electricity and magnetismchunghuastpm.weebly.com/.../chapter13_capacitors.pdf1 13 capacitors...

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13 Capacitors

ELECTRICITY AND MAGNETISM

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Contents 18.1 Capacitance 18.2 Parallel plate capacitors 18.3 Dielectrics 18.4 Capacitors in series and in parallel 18.5 Energy stored in a charged capacitor 18.6 Charging and discharging of a Capacitor

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Objectives a) define capacitance (C = Q/V) b) describe the mechanism of charging a

parallel plate capacitor c) use the formula C = Q/V to derive C =

0A/d for the capacitance of a parallel plate capacitor

d) define relative permittivity r (dielectric constant)

e) describe the effect of a dielectric in a parallel plate capacitor

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Objectives f) use the formula C = r 0A/d g) derive and use the formulae for

effective capacitance of capacitors in series and in parallel

h) use the formulae U = ½ QV, U = ½ Q2/C, U = ½CV2 (Derivations are not required.)

i) describe the charging and discharging process of a capacitor through a resistor

5

Objectives j) define the time constant, and use the

formula = RC k) derive and use the formulae

and for charging a capacitor through a resistor;

Objectives

l) derive and use the formulae , for discharging a capacitor through a

resistor m) solve problems involving charging and discharging of a capacitor through a

resistor. 6

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13.1 Capacitance

8

Defining Capacitors

Short-term Charge Stores

9

Capacitors Device for storing electrical energy which can then be released in a controlled manner A capacitor consists of 2 conductors of any shape placed near each other without touching The region between the 2 conductors is usually filled with an electrically insulating material called a dielectric

10

Capacitors Symbol in circuits is It takes work, which is then stored as potential energy in the electric field that is set up between the two plates, to place charges on the conducting plates of the capacitor Since there is an electric field between the plates there is also a potential difference between the plates

11

Capacitors We usually talk about capacitors in terms of parallel conducting plates They in fact can be any two conducting objects

12

Capacitors When a capacitor

each of its 2 plates carries the same magnitude, q, of charge the potential of the +q plate exceeds that of the q plate by an amount V

The charge q and potential difference V are related by Q = C V where C is the capacitance The SI unit for capacitance is farad (F), where 1 farad = 1 coulomb/volt

13

Dielectric Constant If a dielectric is inserted between the capacitor plates, the electric field E inside the capacitor is weaker than the field E0 inside the empty capacitor, assuming the charge on the plates is unchanged This reduction of the field is described by the dielectric constant , defined as k = E/E0

( or Er= E/E0) 14

Capacitance of Parallel-Plate Capacitor

The capacitance of a parallel-plate capacitor without a dielectric is But if a dielectric is placed between the plates, Thus air or vacuum has = 1

ACd0

0 ACd

16

Capacitance of Capacitor

Capacitance C depends on capacitor on dielectric type NOT on capacitor q or potential difference V

Since > 1, capacitance C increases when a dielectric is present

17

Electric Energy Storage Electric energy can be stored in capacitors

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Key Ideas - Capacitors Capacitors are short-term charge stores. They act as electrical springs. There are a number of electrical parameters (measurements of properties) that are of interest to the electronics engineer. These include working voltage, leakage current, and temperature coefficient. These are written in data tables

19

Comparison to a Battery

Property Battery Capacitor

Charging time Long (hours) Short (fraction of a

second)

Number of charge and discharge cycles

1000 1500 at best Billions of times

Charge held Many coulombs Micro coulombs

How energy is stored Chemical reaction Electric field

analogy Fuel tank Electrical spring

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Example

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Capacitor Sandwich Construction

Two metal plates Separated by insulating material

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Swiss Roll Construction

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Metal Plates + Dielectric

Capacitors consist of two metal plates separated by a layer of insulating material called a dielectric.

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Types of Caps

25

Types of Caps

There are two types of capacitor, electrolytic and non-electrolytic. Electrolytic capacitors hold much more charge Electrolytic capacitors have to be connected with the correct polarity, otherwise they can explode.

26

Schematic Symbol

The symbol for a capacitor is shown below:

27

Comparison of Types

Advantages:

High capacitance Can have high working voltages.

Advantages: Do not lose charge Polarity does not matter Stable up to 106 Hz (or more)

Disadvantages: Polarity important High leakage current Not stable above 10 kHz Can be damaged by AC

Disadvantages: Low capacitance

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Numerical Parameters A working voltage is given. If the capacitor exceeds this voltage, the insulating layer will break down and the component shorts out. The working voltage can be as low as 16 volts, or as high as 1000 V. Leakage current means that the capacitor does not hold its charge indefinitely. A certain amount of current leaks across the dielectric. This is most marked in electrolytic capacitors. Temperature coefficient. The value of a capacitor can change quite markedly with different temperatures.

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Leakage In an electrolytic capacitor there has to be a current to maintain the aluminium oxide layer. This is about 1 mA. Over a period of time the charge leaks away. This is called the leakage current. Also it is important that the polarity of the capacitor is correct, otherwise the aluminium oxide layer is not made and the component will conduct. The resulting heating effect can result in the capacitor exploding.

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Working Voltage

All capacitors have a maximum working voltage. All insulators have a maximum voltage at which they will retain their insulating properties. The breakdown voltage is quoted in units of volts per metre, so it is actually an electric field.

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Working Voltage

The breakdown voltage of air is 3000 V/mm, so a 5 mm gap will insulate up to 15 000 V. The actual voltage at which the breakdown occurs depends on the thickness of the material.

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Working Voltage

The thinner the material, the lower the voltage that is needed before sparking will occur. If sparking occurs over a dielectric, then a hole will be burned in the dielectric and that is the end of the useful life for the capacitor.

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Temperature Coefficient Capacitors, especially electrolytic, can lose their capacitance, i.e. hold less charge, when they get hot. The decrease in capacitance can change the characteristics of the circuit so much that it will not work properly. Therefore it is essential that the temperature in which the circuit is going to operate at is taken into consideration when designing a circuit and choosing the components.

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Temperature Coefficient

Can be tested 35

Temperature Coefficient

From this graph we can see that: Mica capacitors are very stable with temperature. Ceramic bead capacitors have a linear relationship. Other types of capacitor have a temperature at which their capacitance is at a maximum. It falls away either side of the optimum.

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How Capacitors Work When connected to a potential difference (e.g. a battery), the battery tries to push electrons through the wire away from its negative terminal.

complete circuit, you get a flow of electrons to the plate i.e. you get a current without a complete circuit, but only for a short period of time.

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What happens next? As the electrons (the charge) build up on the plate, 2 things happen: 1. The plate becomes more negative and so becomes less attractive to the electrons, so the flow of electrons gradually reduces which means the current gradually reduces. 2. The electrons in the other plate are repelled by the build up of electrons in the first plate. So they move off the second plate. The electrons leaving the second plate complete the circuit.

38 39

If you plot a graph of the potential difference across the plates against charge stored on the plate you find:

V

Q

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How it Works Basic Configuration

Battery

Bulb Capacitor

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Capacitance As charge builds up, so does the pd across the plates Charged stored is directly proportional to the potential difference across the plates. Also, if then, V Q, Where Q = Charge in coulombs Q / V = a constant. Therefore C = Q / V We call the constant which relates the two, C, the capacitance because it is

- the capacity of the plates to store charge. Capacitance is measured in farad, F.

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Capacitance C = Q / V

Q - charge in coulombs C capacitance in farads V - potential difference in volts

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Farads 1F = 1 C V-1 (A capacitance of 1 farad will mean a charge of 1 coulomb can be stored for each volt across the plates). Capacitance is measured in units called farads (F). A farad is a very big unit, and we are much more likely to use microfarads ( F) or nanofarads (nF). Or even picofarads (pF) 1 F = 1 × 10-6 F 1 nF = 1 × 10-9 F 1 pF = 1 × 10-12 F

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Fixed Capacitors

PAPER CAPACITOR

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Fixed Capacitors

MICA CAPACITOR

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Fixed Capacitors

CERAMIC CAPACITOR

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Fixed Capacitors

ELECTROLYTIC CAPACITOR

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Variable Capacitors VARIABLE CAPACITOR

A typical variable capacitor (adjustable capacitor) is the rotor-stator type. It consists of two sets of metal plates arranged so that the rotor plates move between the stator plates. Air is the dielectric. As the position of the rotor is changed, the capacitance value is likewise changed. This type of capacitor is

used for tuning most radio receivers.

49

Variable Capacitors

TRIMMER CAPACITOR

A screw adjustment is used to vary the distance between the plates, thereby changing the capacitance.

50

Color Codes

Although the capacitance value may be printed on the body of a capacitor, it may also be indicated by a color code. The color code used to represent capacitance values is similar to that used to represent resistance values. The color codes currently in use are the Joint Army-Navy (JAN) code and the Radio Manufacturers' Association (RMA) code

NOT IN EXAM

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Color Codes

For each of these codes, colored dots or bands are used to indicate the value of the capacitor. A mica capacitor, it should be noted, may be marked with either three dots or six dots. Both the three- and the six-dot codes are similar, but the six-dot code contains more information about electrical ratings of the capacitor, such as working voltage and temperature coefficient.

52

Color Codes

The capacitor shown in the above figure represents either a mica capacitor or a molded paper capacitor. To determine the type and value of the capacitor, hold the capacitor so that the three arrows point left to right (>). 53

Color Codes

The first dot at the base of the arrow sequence (the left-most dot) represents the capacitor TYPE. This dot is either black, white, silver, or the same color as the capacitor body. Mica is represented by a black or white dot and paper by a silver dot or dot having the same color as the body of the capacitor. The two dots to the immediate right of the type dot indicate the first and second digits of the capacitance value

54

Color Codes

The dot at the bottom right represents the multiplier to be used. The multiplier represents picofarads. The dot in the bottom center indicates the tolerance value of the capacitor

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Color Code - Caps

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Color Code - Example

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Reading Color Code Step 01 Hold the capacitor so the

arrows point left to right

Read the First Dot

Read the second digit dot and apply it to the first digit.

Read the multiplier dot and multiply the first two digits by multiplier (Remember that the multiplier is in picofarads).

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Reading Color Code Step 02 Lastly, read the tolerance dot.

According to the left coding, the capacitor is a mica capacitor whose capacitance is 9100 pF with a tolerance of 6%.

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Ceramic Capacitor Color Code

60

Calculate the Values

61

Calculate the Values

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13.2 Parallel-plate capacitors

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Parallel-plate Capacitors made of two plates each of area A (the

plates are separated by a distance d.

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Parallel-plate Capacitors The electric field is the sum of the electric

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Parallel-plate Capacitors

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67


The electric fields outside the plates cancel out.

68


The electric fields outside the plates cancel out. Make the outside fields disappear.

69

Parallel-plate Capacitors The electric fields between the plates add.

70

Parallel-plate Capacitors The charges move to the inside of the plates.

Move the + and symbols toward the center.

71


The electric field inside is uniform. The electric field outside is small.

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Parallel-Plate Capacitors Assume electric field uniform between the plates, we have

Pictures from Serway & Beichner

E = Q o

= oA

V = E d = Qd oA

C = Q V

Q Qd oA =

C = oA d (As we have argued before)

73

13.3 Dielectrics

74

Dielectrics

A dielectric is an insulator with polar molecules that is placed between the plates of a capacitor.

75

Dielectrics

Polar molecules rotate in the electric field of the capacitor.

76

Dielectrics

The net charge inside the dielectric is zero.

77

Dielectrics

But there is leftover charge on the surfaces of the dielectric.

78

Dielectrics

This charge produces an electric field that opposes the electric field of the plates.

E of dielectric

E of plates

79 01/31/2005 Phys112, Walker Chapter 20 79

A dielectric is an insulating material in which the individual molecules polarize in proportion to the the strength of an external electric field. This reduces the electric field inside the dielectric by a factor , called the dielectric constant.

0CC

Capacitance is increased by .

0EE 0VVand

Dielectrics

For fixed charge Q on plates


Dielectrics Strength

Dielectrics are insulators: charges are not free to move (beyond molecular distances) Above a critical electric field strength, however, the electrostatic forces polarizing the molecules are so strong that electrons are torn free and charge flows. This is called Dielectric Breakdown, and can disturb the mechanical structure of the material


Dielectric Properties of common materials

Material Dielectric Constant:

Dielectric Strength (V/m)

Vacuum 1 2.5·1018 Air (lightening) 1.00059

( -1) Density 3.0·106

Teflon 2.1 60 ·106

Paper 3.7 16 ·106 Mica 5.4 100 ·106

82

Problem Type 1: Fixed Charge

A capacitor is charged with a battery to a charge Q. The battery is removed and a dielectric is inserted. Without the dielectric: With the dielectric:

000 VCQ

00

0

0

00 )(

CVQ

VQC

QCVQ

VdEEV d

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A capacitor is charged with a battery to a charge Q. The battery is removed and a dielectric is inserted.

With the dielectric:

00

0

0

00 )(

CVQ

VQC

QCVQ

VdEEV d

0

0

0

CC

VV

QQ

84


A capacitor is charged with a battery to a charge Q. The battery is removed and a dielectric is inserted.

0

0

0

CC

VV

QQ The electric field of the dielectric reduces the voltage across the capacitor, causing the capacitance to rise.

1

85

Problem Type 2: Fixed Voltage

A capacitor is connected to a battery with voltage V and remains connected as a dielectric is inserted. Without the dielectric:


00VCQ00

0

VCCVQVV

86


A capacitor is connected to a battery with voltage V and remains connected as a dielectric is inserted.


00

0

VCCVQVV

0

0

0

CCVV

QQ

87


A capacitor is connected to a battery with voltage V and remains connected as a dielectric is inserted.

0

0

0

CCVV

QQ The charge on the dielectric pulls additional charge from the battery to the plates, causing the capacitance to rise.

88

Capacitors with dielectrics A dielectrics is an insulating material (rubber, glass, etc.) Consider an insolated, charged capacitor

Q Q Q Q

V0 V

Insert a dielectric

89

Capacitors with dielectrics Notice that the potential difference decreases (k = V0/V) Since charge stayed the same (Q=Q0) capacitance increases

dielectric constant: k = C/C0

Dielectric constant is a material property

0 0 00

0 0

Q Q QC CV V V

90

Capacitors with dielectrics - notes

Capacitance is multiplied by a factor k when the dielectric fills the region between the plates completely E.g., for a parallel-plate capacitor

0ACd

91

Capacitors with dielectrics - notes

The capacitance is limited from above by the electric discharge that can occur through the dielectric material separating the plates In other words, there exists a maximum of the electric field, sometimes called dielectric strength, that can be produced in the dielectric before it breaks down

92

Dielectric constants and dielectric strengths of various materials at room temperature Material Dielectric

constant, k Dielectric strength (V/m)

Vacuum 1.00 -- Air 1.00059 3 106

Water 80 -- Fused quartz 3.78 9 106

93

Take a parallel plate capacitor whose plates have an area of 2.0 m2 and are separated by a distance of 1mm. The capacitor is charged to an initial voltage of 3 kV and then disconnected from the charging source. An insulating material is placed between the plates, completely filling the space, resulting in a decrease in the capacitors voltage to 1 kV. Determine the original and new capacitance, the charge on the capacitor, and the dielectric constant of the material.

Example

94

Take a parallel plate capacitor whose plates have an area of 2 m2 and are separated by a distance of 1mm. The capacitor is charged to an initial voltage of 3 kV and then disconnected from the charging source. An insulating material is placed between the plates, completely filling the space, resulting in a decrease in the capacitors voltage to 1 kV. Determine the original and new capacitance, the charge on the capacitor, and the dielectric constant of the material.

Given:

V1=3,000 V V2=1,000 V

A = 2.00 m2

d = 0.01 m Find: C=? C0=? Q=? k=?

Since we are dealing with the parallel-plate capacitor, the original capacitance can be found as

nFm

mmNC

dAC

181000.1

00.2)/1085.8( 3

22212

00

95

Take a parallel plate capacitor whose plates have an area of 2 m2 and are separated by a distance of 1mm. The capacitor is charged to an initial voltage of 3 kV and then disconnected from the charging source. An insulating material is placed between the plates, completely filling the space, resulting in a decrease in the capacitors voltage to 1 kV. Determine the original and new capacitance, the charge on the capacitor, and the dielectric constant of the material.

Given:

V1=3,000 V V2=1,000 V

A = 2.00 m2

d = 0.01 m Find: C=? C0=? Q=? k=?

9 50 18 10 3000 5.4 10Q C V F V C

The charge on the capacitor can be found to be

The dielectric constant and the new capacitance are

10 0

2

3 18 54VC C C nF nFV

X

96

How does an insulating dielectric material reduce electric fields by producing effective surface charge densities?

Reorientation of polar molecules

Induced polarization of non-polar molecules

Dielectric Breakdown: breaking of molecular bonds/ionization of molecules.

97

13.4 Capacitors in series and in parallel

98

Adding Capacitors

Resistors:

Capacitors:

IRV

CQV 1

CR 1

R and 1/C enter the voltage equations in a similar way. If you replace R with 1/C in series-parallel equations for resistors, you get the correct result for capacitors!

99

Adding Capacitors

Series:

Parallel: 21

2121111,,

CCCVVVQQQ

212121 ,, CCCQQQVVV

100

As with resistors, the voltages across two

capacitors in series add to get the total voltage.

As with resistors, the voltages across two capacitors in parallel are the same.

When we discharge two capacitors in parallel, the total charge that leaves the capacitors is the sum of the charges. (Recall that with resistors the sum of the currents is the total current in parallel.)

101

Why are the charges the same on capacitors in series?

To begin with, there is no charge on either capacitor.

102


Before we start charging the two capacitors, the charge within the dashed box is zero.

103


A the capacitors charge, the charge within the dashed box remains zero.

I

q qq q

104


When the left plate of the left capacitor acquires its final charge +Qcharge is Q.

+Q Q

105


The charge within the box must remain zero, so the right capacitor must have the same charge as the left capacitor.

+Q Q +Q Q

106

Caps in Series - Equation

RnRR

RT 1...111

21

CnCC

CT 1...111

21

107

2 Caps in Series - Equation

21

21CCCCCT

108

CT in Series - Example

Determine the total capacitance of a series circuit containing three capacitors whose values are 0.01 µF, 0.25 µF, and 50,000 pF,

109

CT in Series Solution

110

Caps in Parallel - Equation

CT = C1 + C2 + C3 n

111

CT in Parallel - Example

Determine the total capacitance in a parallel capacitive circuit containing three capacitors whose values are 0.03 µF, 2.0 µF, and 0.25 µF, respectively

112

CT in Parallel - Solution

113

Questions Parallel

What is the single capacitor equivalent of this circuit below?

What is the charge on each capacitor?

12 V C1 4 F 6 F C2

114

Answer Parallel

Ctotal = C1 + C2 =

C1

Charge on C2 , Q2 = 6 10-6 F 12 V = 7.2 10-5C = 72 mC Total charge, Q = 48 mC + 72 mC = 120 mC

12 V C1 4 F 6 F C2

115

Questions Series What is the single capacitor equivalent of this circuit below?

What is the charge on each capacitor?

What are the voltmeter readings?

12 V C1 4 F

C2 6 F

116

Answer Series

117

13.5 Energy stored in a charged capacitor

118

Energy Stored in a Battery

119

Energy Stored in a Capacitor

q q+ q Q Charge on plates

120

Stored Energy Equation

Energy stored by the capacitor, W = 1/2 QV equation may be written as W = 1/2 CV2

121

Measuring Stored Energy

122

Measuring Stored Energy

A joulemeter is used to measure the energy transfer from a charged capacitor to a light bulb when the capacitor discharges. The capacitor p.d. V is measured and the joulemeter reading recorded before the discharge starts. When the capacitor has discharged, the joulemeter reading is recorded again. The difference of the two joulemeter readings is the energy transferred from the capacitor during the discharge process. This is the total energy stored in the capacitor before it discharged. This can be compared with the calculation of the energy stored using W = ½CV2.

123

Stored Energy Question 1

Calculate the charge and energy stored in a l0 F capacitor charged to a potential difference of: a.) 3 V b.) 6 V

Q= 30 C, W = 45 J Q= 60 C, W = 180 J

124

Stored Energy Question 2

A 50k F capacitor is charged from a 9 V battery then discharged through a light bulb in a flash of light lasting 0.2 s. Calculate: a) the charge and energy stored in the capacitor before discharge, b) the average power supplied to the light bulb. a) = .45 C b) = 10 W

125

Energy in a Capacitor Start with two parallel plates with no charge. Move one charge from one plate to the other. There is no electric field and no force, so it

requires no work.

126

Energy in a Capacitor After the charge is transferred, the capacitor has

a small charge and a small field. The field causes a force on the next charge we

move, forcing us to do work.

127

Energy in a Capacitor

When the charge on a capacitor is q, the voltage is q/C and the electric field is V/d=q/Cd.

The force on a small charge dq is

dqCdqEdqF )(

128

Energy in a Capacitor The work done in moving the charge is

qdqC

FddW 1

129

Energy in a Capacitor The work done in charging the capacitor to

its final charge Q is:

UCVC

QqdqC

dWWQ

22

0 21

21

130

Energy in a Capacitor

2

21 CVU

131

Energy Density

Energy per unit volume in a an electric field.

In a parallel-plate capacitor of volume v=Ad :

AdEEddACVU 2

0202

21

21

21

202

1 EvUu

132

Energy Density

The density of the energy stored in any electric field, not just a capacitor, is:

202

1 Eu

133

13.6 Charging and discharging of a Capacitor

134

Capacitors in Circuits

In DC circuits, capacitors just charge or discharge. No current flows after a capacitor is fully charged or discharged.

135


Describe what happens in this circuit after the switch is closed.

20

12 V

5

1

136


Initially positive charge on the right plate of the capacitor pushes charge off the left plate. It is as if the capacitor were replaced by a wire. 20

12 V

5

1

137


When the capacitor starts charging, it behaves like a battery that opposes the flow of current. + 20

12 V

5

1

138

20

12 V

5

1

Capacitors in Circuits Eventually, the capacitor becomes fully charged. No more current flows. What is the final voltage on the capacitor?

+

139

20

12 V

5

1

Capacitors in Circuits First, ignore the branch with the capacitor. Rtotal=3 . I = 4 A. V across the 1 resistor is IR = 4 V.

+

140

20

12 V

5

1

Capacitors in Circuits V across the 5 resistor is 0. Why? V across the capacitor is 4 V. Q on the capacitor = CV = 80 C

+

141


Summary: In steady state, no current flows through the capacitor. Just find the voltage across the capacitor and you can determine the charge.

142

RC Discharging

Charge a capacitor with a battery to a voltage V. Disconnect the capacitor and attach it to a resistor. The initial charge is Q=CV. The charge decays to zero but what is Q(t)?

Q(t)

Q0

I

t

143

RC Discharging Look at the voltage

around the circuit. We

rule:

0IRCQ

I

CQVC

IRVR

I

144

RC Discharging

The minus sign comes from:

1)I > 0

2)Q is the charge on the capacitor

3)The capacitor is discharging so

dtdQI

QI

IRCQ

0,0

0

0dtdQ

I

CQVC

IRVR

145

RC Discharging

)(1

0

0,0

0

tQRCdt

dQdtdQR

CQ

dtdQI

QI

IRCQ

I

CQVC

IRVR

146

RC Discharging

)(1 tQRCdt

dQ

This is a differential equation, but it is a really easy one to solve.

equations.

i

f

QQ

dtRCQ

dQdtRCQ

dQ ln11

147

RC Discharging

/0

0

0

)(

)(

,1)(ln

1

1

0

t

ttQ

Q

eQtQ

RCtRCQ

tQ

dtRCQ

dQ

dtRCQ

dQ

148

RC Time Constant

= RC. has units of seconds. When is big, capacitors charge and discharge slowly. If R is large, not much current flows, so is big.

If C is large, there is a lot of charge that has to flow, so is big.

149

RC Discharging

=1 s =2 s

=3 s

e1

Discharging capacitors with three different time constants. The time constant is the time it takes the charge to drop to 1/e of its original value.

150

RC Charging

A capacitor is initially uncharged.

Use a battery with voltage V0 to charge the capacitor. The voltage increases to V0.

The charge increases to Q=CV0.

V0 C

R

151

RC Charging

We again use

00 CQIRV

I

CQVC

IRVR

0V

152

RC Charging

We again use

0

0

0

0

0

CQ

dtdQRV

dtdQI

CQIRV

I

CQVC

IRVR

0V

153

RC Charging

This differential equation has the solution:

00 CQ

dtdQRV

RC

CVQeQtQ

f

tf

0

/1)(

Try plugging the solution into the differential equation and see if it works!

154

RC Charging

=1 s

=2 s

=3 s

e11

Charging capacitors with three different time constants. The time constant is the time it takes the charge to rise to 1-1/e of its final value.

155

Charging and Discharging

Charging Capacitors What happens to current as time passes?

Current falls away as it becomes less attractive for electrons to move to the plate from the cell. Note: The area under the current-time graph is equal to the amount of charge stored on the plates.

156

What happens to the charge on the plate?

Charge builds up - quickly at first (a lot of electrons arriving each second) and then more slowly. The potential difference is proportional to charge, so the p.d.-time graph is exactly the same shape as the charge-time graph.

157

Charging Capacitors When the capacitor is fully charged, the pd across the plates will equal the emf of the cell charging it.

Look at the diagram.

The cell is trying to push electrons clockwise (with its

capacitor is trying to push electrons anticlockwise (with its push of 2V). Neither wins so no charge flows.

158

Discharging Capacitors Initially there is a large current due to the large potential difference across the plates. The current drops as pd drops.

Notice that the electrons are now moving the opposite way round the circuit so the graph shows the current as negative to show this.

159

Discharging Capacitors

Charge drops quickly at first (due to the large current - which is of course, a large flow of charge). As the charge and therefore the pd across the plates drops, so the charge drops more slowly.

160

Discharging Capacitors

As the potential difference across the plates is directly proportional to the charge on the plates, the p.d.-time graph is the same shape as the charge-time graph as before.

161

Charge/Discharge Cycle

162

R-C circuits So far we have assumed

that all emfs and resistances are constant, time-independent quantities. This assumption fails when we consider a circuit with a capacitor. Lets assume an ideal

source with emf connected to a resistor of resistance R and capacitor with capacitance C.

163

R-C circuits

When the switch is open i is zero, as is the charge on the capacitor q. When the switch is

initially closed the current is maximized, and q is zero. As the capacitor charges i decreases as q increased until q is maximized and i is zero.

164

Decay

165

Decay

166

Charging a Capacitor The capacitor in the figure

is initially uncharged. That means that vbc is zero at time t = 0. According to the loop rule

the voltage across the resistor, vab, is equal to the emf of the source E. The initial current through

the resistor: Io = vab/R = E/R

167

Charging a Capacitor As the capacitor charges,

its voltage vbc increases and the potential difference across the resistor, vab, decreases. The sum of these two

voltages must always be equal to E.

168

Charging a Capacitor When the capacitor is fully charged, vbc = E. Why does the current decrease as the

capacitor charges? Let q represent the charge on the capacitor

and i the current in the circuit as some time t after the switch has been closed. The current will flow counter clockwise. The instantaneous voltages for the resistor and capacitor are:

iRvab Cqvbc

169

Charging a Capacitor

The potential drops by iR from a to b and

by q/C from b to c. Solving the above equation for i we get:

RCq

REi

0CqiRE

170

Charging a Capacitor When the switch is closed at t = 0 the

capacitor is not yet charged, and the initial current is, as we have stated before, E/R. Without the capacitor, this would be the constant value of the current. As q increases the voltage drop across the

capacitor increases. The drop equals E when the charge reaches its final value Qf. The current eventually reaches zero. When this happens Qf = EC.

171

Charging a Capacitor The current and the

capacitor charge are given as functions of time to the right. Current jumps from 0 to Io

= E/R at t = 0, then gradually approaches zero. The charge starts at zero and gradually approaches Qf.

172

Charging a Capacitor We can derive general

expressions for the charge and current as functions of time. Due to our choice of the direction of current, i equals the rate at which positive charge arrives at the left-hand plate of the capacitor, so i = dq/dt. Putting this into equation,

CEqRCRC

qRE

dtdq 1

173

Charging a Capacitor The previous equation rearranged: And then integrate both sides. Lets

change the integration variables to and so we can use q and t as the upper

limits:

RCdt

CEqdq

tq

RCtd

CEqqd

00

174

Charging a Capacitor When we carry out the integration we get: Exponentiating both sides, taking the inverse

logarithm, and solving for q: The instantaneous current i is just the time

derivative:

RCt

CECEqln

)1()1( RCt

fRCt

eQeCEq

RCt

oRCt

eIeRE

dtdqi

175

Time Constant After a time equal to RC,

the current in the R-C circuit has decreased to 1/e of its initial value and the charge has reached (1 1/e) of its final value. The product RC is called the time constant or relaxation time of the circuit, = RC. It is a measure of

how quickly the capacitor charges or discharges.

176

Time Constant

When tau is small, the capacitor charges quickly. When it is large the capacitor takes a longer time to charge. Hence, tau is the factor RC.

177

Discharging a Capacitor Suppose that after the capacitor in figure 26.20 has

acquired a charge Qf we remove the battery and connect points a and c to an open switch. When we close the switch, t = 0 and q = Qo, the

capacitor discharges through the resistor and the charge again returns to zero. Let i and q represent the time varying current and

charge at some instant after the switch is closed. We choose our current direction to be the same as we did before so we can use equation 26.10 for

E = 0. This gives:

RCq

dtdqi

178

Discharging a Capacitor The initial current, when t =0, Io = -Qo/RC. To find q as a function of time we take the same steps as

charging:

RCq

dtdqi

tqtd

RCqqd

00

1

RCt

Qq

o

ln

RCt

oeQq

RCt

o

RCto

eI

eRCQdtdqi

RCq

dtdqi

179

180

181

Summary

By Carsten Denker

182

Capacitance

Capacitance [F = C/V]

A battery maintains a potential difference between its terminals. It sets up an electric field which drives electrons through the wire towards the positive terminal.

E

q CV

183

Calculating the Capacitance Parallel-plate capacitor

Cylindrical capacitor

Spherical capacitor

Isolated sphere

0 ACd

02ln /

LCb a

04 abCb a

04C R

184

Capacitors in Parallel & in Series

capacitor in parallel capacitors

in series

When a potential difference is applied across several capacitors connected in parallel, that potential difference is applied across each capacitor. The total charge stored on the capacitors is the sum of the charges stored on all the capacitors. When a potential difference is applied across several capacitors connected in series, the capacitors have identical charges . The sum of the potential

differences across all the capacitors is equal to the applied potential difference .

eq1

n

jj

C C

1eq

1 1n

j jC C

185

Energy Stored in an Electric Field

Potential energy

Potential energy

Energy density

The potential energy of a charged capacitor may be viewed as being stored in the electric field between its plates.

Material

Dielectric Constant

Dielectric Strength (kV/mm)

Air 1.00054 3 Polystyrene 2.6 24 Paper 3.5 16 Transformer Oil 4.5 Pyrex 4.7 14 Ruby Mica 5.4 Porcelain 6.5 Silicon 12 Germanium 16 Ethanol 25 Water (20º C) 80.4 Water (50º C) 78.5 Titania Ceramic 130 Strontium Titanate 310 8

2

2qUC

212

U CV

20

12

u E

186

Dielectric

In a region completely filled by a dielectric material of dielectric constant , all electrostatic equations containing

the permittivity constant are to be modified by replacing with .

Polar and non-polar dielectrics

dielectric

0 E dA q

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