Chapter 2: Capacitor and Dielectrics - YSL Physicsyslphysics.weebly.com/uploads/4/8/2/6/48261583/chapter_2.pdf · 2019-11-10 · CHAPTER 2: CAPACITOR AND DIELECTRICS prepared by Yew

CHAPTER 2: CAPACITOR AND DIELECTRICS

prepared by Yew Sze Ling@Fiona, KMPP

Chapter 2: Capacitor and Dielectrics

Subtopic C1 & C2 C3 & C4

2.1 Capacitance

and capacitors

in series and

parallel

Define capacitance V

QC

Derive the effective

capacitance of capacitors in

series and parallel

Derive energy stored in a

capacitor

C

QQVCVU

22

2

1

2

1

2

1

Use capacitance V

QC

Determine the effective

capacitance of capacitors in series

and parallel

Use energy stored in a capacitor

C

QQVCVU

22

2

1

2

1

2

1

2.2 Charging and

discharging of

capacitors

State the physical meaning of

time constant

Explain characteristic of Q-t

and I-t graph for charging and

discharging of capacitor

Use time constant RC

Sketch Q-t and I-t graph for

charging and discharging of

capacitor

Use

o RC

t

eQQ

0 for discharging

o

RC

t

eQQ 10 for charging

2.3 Capacitors

with

dielectrics

Define dielectric constant

0

r

Describe the effect of dielectric

on a parallel plate capacitor

Calculate capacitance of air filled

parallel plate capacitor d

AC 0

0

Use dielectric constant 0

r

Use capacitance with dielectric

0CC r



Introduction

A capacitor is a device which stores electric charge. Capacitors vary in shape and size, but the basic

configuration is two conductors carrying equal but opposite charges. Capacitors have many important

applications in electronics. The most common use of capacitor is storing electric potential energy.

In general, a capacitor consists of two conductors or plates of any shape placed near one another

without touching. These two conductors are separated by a small air gap or a thin insulator called

a dielectric. The symbol for capacitor:

Types of capacitors:

Parallel plates capacitor

Rolled up parallel plates capacitor

Cylindrical capacitor

Spherical capacitor

2.1 Capacitance and Capacitors in Series and Parallel

For a given capacitor, it is found that the amount of charge Q acquired by each plate is proportional to the magnitude of the potential difference V between the plates:

VQ

Mathematically,

CVQ

The proportionality constant C, which is the capacitance of the capacitor does not in general depend

on Q or V. Its value depends only on the size, shape, and relative position of the two conductors

or plates, and also on the material that separates them (dielectric).

OR



The capacitance C of a capacitor is defined as the ratio of the magnitude of the charge on either conductor/plate to the magnitude of the potential difference between the conductors/plates:

V

QC

SI unit of capacitance: farad (F) or coulomb per volt (C V-1).

1 farad is defined as the charge of 1 coulomb stored on each of the conducting plates as a result of a

potential difference of 1 volt between the two plates.

Capacitance is a scalar quantity, and its value is always positive. It reflects the ability of the capacitor

to store charge, in the sense that a larger capacitance C allows more charge q to be put onto the plates

for a given value of the potential difference V. A typical capacitance is in the picofarad (pF) to

millifarad (mF) range.

Capacitor in Series Capacitor in Parallel

21 QQQ

21 VVV

21

111

CCCeff

21 QQQ

21 VVV

21 CCC



Example

Determine the equivalent capacitance of the configuration shown in figure below. All the capacitors are identical and each has capacitance of 1 µF.

Label all the capacitors in the circuit.

To calculate the effective capacitance, it is easier to solve it from the end of the circuit (left) to the

terminal (right). Capacitors C1, C2 and C3 are connected in series.

321

1111

CCCCX

4CCC XY

65

1111

CCCC YZ

7CCC Zeq

F 67.1 eqC



Energy Stored in a Capacitor

A charged capacitor stores electrical energy in electric field between the plates.

The energy stored in a capacitor will be equal to the work done to charge it.

A capacitor does not become charged instantly. It takes time.

Initially, when the capacitor is uncharged, it requires no work to move the first bit of charge over.

When some charge is on each plate, it requires work to add more charge of the same sign because of the electric repulsion.

When a switch in figure is closed, charges begin to accumulate on the plates.

A small amount of work (dW) is done in bringing a small amount of charge (dQ) from the battery to the capacitor. This is given by

VdQdW and C

QV

dQC

QdW

The total work W required to increase the accumulated charge from zero to Q is given by

QW

dQC

QdW

00

C

QW

2

2

1

This is equal to the electrical potential energy stored in the capacitor:

QVCVC

QU

2

1

2

1

2

1 22

It is a scalar quantity and the unit for work is joule (J).

Graphically, energy stored can be calculated from area under V-Q graph:

QVU2

1

QVU

Energy lost

= Energy produced by a cell

– Energy stored in capacitor



Example

A 2 µF capacitor is charged to 200V using a battery. Calculate the

a. charge delivered by the battery

C 104

200102

4

6

CVQ

b. energy supplied by the battery

J 108

200104

2

4

supplied

QVU

c. energy stored in the capacitor

J 104

2001042

1

2

1

2

4

stored

QVU

d. energy dissipated as heat

2

22

storedsupplieddissipated

104

104108

UUU

2.2 Charging and Discharging of Capacitors

Time Constant, τ

The quantity RC that appears in the exponent for all equation is called time constant or

relaxation time of the circuit or mathematically

It is a scalar quantity.

Its unit is second (s).

It is a measure of how quickly the capacitor charges or discharges.

Charging Discharging

The time constant for a circuit used to charge a

capacitor is defined as the time required for the

capacitor’s charge (or voltage) to reach

e

11

= 0.63 or 63% of its maximum value.

0

0

0

63.0

1

, 1

Q

eQ

RCteQQ

RC

RC

RC

t

The time constant for a circuit used to discharge

a capacitor is defined as the time taken for the

charge (or voltage) of the capacitor to decrease to

e

1 = 0.37 or 37% of its initial value.

0

0

0

37.0

,

Q

eQ

RCteQQ

RC

RC

RC

t

RC




Originally, both plates are neutral

When switch S is closed, current I0 immediately

begins to flow through the circuit.

Electrons will flow out from the negative

terminal of the battery and accumulate on the

plate B of the capacitor.

Then electrons will flow into the positive

terminal of the battery through the resistor R ,

leaving a positive charges on the plate A

As charges accumulate on the capacitor, the

potential difference across it increases and the

current is reduced until eventually the

maximum voltage across the capacitor Vc equals

the voltage supplied by the battery, V0.

At this time, no further current flows

(I = 0) through the resistor R and the charge Q on

the capacitor thus increases gradually and

reaches a maximum value Q0.

When switch S is closed, electrons from plate B

begin to flow through the resistor R and neutralize

positive charges at plate A.

Initially, the potential difference (voltage) across the

capacitor is maximum, V0 and then a maximum

current I0 flows through the resistor R.

When part of the positive charges on plate A is

neutralized by the electrons, the voltage across the

capacitor is reduced.

The process continues until the current through the

resistor is zero.

At this moment, all the charges at plate A is fully

neutralized and the voltage across the capacitor

becomes zero.

RC

t

eQQ 10

RC

t

eVV 10

RC

t

eQQ

0

RC

t

eVV

0

, Vc




I0 can be determined by:

OR

Note:

The negative sign indicates that as the capacitor

discharges, the current direction is opposite to its

direction when the capacitor is being charged. For calculation of current in discharging process,

ignore the negative sign in the formula.

2.3 Capacitors with Dielectrics

Dielectric is a non-conducting (insulating) material placed between the plates of a capacitor. Placing

solid dielectric between the plates of a capacitor serves three functions:

It solves mechanical problem of maintaining two large metal sheets at a very small separation without actual contact/ without touching.

Using a dielectric increases the maximum operating voltage. Any insulating material, when

subjected to sufficiently large electric field, experiences a partial ionization that permits

conduction through it. This phenomenon is called dielectric breakdown. Many dielectric

materials can tolerate stronger electric fields without breakdown compare to air (without

dielectric). Thus, using dielectric allows a capacitor to sustain higher potential difference and so

store greater amounts of charge and energy.

Using a dielectric increase the capacitance of a capacitor by a factor εr, known as dielectric constant.

The simplest form of capacitor consists of two parallel conducting plates, each with area A, separated by a distance d that is small in comparison with their dimension.

Circular parallel-plates capacitor Rectangular parallel-plates capacitor

RC

t

eII

0

RC

t

eII

0

R

VI 0

0 RC

QI 0

0

–



When the capacitor is charged, its plates have charges of equal

magnitudes but opposite signs (+Q and −Q) and then the

potential difference V across the plates is produced. Since d « A so that the electric field strength E is uniform between the plates.

The capacitance of air-filled parallel plate capacitor is:

d

AC 0

0

where ε0 : permittivity of free space, 8.85×10-12 C2 N-1 m-2

A : area of each plate

d : distance between two plates

By inserting a dielectric between the two parallel plates, it increases the capacitance by a factor εr, known as dielectric constant. Thus, for parallel plate capacitor with dielectric:

d

AC r 0

d

AC

where 0 r

ε : permittivity of dielectric material

ε0 : permittivity of free space

εr : dielectric constant

Dielectric constant (also known as relative permittivity) is defined as the ratio between the permittivity of dielectric material to the permittivity of free space.

0

r

It is a dimensionless constant (no unit). Since it increases the capacitance by a factor εr, it can also

be written as

0C

Cr

From the definition of capacitance,

V

V

VQ

VQ

r0

0

From the relationship between E and V for uniform electric field,

E

E

Ed

dEr

00

As conclusion: E

E

V

V

C

Cr

00

00

Battery disconnected

Q remains constant



The Effect of Dielectric on a Parallel Plate Capacitor

Battry is Disconnected when Inserted Dielectric

The battery is now removed and the charge on the plates remains constant.

The region between the charged plates is empty. The field lines point from the positive toward the

negative plate.

The electric field produced by the charges on the plates aligns the molecular dipoles within the

dielectric with their positive ends

pointing toward the negatively

charged plate and their negative ends pointing toward the positively

charged plated.

Due to the surface charge, not all the electric field lines generated by the charges on the plates pass

through the dielectric. The

applied electric field E0 is

partially cancelled. Thus, the electric field inside the dielectric

E is less strong than the electric

field inside the empty capacitor E0.

Because the new electric field strength (E < E0) is less than the potential difference, V across the

plates is less as well.

EdV , when VE ,

Since V is smaller while Q remains the same,

V

QC , when QV ,

The capacitance is increased by the dielectric.

Positive

surface

charge

Negative

surface

charge



Battry is Connected when Inserted Dielectric

If battery is connected, the potential difference between two plates remains constant.

Voltage across capacitor remains constant (equal to the supplied voltage Vc = Vbattery) when inserting

dieletric with battery connect.

Since capacitance must be increased when dielectric is inserted, 0CC r .

If V is constant, C increases, Q will also increase.

V

QC , QC ,

Charge increased by a factor εr, dielectric constant.

000 Q

Q

VQ

VQ

C

Cr

0QQ r

Battery connected

V remains constant

Documents

Chapter 2: Capacitor and Dielectrics - YSL Physicsyslphysics.weebly.com/uploads/4/8/2/6/48261583/chapter_2.pdf · 2019-11-10 · CHAPTER 2: CAPACITOR AND DIELECTRICS prepared by Yew