Topic 20: Capacitance and Dielectrics

The charge on the square plates of a parallel‐plate capacitor is Q. The potential across the plates is maintained with constant voltage by a battery as they are pulled apart to twice their original separation, which is small compared to the dimensions of the plates. The amount of charge on the plates is now equal to

A) 4Q.

B) 2Q.

C) Q.

D) Q/2.

E) Q/4.

D) Q/2.

The electric field between square the plates of a parallel‐plate capacitor has magnitude E. The potential across the plates is maintained with constant voltage by a battery as they are pulled apart to twice their original separation, which is small compared to the dimensions of the plates. The magnitude of the electric field between the plates is now equal to

A) 4E.

B) 2E.

C) E.

D) E/2.

E) E/4.

D) E/2.

Equal but opposite charges Q are placed on the square plates of an air‐filled parallel‐plate capacitor. The plates are then pulled apart to twice their original separation, which is small compared to the dimensions of the plates. Which of the following statements about this capacitor are true? (There may be more than one correct choice.)

A) The energy stored in the capacitor has doubled.

B) The energy density in the capacitor has increased.

C) The electric field between the plates has increased.

D) The potential difference across the plates has doubled.

E) The capacitance has doubled.

A) The energy stored in the capacitor has doubled.

D) The potential difference across the plates has doubled.

When two or more capacitors are connected in series across a potential difference

A) the potential difference across the combination is the algebraic sum of the potential differences across the

individual capacitors.

B) each capacitor carries the same amount of charge.

C) the equivalent capacitance of the combination is less than the capacitance of any of the capacitors.

D) All of the above choices are correct.

E) None of the above choices are correct.

D) All of the above choices are correct.

When two or more capacitors are connected in parallel across a potential difference

A) the potential difference across each capacitor is the same.

B) each capacitor carries the same amount of charge.

C) the equivalent capacitance of the combination is less than the capacitance of any of the capacitors. D) All of the above choices are correct.

E) None of the above choices are correct.

A) the potential difference across each capacitor is the same.

In the circuit shown in the figure, the capacitors are initially uncharged. The switch is first thrown to position A and kept there for a long time. It is then thrown to position B. Let the charges on the capacitors be Q1, Q2, and

Q3 and the potential differences across them be V1, V2, and V3. Which of the following conditions must be true with the switch in position B?

A) V1 = V2 = V3

B) V1 + V2 = V3

C) V3 = V0

D) Q1 = Q2 = Q3

E) Q1 + Q2 = Q3

B) V1 + V2 = V3

An ideal parallel‐plate capacitor consists of a set of two parallel plates of area A separated by a very small distance d. When this capacitor is connected to a battery that maintains a constant potential difference between the plates, the energy stored in the capacitor is U0. If the separation between the plates is doubled, how much

energy is stored in the capacitor?

A) 4U0

B) 2U0

C) U0

D) U0/2

E) U0/4

D) U0/2

An ideal parallel‐plate capacitor consists of a set of two parallel plates of area A separated by a very small distance d. When the capacitor plates carry charges +Q and -Q, the capacitor stores energy U0. If the separation

between the plates is doubled, how much electrical energy is stored in the capacitor?

A) 4U0

B) 2U0

C) U0

D) U0/2

E) U0/4

B) 2U0

An ideal air‐filled parallel‐plate capacitor has round plates and carries a fixed amount of equal but opposite charge on its plates. All the geometric parameters of the capacitor (plate diameter and plate separation) are now DOUBLED. If the original capacitance was C0, what is the new capacitance?

A) 4C0

B) 2C0

C) C0

D) C0/2

E) C0/4

B) 2C0

An ideal air‐filled parallel‐plate capacitor has round plates and carries a fixed amount of equal but opposite charge on its plates. All the geometric parameters of the capacitor (plate diameter and plate separation) are now DOUBLED. If the original energy stored in the capacitor was U0, how much energy does it now store?

A) 4U0

B) 2U0

C) U0

D) U0/2

E) U0/4

D) U0/2

An ideal air‐filled parallel‐plate capacitor has round plates and carries a fixed amount of equal but opposite charge on its plates. All the geometric parameters of the capacitor (plate diameter and plate separation) are now DOUBLED. If the original energy density between the plates was u0, what is the new energy density?

A) 16u0

B) 4u0

C) u0

D) u0/4

E) u0/16

E) u0/16

A charged capacitor stores energy U. Without connecting this capacitor to anything, dielectric having dielectric constant K is now inserted between the plates of the capacitor, completely filling the space between them. How much energy does the capacitor now store?

A) 2KU

B) KU

C) U

D) U/K

E) U/2K

D) U/K

Two capacitors, C1 and C2, are connected in series across a source of potential difference. With the potential source still connected, a dielectric is now inserted between the plates of capacitor C1. What happens to the charge on capacitor C2?

A) The charge on C2 increases.

B) The charge on C2 decreases.

C) The charge on C2 remains the same.

A) The charge on C2 increases.

An air‐filled parallel‐plate capacitor is connected to a battery and allowed to charge up. Now a slab of dielectric material is placed between the plates of the capacitor while the capacitor is still connected to the battery. After this is done, we find that

A) the energy stored in the capacitor had decreased.

B) the voltage across the capacitor had increased.

C) the charge on the capacitor had increased.

D) the charge on the capacitor had not changed.

E) None of these choices are true.

C) the charge on the capacitor had increased.

Each plate of a parallel-plate air-filled capacitor has an area of 0.0020 m2, and the separation of the plates is 0.020 mm. An electric field of 3.9 × 106 V/m is present between the plates. What is the surface charge density on the plates? (ε0 = 8.85 × 10^-12 C2^/N · m^2)

A) 35 μC/m^2

B) 73 μC/m^2

C) 17 μC/m^2

D) 52 μC/m^2

E) 87 μC/m^2

A) 35 μC/m^2

A metal cylinder of radius 2.0 mm is concentric with another metal cylinder of radius 5.0 mm. If the space between the cylinders is filled with air and the length of the cylinders is 50 cm, what is the capacitance of this

arrangement? (k = 1/4πε0 = 8.99 × 10^9 N · m^2/C^2)

A) 33 pF

B) 60 pF

C) 22 pF

D) 30 pF

E) 11 pF

D) 30 pF

The capacitance per unit length of a very long coaxial cable, made of two concentric cylinders, is 50 pF/m. What is the radius of the outer cylinder if the radius of the inner one is 1.0 mm? (k = 1/4πε0 = 8.99 × 10^9 N · m^2/C^2)

A) 3.0 mm

B) 2.0 mm

C) 4.0 mm

D) 1.0 mm

E) 0.50 mm

A) 3.0 mm

A cylindrical capacitor is made of two thin-walled concentric cylinders. The inner cylinder has radius r1 = 4.0 mm, and the outer one a radius r2= 8.0 mm. The common length of the cylinders is L = 150 m. What is the potential energy stored in this capacitor when a potential difference 4.0 V is applied between the inner and outer cylinder? (k = 1/4πε0 = 8.99 × 10^9 N · m2/C2)

A) 9.6 × 10^-8 J

B) 1.3 × 10^-8 J

C) 6.3 × 10^-8 J

D) 0.34 × 10^-8 J

E) 4.6 × 10^-8 J

A) 9.6 × 10^-8 J

A 1.0 m long piece of coaxial cable has a wire with a radius of 1.1 mm and a concentric conductor with inner radius 1.3 mm. The area between the cable and the conductor is filled with a dielectric. If the voltage drop across the capacitor is 6000 V when the line charge density is 8.8 μC/m, find the value of the dielectric constant.

(k=1/4πε0 =8.99×10^9 N·m^2/C^2)

A) 4.4

B) 4.8

C) 5.3

D) 5.7

A) 4.4

An air-filled capacitor is formed from two long conducting cylindrical shells that are coaxial and have radii of

48 mm and 84 mm. The electric potential of the inner conductor with respect to the outer conductor is -400 V. (k = 1/4πε0 = 8.99 × 10^9 N · m2/C2) The energy stored in a 1.0-m length of this capacitor is closest to

A) 8.0 μJ.

B) 5.7 μJ.

C) 11 μJ.

D) 16 μJ.

E) 22 μJ.

A) 8.0 μJ.

A 6.00-μF parallel-plate capacitor has charges of ±40.0 μC on its plates. How much potential energy is stored in this capacitor?

A) 103 μJ

B) 113 μJ

C) 123 μJ

D) 133 μJ

E) 143 μJ

D) 133 μJ

A charge of 2.00 μC flows onto the plates of a capacitor when it is connected to a 12.0 -V potential source. What is the minimum amount of work that must be done in charging this capacitor?

A) 6.00 μJ

B) 24.0 μJ

C) 12.0 μJ

D) 144 μJ

E) 576 μJ

C) 12.0 μJ

A 1.0 μF capacitor has a potential difference of 6.0 V applied across its plates. If the potential difference across its plates is increased to 8.0 V, how much ADDITIONAL energy does the capacitor store?

A) 14 μJ

B) 28 μJ

C) 2.0 μJ

D) 4.0 μJ

A) 14 μJ

Two square air-filled parallel plates that are initially uncharged are separated by 1.2 mm, and each of them has an area of 190 mm2. How much charge must be transferred from one plate to the other if 1.1 nJ of energy are to be stored in the plates? (ε00 = 8.85 × 10^-12 C^2/N · m^2)

A) 56 pC

B) 39 pC

C) 78 pC

D) 3.5 μC

A) 56 pC

Each plate of an air-filled parallel-plate air capacitor has an area of 0.0040 m^2, and the separation of the plates is 0.080 mm. An electric field of 5.3 × 10^6 V/m is present between the plates. What is the energy density between the plates? (ε0 = 8.85 × 10^-12 C^2/N · m^2)

A) 124 J/m^3

B) 84 J/m^3

C) 170 J/m^3

D) 210 J/m^3

E) 250 J/m3

A) 124 J/m^3

A parallel-plate capacitor with plate separation of 1.0 cm has square plates, each with an area of 6.0 × 10^-2 m^2. What is the capacitance of this capacitor if a dielectric material with a dielectric constant of 2.4 is placed

between the plates, completely filling them? (ε0 = 8.85 × 10^-12 C^2/N · m2)

A) 15 × 10^-12 F

B) 15 × 10^-14 F

C) 64 × 10^-14 F

D) 1.3 × 10^-12 F

E) 1.3 × 10^-10 F

E) 1.3 × 10^-10 F

A parallel-plate capacitor has a capacitance of 10 mF and is charged with a 20-V power supply. The power supply is then removed and a dielectric material of dielectric constant 4.0 is used to fill the space between the plates. What is the voltage now across the capacitor?

A) 80 V

B) 20 V

C) 10 V

D) 5.0 V

E) 2.5 V

D) 5.0 V

A 6.0-μF air-filled capacitor is connected across a 100-V voltage source. After the source fully charges the capacitor, the capacitor is immersed in transformer oil (of dielectric constant 4.5). How much ADDITIONAL charge flows from the voltage source, which remained connected during the process?

A) 1.2 mC

B) 1.5 mC

C) 1.7 mC

D) 2.1 mC

E) 2.5 mC

D) 2.1 mC

A parallel-plate capacitor has a capacitance of 10 mF and charged with a 20 -V power supply. The power supply is then removed and a dielectric material of dielectric constant 4.0 is used to fill the space between the plates. How much energy is now stored by the capacitor?

A) 250 mJ

B) 125 mJ

C) 500 mJ

D) 62.5 mJ

E) 1200 mJ

C) 500 mJ

A parallel-plate capacitor consists of two parallel, square plates that have dimensions 1.0 cm by 1.0 cm. If the plates are separated by 1.0 mm, and the space between them is filled with teflon, what is the capacitance of this

capacitor? (The dielectric constant for teflon is 2.1, and ε0 = 8.85 × 10^-12 C^2/N · m^2.)

A) 1.9 pF

B) 0.44 pF

C) 2.1 pF

D) 0.89 pF

A) 1.9 pF

An air-filled capacitor stores a potential energy of 6.00 mJ due to its charge. It is accidentally filled with water in such a way as not to discharge its plates. How much energy does it continue to store after it is filled? (The dielectric constant for water is 78 and for air it is 1.0006.)

A) 0.077 mJ

B) 468 mJ

C) 0.040 mJ

D) 6.00 mJ

A) 0.077 mJ