Capacitance and Dielectrics

1.CapacitanceDefinitionHow to calculate the capacitance

2.Capacitor3.Energy stored in a capacitor4.Capacitor with dielectrics5.Dielectrics explained in an atomic

view

Lect. (4)

1

Capacitance: the definition. The capacitance, C, is defined as the ratio

of the amount of the charge Q on the conductor to the potential increase ΔV of the conductor because of the charge:

This ratio is an indicator of the capability that the object can hold charges. It is a constant once the object is given, regardless there is charge on the object or not. This is like the capacitance of a mug which does not depend on there being water in it or not.

The SI unit of capacitance is the farad (F)

V

QC

QV

C=

1V

1C F1

2

More About Capacitance

Capacitance will always be a positive quantity The capacitance of a given capacitor is constant The capacitance is a measure of the capacitor’s

ability to store charge The farad is an extremely large unit, typically you

will see

microfarads (mF=10-6F),

nanofarads (nF=10-9F), and

picofarads (pF=10-12F)

3

Capacitors are devices that store electric charge Any conductors can store electric charge, but Capacitors are specially designed devices to store a lot of

charge

Examples of where capacitors are used include: radio receivers filters in power supplies to eliminate sparking in

automobile ignition systems energy-storing devices in

electronic flashes

4

How to Make a Capacitor?

Requirements: Hold charges The potential increase

does not appear outside of the device, hence no influence on other devices.

For example, a `parallel plate’ capacitor, has capacitance

area surface theis

,0

0

A

d

A

d

A

Ed

A

V

QC

02

E

00 22

E

0E0

E 0E

d

EdV 5

A Real Parallel Plate Capacitor charged up with a battary Each plate is connected to a terminal of the battery

The battery is a source of potential difference If the capacitor is initially uncharged, the battery

establishes an electric field in the connecting wires This field applies a force on electrons in the wire just

outside of the plates The force causes the electrons to move onto the

negative plate This continues until equilibrium is achieved

The plate, the wire and the terminal are all at the same potential

At this point, there is no field present in the wire and the movement of the electrons ceases

The plate is now negatively charged A similar process occurs at the other plate, electrons

moving away from the plate and leaving it positively charged

In its final configuration, the potential difference across the capacitor plates is the same as that between the terminals of the battery

V

6

Energy stored in a charged capacitor

Consider the circuit to be a

system Before the switch is closed, the

energy is stored as chemical energy in the battery

When the switch is closed, the energy is transformed from chemical to electric potential energy

The electric potential energy is related to the separation of the positive and negative charges on the plates

A capacitor can be described as a device that stores energy as well as charge

7

How Much Energy Stored in a Capacitor?

q -q

dq

To study this problem, recall that the work the field force does equals the electric potential energy loss:

VQUWE

E

V

dqC

qVdqdWB

When the charge buildup is q, move a dq, the work is

This also means that when the battery moves a charge dq to charge the capacitor, the work the battery does equals to the buildup of the electric potential energy:

UWB

We now have the answer to the final charge Q:

UC

Qdq

C

qdWW

QQ

BB 2

2

008

Energy in a Capacitor, the formula

When a capacitor has charge stored in it, it also stores electric potential energy that is

This applies to a capacitor of any geometry The energy stored increases as the charge

increases and as the potential difference (voltage) increases

In practice, there is a maximum voltage before discharge occurs between the plates

22

)(2

1

2VC

C

QU E

9

Energy in a Capacitor, final discussion

The energy can be considered to be stored in the electric field

For a parallel-plate capacitor, the energy can be expressed in terms of the field as

U = ½ (εoAd)E2

It can also be expressed in terms of the energy density (energy per unit volume)

uE = ½ eoE2

10

11

Circuit Symbols

A circuit diagram is a simplified representation of an actual circuit

Circuit symbols are used to represent the various elements

Lines are used to represent wires

The battery’s positive terminal is indicated by the longer line

12

Capacitors are in Series:

When capacitors are in series,

the charge is the same on each capacitor.

321 VVVVt C

QVCVQ

3

3

2

2

1

1

C

Q

C

Q

C

Q

C

Q

t

t

321 QQQQt

321

1111

CCCCt

Capacitors are in Parallel

When capacitors are in parallel ,

the total charge is the sum of that on each capacitor.

321 QQQQt CVQ

332211 VCVCVCVC tt

321 VVVVt

321 CCCCt

Equivalent Capacitance, Example

The 1.0-mF and 3.0-mF capacitors are in parallel as are the 6.0-mF and 2.0-mF capacitors

These parallel combinations are in series with the capacitors next to them

The series combinations are in parallel and the final equivalent capacitance can be found 15

A dielectric is a nonconducting material, an electrical insulator, such as rubber, glass, or waxed paper.

When a dielectric is inserted between the plates of a capacitor, the capacitance increases by a dimensionless factor , which is called the dielectric constant of the material.

The dielectric constant is the ratio of the field without the dielectric (Eo) to the net field (E) with the dielectric:

 = Eo /E

Dielectrics

16

• Dielectrics

• The dielectric constant varies from one material to another.

The voltages with and without the dielectric are related by the factor as follows:

17

if E0 is the electric field without the dielectric, the field in the presence of a dielectric is

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