q and 2V, respectively

36. Two identical capacitors, each with capacitance C, are connected in parallel and the

combination is connected in series to a third identical capacitor. The equivalent capacitance of

this arrangement is:

A) 2C/3

B) C

C) 3C/2

D) 2C

E) 3C

37. The diagram shows six 6-



F capacitors. The capacitance between points a and b is:

A) 1



F

B) 3



F

C) 4



F

D) 6



F

E) 9



F

38. Each of the three 25-



F capacitors shown is initially uncharged. How many coulombs of

charge pass through the ammeter A after the switch S is closed?

A) 0.033 C

B) 0.10 C

C) 0.30 C

D) 10 C

E) none of these

39. Two parallel-plate capacitors with the same plate separation but different capacitance are

connected in parallel to a battery. Both capacitors are filled with air. The quantity that is NOT

the same for both capacitors when they are fully charged is:

A) potential difference

B) energy density

C) electric field between the plates

D) charge on the positive plate

E) dielectric constant

40. Capacitor C1 is connected alone to a battery and charged until the magnitude of the charge

on each plate is 4.0  10−8 C. Then it is removed from the battery and connected to two other

capacitors C2 and C3, as shown. The charge on the positive plate of C1 is then 1.0  10−8 C.

The charges on the positive plates of C2 and C3 are:

A) q2 = 3.0  10−-8 C and q3 = 3.0  10−8 C

B) q2 = 2.0  10−8 C and q3 = 2.0  10−8 C

C) q2 = 5.0  10−8 C and q3 = 1.0  10−8 C

D) q2 = 3.0  10−8 C and q3 = 1.0  10−8 C

E) q2 = 1.0  10−8 C and q3 = 3.0  10−8 C

41. Let Q denote charge, V denote potential difference and U denote stored energy. Of these

quantities, capacitors in series must have the same:

A) Q only

B) V only

C) U only

D) Q and U only

E) V and U only

42. Let Q denote charge, V denote potential difference and U denote stored energy. Of these

quantities, capacitors in parallel must have the same:

A) Q only

B) V only

C) U only

D) Q and U only

E) V and U only

43. A 20-µF capacitor is charged to 200 V. Its stored energy is:

A) 4000 J

B) 4 J

C) 0.4 J

D) 0.1 J

E) 0.004 J

44. A charged capacitor stores 10 C at 40 V. Its stored energy is:

A) 400 J

B) 200 J

C) 4 J

D) 2.5 J

E) 1.25 J

45. A 2-



F and a 1-



F capacitor are connected in series and charged by a battery. They store

energies P and Q, respectively. When disconnected and charged separately using the same

battery, they have energies R and S, respectively. Then:

A) R > P > S > Q

B) P > Q > R > S

C) R > P > Q > S

D) P > R > S > Q

E) R > S > Q > P

46. Capacitors A and B are identical. Capacitor A is charged so it stores 4 J of energy and

capacitor B is uncharged. The capacitors are then connected in parallel. The total stored energy

in the capacitors is now:

A) 16 J

B) 8 J

C) 4 J

D) 2 J

E) 1 J

47. To store a total of 0.040 J of energy in the two identical capacitors shown, each should

have a capacitance of:

A) 0.50



F

B) 1.0



F

C) 1.5



F

D) 2.0



F

E) 4.0



F

48. A certain capacitor has a capacitance of 5.0



F. After it is charged to 5



C and isolated, the

plates are brought closer together so its capacitance becomes 10



F. The work done by the agent

is about:

A) 0 J

B) 1.25  10–6 J

C) −1.25  10–6 J

D) 8.3  10–7 J

E) −8.3  10–7 J

49. A battery is used to charge a parallel-plate capacitor, after which it is disconnected. Then

the plates are pulled apart to twice their original separation. This process will double the:

A) capacitance

B) surface charge density on each plate

C) stored energy

D) electric field between the two places

E) charge on each plate

50. A parallel-plate capacitor has a plate area of 0.30 m2 and a plate separation of 0.10 mm. If the

charge on each plate has a magnitude of 5.0  10–6 C, what is the energy density in its electric

field?

A) 0.16 J/m3

B) 3.5 J/m3

C) 7.8 J/m3

D) 16 J/m3

E) 24 J/m3

51. The quantity (1/2)0E2 has the dimensions of:

A) energy/farad

B) energy/coulomb

C) energy

D) energy/volume

E) energy/volt

52. There are approximately 10 explosions in the US every year due to agricultural grain (corn

and soybean) dust. Why is corn dust so dangerous?

A) Corn contains a lot of oil and is therefore very flammable.

B) Grain elevators use a lot of flammable material while processing the corn.

C) Corn dust has a lot of surface area and can be ignited by even a very small spark, which can

be created simply by a person walking around.

D) Due to the presence of lots of agricultural material, grain elevators are especially rich in

oxygen.

E) Workers are not careful about where they throw their cigarette butts, and sometimes set the

corn on fire.

53. An air-filled parallel-plate capacitor has a capacitance of 1 pF. The plate separation is then

doubled and a wax dielectric is inserted, completely filling the space between the plates. As a

result, the capacitance becomes 2 pF. The dielectric constant of the wax is:

A) 0.25

B) 0.50

C) 2.0

D) 4.0

E) 8.0

54. One of the materials listed below is to be placed between two identical metal sheets, with

no air gap, to form a parallel-plate capacitor. Which produces the greatest capacitance?

A) material of thickness 0.1 mm and dielectric constant 2

B) material of thickness 0.2 mm and dielectric constant 3

C) material of thickness 0.3 mm and dielectric constant 2

D) material of thickness 0.4 mm and dielectric constant 8

E) material of thickness 0.5 mm and dielectric constant 11

55. Which of the following is not a dielectric?

A) silicon

B) germanium

C) polystyrene

D) titanium

E) paper

56. A dielectric slab is slowly inserted between the plates of a parallel plate capacitor while the

capacitor is connected to a battery. As it is being inserted:

A) the capacitance, the potential difference between the plates, and the charge on the positive

plate all increase

B) the capacitance, the potential difference between the plates, the charge on the positive plate

all decrease

C) the potential difference between the plates increases, the charge on the positive plate

decreases, and the capacitance remains the same

D) the capacitance and the charge on the positive plate decrease but the potential difference

between the plates remains the same

E) the capacitance and the charge on the plate increase but the potential difference between the

plates remains the same

57. A parallel-plate capacitor, with air dielectric, is charged by a battery, after which the

battery is disconnected. A slab of glass dielectric is then slowly inserted between the plates. As it

is being inserted:

A) a force repels the glass out of the capacitor

B) a force attracts the glass into the capacitor

C) no force acts on the glass

D) a net charge appears on the glass

E) the glass makes the plates repel each other

58. What is the difference between a polar dielectric and a nonpolar dielectric?

A) A polar dielectric has a permanent electric field.

B) A nonpolar dielectric never has an internal electric field.

C) The molecules of a polar dielectric have a permanent electric dipole moment.

D) A nonpolar dielectric can have an induced electric field in any direction.

E) A polar dielectric is always aligned with the Earth’s electric field.

59. What happens to the atoms in a dielectric when it is placed between the plates of a charged

capacitor?

A) They begin to conduct electricity.

B) They create an induced electric field that is in the opposite direction of the field due to the

charges on the plates.

C) They create an induced electric field that is in the same direction as the field due to the

charges on the plates.

D) They completely cancel the electric field due to the charges on the plates.

E) They rotate so their positive ends are towards the positively charged plate.

60. An air-filled capacitor is charged, and then a dielectric is inserted. As a result, there is an

induced charge on the dielectric. What is the difference between induced charge and free charge?

A) There is no difference.

B) Induced charge does not result in a net charge on the dielectric.

C) Induced charge is smaller than free charge.

D) Free charge creates an electric field, but induced charge does not.

E) Free charge creates an electric potential, but induced charge does not.

61. Two capacitors are identical except that one is filled with air and the other with oil. Both

capacitors carry the same charge. The ratio of the electric fields Eair/Eoil is:

A) between 0 and 1

B) 0

C) 1

D) between 1 and infinity

E) infinite