Copyright © 2009 Pearson Education, Inc. Chapter 24 Capacitance, Dielectrics, Electric Energy Storage

Copyright © 2009 Pearson Education, Inc.

Chapter 24Capacitance, Dielectrics, Electric Energy Storage


• Capacitors

• Determination of Capacitance

• Capacitors in Series and Parallel

• Electric Energy Storage

• Dielectrics

• Molecular Description of Dielectrics

Units of Chapter 24


A capacitor consists of two conductors that are close but not touching. A capacitor has the ability to store electric charge.

24-1 Capacitors


Parallel-plate capacitor connected to battery. (b) is a circuit diagram.

24-1 Capacitors


When a capacitor is connected to a battery, the charge on its plates is proportional to the voltage:

The quantity C is called the capacitance.

Unit of capacitance: the farad (F):

1 F = 1 C/V.

24-1 Capacitors


24-2 Determination of CapacitanceFor a parallel-plate capacitor as shown, the field between the plates is

E = Q/ε0A.

Integrating along a path between the plates gives the potential difference:

Vba = Qd/ε0A.

This gives the capacitance:


24-2 Determination of CapacitanceExample 24-1: Capacitor calculations.

(a) Calculate the capacitance of a parallel-plate capacitor whose plates are 20 cm × 3.0 cm and are separated by a 1.0-mm air gap. (b) What is the charge on each plate if a 12-V battery is connected across the two plates? (c) What is the electric field between the plates? (d) Estimate the area of the plates needed to achieve a capacitance of 1 F, given the same air gap d.


24-2 Determination of CapacitanceCapacitors are now made with capacitances of 1 farad or more, but they are not parallel-plate capacitors. Instead, they are activated carbon, which acts as a capacitor on a very small scale. The capacitance of 0.1 g of activated carbon is about 1 farad.

Some computer keyboards use capacitors; depressing the key changes the capacitance, which is detected in a circuit.


24-2 Determination of Capacitance

Example 24-2: Cylindrical capacitor.A cylindrical capacitor consists of a cylinder (or wire) of radius Rb surrounded by a coaxial cylindrical shell of inner radius Ra. Both cylinders have length l which we assume is much greater than the separation of the cylinders, so we can neglect end effects. The capacitor is charged (by connecting it to a battery) so that one cylinder has a charge +Q (say, the inner one) and the other one a charge –Q. Determine a formula for the capacitance.

l


24-2 Determination of CapacitanceExample 24-3: Spherical capacitor.

A spherical capacitor consists of two thin concentric spherical conducting shells of radius ra and rb as shown. The inner shell carries a uniformly distributed charge Q on its surface, and the outer shell an equal but opposite charge –Q. Determine the capacitance of the two shells.


24-2 Determination of Capacitance

Example 24-4: Capacitance of two long parallel wires.

Estimate the capacitance per unit length of two very long straight parallel wires, each of radius R, carrying uniform charges +Q and –Q, and separated by a distance d which is large compared to R (d >> R).


Capacitors in parallel have the same voltage across each one. The equivalent capacitor is one that stores the same charge when connected to the same battery:

24-3 Capacitors in Series and Parallel


Capacitors in series have the same charge. In this case, the equivalent capacitor has the same charge across the total voltage drop. Note that the formula is for the inverse of the capacitance and not the capacitance itself!



24-3 Capacitors in Series and ParallelExample 24-5: Equivalent capacitance.

Determine the capacitance of a single capacitor that will have the same effect as the combination shown.



Example 24-6: Charge and voltage on capacitors.

Determine the charge on each capacitor and the voltage across each, assuming C = 3.0 μF and the battery voltage is V = 4.0 V.


24-3 Capacitors in Series and ParallelExample 24-7: Capacitors reconnected.

Two capacitors, C1 = 2.2 μF and C2 = 1.2 μF, are connected in parallel to a 24-V source as shown. After they are charged they are disconnected from the source and from each other and then reconnected directly to each other, with plates of opposite sign connected together. Find the charge on each capacitor and the potential across each after equilibrium is established.


A charged capacitor stores electric energy; the energy stored is equal to the work done to charge the capacitor:

24-4 Electric Energy Storage



Example 24-8: Energy stored in a capacitor.

A camera flash unit stores energy in a 150-μF capacitor at 200 V. (a) How much electric energy can be stored? (b) What is the power output if nearly all this energy is released in 1.0 ms?



Conceptual Example 24-9: Capacitor plate separation increased.

A parallel-plate capacitor carries charge Q and is then disconnected from a battery. The two plates are initially separated by a distance d. Suppose the plates are pulled apart until the separation is 2d. How has the energy stored in this capacitor changed?



Example 24-10: Moving parallel capacitor plates.

The plates of a parallel-plate capacitor have area A, separation x, and are connected to a battery with voltage V. While connected to the battery, the plates are pulled apart until they are separated by 3x. (a) What are the initial and final energies stored in the capacitor? (b) How much work is required to pull the plates apart (assume constant speed)? (c) How much energy is exchanged with the battery?


The energy density, defined as the energy per unit volume, is the same no matter the origin of the electric field:

The sudden discharge of electric energy can be harmful or fatal. Capacitors can retain their charge indefinitely even when disconnected from a voltage source – be careful!



Heart defibrillators use electric discharge to “jump-start” the heart, and can save lives.



A dielectric is an insulator, and is characterized by a dielectric constant K.

Capacitance of a parallel-plate capacitor filled with dielectric:

24-5 Dielectrics

Using the dielectric constant, we define the permittivity:


Dielectric strength is the maximum field a dielectric can experience without breaking down.

24-5 Dielectrics


24-5 DielectricsHere are two experiments where we insert and remove a dielectric from a capacitor. In the first, the capacitor is connected to a battery, so the voltage remains constant. The capacitance increases, and therefore the charge on the plates increases as well.


24-5 DielectricsIn this second experiment, we charge a capacitor, disconnect it, and then insert the dielectric. In this case, the charge remains constant. Since the dielectric increases the capacitance, the potential across the capacitor drops.


24-5 DielectricsExample 24-11: Dielectric removal.

A parallel-plate capacitor, filled with a dielectric with K = 3.4, is connected to a 100-V battery. After the capacitor is fully charged, the battery is disconnected. The plates have area A = 4.0 m2 and are separated by d = 4.0 mm. (a) Find the capacitance, the charge on the capacitor, the electric field strength, and the energy stored in the capacitor. (b) The dielectric is carefully removed, without changing the plate separation nor does any charge leave the capacitor. Find the new values of capacitance, electric field strength, voltage between the plates, and the energy stored in the capacitor.


The molecules in a dielectric, when in an external electric field, tend to become oriented in a way that reduces the external field.

24-6 Molecular Description of Dielectrics


This means that the electric field within the dielectric is less than it would be in air, allowing more charge to be stored for the same potential. This reorientation of the molecules results in an induced charge – there is no net charge on the dielectric, but the charge is asymmetrically distributed.

The magnitude of the induced charge depends on the dielectric constant:

24-6 Molecular Description of Dielectrics


Review Questions

Capacitor Capacitor CC11 is connected across is connected across

a battery of a battery of 5 V5 V. An identical . An identical

capacitor capacitor CC22 is connected across is connected across

a battery of a battery of 10 V10 V. Which one has . Which one has

more charge?more charge?

1) CC11

2) CC22

3) both have the same charge

4) it depends on other factors

ConcepTest 24.1ConcepTest 24.1 CapacitorsCapacitors

Since QQ = = CVCV and the two capacitors are

identical, the one that is connected to the

greater voltagegreater voltage has more charge charge, which

is CC22 in this case.

Capacitor Capacitor CC11 is connected across is connected across

a battery of a battery of 5 V5 V. An identical . An identical

capacitor capacitor CC22 is connected across is connected across

a battery of a battery of 10 V10 V. Which one has . Which one has

more charge?more charge?

1) CC11

2) CC22

3) both have the same charge

4) it depends on other factors

ConcepTest 24.1ConcepTest 24.1 CapacitorsCapacitors

1) increase the area of the plates increase the area of the plates

2) decrease separation between the platesdecrease separation between the plates

3) decrease the area of the plates

4) either (1) or (2)


What must be done to What must be done to

a capacitor in order to a capacitor in order to

increase the amount of increase the amount of

charge it can hold (for charge it can hold (for

a constant voltage)?a constant voltage)?

+Q –Q

ConcepTest 24.2aConcepTest 24.2a Varying Capacitance IVarying Capacitance I

Since Q = CVQ = CV, in order to increase the charge

that a capacitor can hold at constant voltage,

one has to increase its capacitanceincrease its capacitance. Since the

capacitance is given by , that can be

done by either increasing increasing AA or decreasing decreasing dd.

1) increase the area of the plates increase the area of the plates

2) decrease separation between the platesdecrease separation between the plates

3) decrease the area of the plates



dAC 0

What must be done to What must be done to

a capacitor in order to a capacitor in order to

increase the amount of increase the amount of

charge it can hold (for charge it can hold (for

a constant voltage)?a constant voltage)?

+Q –Q

ConcepTest 24.2aConcepTest 24.2a Varying Capacitance IVarying Capacitance I

+Q –Q

A parallel-plate capacitor A parallel-plate capacitor

initially has a voltage of initially has a voltage of 400 V400 V

and and stays connected to the stays connected to the

batterybattery. If the plate spacing is . If the plate spacing is

now now doubled,doubled, what happens? what happens?

1) the voltage decreasesthe voltage decreases

2) the voltage increasesthe voltage increases

3) the charge decreasesthe charge decreases

4) the charge increasesthe charge increases

5) both voltage and charge changeboth voltage and charge change

ConcepTest 24.2bConcepTest 24.2b Varying Capacitance IIVarying Capacitance II

Since the battery stays connected, the Since the battery stays connected, the

voltage must remain constant!voltage must remain constant! Since

, when the spacing d

is doubled, the capacitance C is halved.

And since QQ = = CVCV, that means the

charge must decreasecharge must decrease.

+Q –Q

dAC 0

A parallel-plate capacitor A parallel-plate capacitor

initially has a voltage of initially has a voltage of 400 V400 V

and and stays connected to the stays connected to the

batterybattery. If the plate spacing is . If the plate spacing is

now now doubled,doubled, what happens? what happens?

1) the voltage decreasesthe voltage decreases

2) the voltage increasesthe voltage increases

3) the charge decreasesthe charge decreases

4) the charge increasesthe charge increases

5) both voltage and charge changeboth voltage and charge change

ConcepTest 24.2bConcepTest 24.2b Varying Capacitance IIVarying Capacitance II

Follow-up:Follow-up: How do you increase the charge? How do you increase the charge?

A parallel-plate capacitor initially has A parallel-plate capacitor initially has

a potential difference of a potential difference of 400 V400 V and is and is

then disconnected from the charging then disconnected from the charging

battery. If the plate spacing is now battery. If the plate spacing is now

doubleddoubled (without changing (without changing QQ), what ), what

is the new value of the voltage?is the new value of the voltage?

1) 100 V 100 V

2) 200 V200 V

3) 400 V

4) 800 V

5) 1600 V

+Q –Q

ConcepTest 24.2cConcepTest 24.2c Varying Capacitance IIIVarying Capacitance III

Once the battery is disconnected, Once the battery is disconnected, QQ has to has to

remain constantremain constant, since no charge can flow

either to or from the battery. Since

, when the spacing d is doubled,

the capacitance C is halved. And since QQ = =

CVCV, that means the voltage must doublevoltage must double.

A parallel-plate capacitor initially has A parallel-plate capacitor initially has

a potential difference of a potential difference of 400 V400 V and is and is

then disconnected from the charging then disconnected from the charging

battery. If the plate spacing is now battery. If the plate spacing is now

doubleddoubled (without changing (without changing QQ), what ), what

is the new value of the voltage?is the new value of the voltage?

1) 100 V 100 V

2) 200 V200 V

3) 400 V

4) 800 V

5) 1600 V

+Q –Q

dAC 0

ConcepTest 24.2cConcepTest 24.2c Varying Capacitance IIIVarying Capacitance III

ConcepTest 24.3aConcepTest 24.3a Capacitors ICapacitors I

o

o

C CC

Ceq

1) 1) CCeq eq = 3/2= 3/2CC

2) 2) CCeq eq = 2/3= 2/3CC

3) 3) CCeq eq = 3= 3CC

4) 4) CCeq eq = 1/3= 1/3CC

5) 5) CCeq eq = 1/2= 1/2CC

What is the equivalent capacitance, What is the equivalent capacitance,

Ceq , of the combination below? , of the combination below?

The 2 equal capacitors in seriesseries add

up as inversesinverses, giving 1/21/2CC. These

are parallelparallel to the first one, which

add up directlydirectly. Thus, the total

equivalent capacitance is 3/23/2CC.

ConcepTest 24.3aConcepTest 24.3a Capacitors ICapacitors I

o

o

C CC

Ceq

1) 1) CCeq eq = 3/2= 3/2CC

2) 2) CCeq eq = 2/3= 2/3CC

3) 3) CCeq eq = 3= 3CC

4) 4) CCeq eq = 1/3= 1/3CC

5) 5) CCeq eq = 1/2= 1/2CC

What is the equivalent capacitance, What is the equivalent capacitance,

Ceq , of the combination below? , of the combination below?

ConcepTest 24.3bConcepTest 24.3b Capacitors IICapacitors II

1) 1) VV11 = = VV22

2) 2) VV11 > > VV22

3) 3) VV11 < < VV22

4) all voltages are zero4) all voltages are zero

CC11 = 1.0 = 1.0 FF CC33 = 1.0 = 1.0 FF

CC22 = 1.0 = 1.0 FF

10 V10 V

How does the voltage How does the voltage VV11 across across

the first capacitor (the first capacitor (CC11) compare to ) compare to

the voltage the voltage VV22 across the second across the second

capacitor (capacitor (CC22)?)?

ConcepTest 24.3bConcepTest 24.3b Capacitors IICapacitors II

1) 1) VV11 = = VV22

2) 2) VV11 > > VV22

3) 3) VV11 < < VV22

4) all voltages are zero4) all voltages are zero

CC11 = 1.0 = 1.0 FF CC33 = 1.0 = 1.0 FF

CC22 = 1.0 = 1.0 FF

10 V10 V

The voltage across C1 is 10 V.

The combined capacitors C2 +

C3 are parallel to C1. The

voltage across C2 + C3 is also

10 V. Since C2 and C3 are in

series, their voltages add.

Thus the voltage across C2

and C3 each has to be 5 V,

which is less than V1.

How does the voltage How does the voltage VV11 across across

the first capacitor (the first capacitor (CC11) compare to ) compare to

the voltage the voltage VV22 across the second across the second

capacitor (capacitor (CC22)?)?

Follow-up:Follow-up: What is the current in this circuit??

ConcepTest 24.3cConcepTest 24.3c Capacitors IIICapacitors III

CC11 = 1.0 = 1.0 FF CC33 = 1.0 = 1.0 FF

CC22 = 1.0 = 1.0 FF

10 V10 V

1) 1) QQ11 = = QQ22

2) 2) QQ11 > > QQ22

3) 3) QQ11 < < QQ22

4) all charges are zero4) all charges are zero

How does the charge How does the charge QQ11 on the first on the first

capacitor (capacitor (CC11) compare to the charge ) compare to the charge

QQ22 on the second capacitor ( on the second capacitor (CC22)?)?

ConcepTest 24.3cConcepTest 24.3c Capacitors IIICapacitors III

CC11 = 1.0 = 1.0 FF CC33 = 1.0 = 1.0 FF

CC22 = 1.0 = 1.0 FF

10 V10 V

We already know that the

voltage across C1 is 10 V

and the voltage across both

C2 and C3 is 5 V each. Since

QQ = = CVCV and C is the samesame for

all the capacitors, we have have

VV11 > V > V22 and therefore QQ11 > Q > Q22.

1) 1) QQ11 = = QQ22

2) 2) QQ11 > > QQ22

3) 3) QQ11 < < QQ22

4) all charges are zero4) all charges are zero

How does the charge How does the charge QQ11 on the first on the first

capacitor (capacitor (CC11) compare to the charge ) compare to the charge

QQ22 on the second capacitor ( on the second capacitor (CC22)?)?


Ch. 24 Homework

• Read Ch. 24 and begin looking over Ch. 25.

• Group problems #’s 2, 12, 22

• HW problems #’s 3, 5, 7, 11, 13, 17, 23, 25, 29, 35, 41, 47, 59

Documents

Copyright © 2009 Pearson Education, Inc. Chapter 24 Capacitance, Dielectrics, Electric Energy Storage