Capacitor

Any two pieces of metal (conductor)

brought near each other.

Parallel plate capacitor

L

W

Area A = L x W

Charged capacitor

“Charged Capacitor” has no net charge.

+++++++++++++++++++++++

-----------------------------------------

+Q

-Q

E=0

E=0

E

Electric Field (treating plates as

infinite) and having area A.

⎟⎠

⎞⎜⎝

⎛===A

QEE

00

1||

εεσv

d

Capacitance

+++++++++++++++++++++++++++

----------------------------------------------

+Q

-Q

⎟⎠

⎞⎜⎝

⎛==A

QE

00

1

εεσ

EdrdEVV =⋅=∆= ∫v

v

)( 0 dA

QV

ε=→

We define “capacitance”

V

QC ≡

For parallel plate capacitor

d

AC 0ε≡

d

Capacitance

V

QC ≡

Can think of capacitor as a device for storing charge

(and also energy).

Units [Capacitance] = [Coulomb/Volt] = [Farad]

L

L

d

A parallel-plate capacitor has square plates of edge

length L, separated by a distance d.

If we doubled the dimension L and halve the dimension

d, by what factor have we changed the capacitance?

A) 1.0

B) 2.0

C) 4.0

D) 8.0

E) 16.0( )22 2oo o o

2LA L LC 8

d d d / 2 d

εε ε ε= = → =

CPS Question

How big is a one Farad Capacitor?

CPS question

Assume you have two parallel plates that are separated by 1 mm.

If you estimate ε0

~ 10-11, what is the area of the plates to have a

1.0 Farad capacitance?

A) 100,000,000 square meters

B) 1,000 square meters

C) 1 square meters

D) 0.1 square meters

E) 0.001 square meters

d

AC 0ε≡

Commercial capacitors

Micro-farad capacitors in small packages are

made by making d very small ~ atomic

dimensions.

Energy Storage

Capacitor stores both charge and energy.

Suppose the capacitor is not fully charged. How much work is

required to move a little bit +dq across?

+++++++++++++++++++++++++++

----------------------------------------------

+Q

-Q

dqVdqdUdWWork ===

dqC

qdU ⎟

⎠

⎞⎜⎝

⎛=

C

Qqd

C

qdUU

Q

2

2

0

=⎟⎠

⎞⎜⎝

⎛== ∫∫

Each next piece +dq gets harder

and harder.

QVCVC

QU

2

1

2

1

2

1 22

===Potential Energy stored in the

capacitor.

CPS Question

A parallel plate capacitor is charge up (+Q on one plate and

–Q on the other). The plates are isolated so the charge Q

cannot change. The plates are then pulled apart so that the

plate separation d increases. The total electrostatic energy

stored in the capacitor?

A) Increases

B) Decreases

C) Remains constant

+++++++++++++++++++++++

-----------------------------------------

+Q

-Q

Ed QVCVC

QU

2

1

2

1

2

1 22

===

Where is this energy U?

Answer = the electric field has energy.

Energy density of E-field = uVol

U

volume

energy ==.

( )dA

Edd

A

Vol

CV

⋅

⎟⎠

⎞⎜⎝

⎛

==

202

21

.21 ε

202

1Eε=

Turns out work to charge capacitor is equal to energy stored.

Electric Field energy

This is valid for any electric field configuration in vacuum:

“empty” space contains energy!!!

Spring analogy

+++++++++++++++++++++++++++

----------------------------------------------

+Q

-Q

21

2

1Q

CU ⎟

⎠

⎞⎜⎝

⎛=2

2

1kxU =

Electric field energy Elastic energy

What is this device?

Applications

Applications (cont)

Capacitors are good for storing large amounts of energy, which

can then be accessed quickly (e.g. camera flash).

Capacitors are also ideal transducers.

Devices that covert physical quantities into electrical

signals (e.g. computer keyboard, elevator buttons).

A

d d

AC 0ε≡

Circuits with capacitors

Circuit symbol for a capacitor

More than one capacitor?...

Capacitors in parallel

( )

21

21

2121

2211

21

;

CCC

CCV

Q

VCCQQQ

VCQVCQ

VVVV

eq

total

ab

+=→

+=→

+=+===→

===

Capacitors in parallel (cont)

Think of two capacitors in parallel as adding the

area of the two capacitors

...321 +++= CCCCeq

Many capacitors in parallel:

Capacitors in series

21

2121

22

11

21

111

11

;

CCCQ

V

CCQVVVV

C

QVV

C

QVV

QQ

eq

ab

cbac

+==→

⎟⎟⎠

⎞⎜⎜⎝

⎛+=+==

====→

=

Capacitors in series (cont)

21

21

111

or

111

CC

C

CCC

eq

eq

+=

+=

...1111

321

+++=CCCCeq

Many capacitors in series:

19

CPS question

A bank of three capacitors (C1, C

2, C

3) is connected as follows.

C1=2 F

C3=1 F

C2=1 F

What is the effective total

capacitance of the bank?

A) C = 0 Farad

B) C = 1 Farad

C) C = 1.5 Farad

D) C = 2.0 Farad

E) C = 0.5 Farad

DielectricsWhat if the gap between the capacitor plates is not empty?

Dielectrics (cont)

0/VQCwithout =

withoutCC

VV

VQC

>→<

=

0but

/

Without dielectric:

With dielectric:

Increases the capacitance!!!

Dielectric constant

Dielectric constant:

V

V

C

CK 0

0

==

K

VV 0=

With the dielectric present, the

potential difference for a given charge Qis reduced by a factor K

Dielectric constants for some materials

Induced charge and polarization

PolarizationWe assume that the induced surface charge

density is directly proportional to the electric

field magnitude in the material

Without dielectric:

00 ε

σ=E

0

0

εσσ induced

K

EE

−==With dielectric:

⎟⎠

⎞⎜⎝

⎛ −=Kinduced

11σσFrom these we

obtain:

Permittivity

0εε K=

We have found:

εσ=E

0

0

εσ

KK

EE ==

Defining:

We obtain:

permittivity

and

220

00

2

1

2

1EEKu

d

A

d

AKKCC

εε

εε

==

===Capacitance

Energy density

Easy rule of thumb: replace ε by ε0

in all the equations