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Capacitance and Dielectrics 1 Capacitance and Dielectrics Capacitance and Dielectrics 2 Capacitance A capacitor is a charge storage device. It stores energy as potential energy in an electric field. A capacitor consists of two nontouching conductors, which hold equal but opposite charges. Capacitance and Dielectrics 3 Capacitance C = Q "V Capacitance has units of coulombs per volt, which is defined to be a farad (F). Capacitance (C) is the ratio of the charge Q to the potential difference !V between the conductors. or Q = C"V The capacitance is constant for a given capacitor, and depends only upon the structure and dimensions of the device. "V = Q C Capacitance and Dielectrics 4 Parallel-Plate Capacitors A parallel-plate capacitor consists of two conductive plates with area A, separated by a distance d. The capacitance is given by : C = " o A d ! o - the permittivity of a vacuum (! o = 8.85 x 10 -12 C 2 /N·m 2 ) A - area of one plate (m 2 ) d - spacing between plates (m) C - capacitance (F) 5 Parallel conducting sheets total area A, spacing d, and charge Q Between the sheets : "V = V + # V # = Ed (uniform field) "V = # $ 0 d C = Q "V v E " d v A # = q enc $ 0 EA = "A # 0 E = " # 0 = Q " 0 A d = " 0 A d " -" d A E E = "V d Parallel-Plate Capacitors 6 r i Q -Q r o Concentric spherical conducting shells E = Q 4"# 0 r 2 Between the spheres : "V = V i # V o = # v E $ d v r r o r i % "V = # Q 4$% 0 r 2 r o r i & dr = Q 4"# 0 1 r i $ 1 r o % & ' ( ) * "V = Q 4#$ 0 r o % r i r i r o & ' ( ) * + C = Q "V = 4"# 0 r i r o r o $ r i % & ' ( ) * v E " d v A # = q enc $ 0 E 4"r 2 = Q # 0 Spherical Capacitors

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Capacitance and Dielectrics 1

Capacitance and Dielectrics

Capacitance and Dielectrics 2

Capacitance

A capacitor is a charge storage device. It storesenergy as potential energy in an electric field. Acapacitor consists of two nontouching conductors,which hold equal but opposite charges.

Capacitance and Dielectrics 3

Capacitance

!

C =Q"V

Capacitance has units of coulombs per volt, which isdefined to be a farad (F).

Capacitance (C) is the ratio of the charge Q to the potentialdifference !V between the conductors.

!

or Q = C"V

The capacitance is constant for a given capacitor, anddepends only upon the structure and dimensions of thedevice.

!

"V =QC

Capacitance and Dielectrics 4

Parallel-Plate Capacitors

A parallel-plate capacitor consists of two conductive plateswith area A, separated by a distance d. The capacitance isgiven by :

!

C ="oAd

!o - the permittivity of a vacuum (!o = 8.85 x 10-12 C2/N·m2)A - area of one plate (m2)d - spacing between plates (m)C - capacitance (F)

5

!

Parallel conducting sheets total area A, spacing d, and charge Q

!

Between the sheets :

!

"V =V+ #V# = Ed (uniform field)

!

"V =#$0d

!

C =Q"V

!

v E "d

v A # =

qenc$0

!

EA ="A#0

!

E ="#0 !

=Q"0A

d

!

="0Ad

"

-"d

AE

!

E ="Vd

Parallel-Plate Capacitors

6

riQ -Q

ro

!

Concentric spherical conducting shells

!

E =Q

4"#0r2

!

Between the spheres :

!

"V = Vi #Vo = #v E $dv r

ro

ri%

!

"V = #Q

4$%0r2

ro

ri& dr

!

=Q4"#0

1ri$1ro

%

& '

(

) *

!

"V =Q4#$0

ro % ririro

&

' (

)

* +

!

C =Q"V

!

= 4"#0riroro $ ri

%

& '

(

) *

!

v E "d

v A # =

qenc$0

!

E4"r2 =Q#0

Spherical Capacitors

7

!

Coaxial cylindrical conducting shells total length L and charge Q

!

E ="

2#$0r

!

Between the cylinders :

!

"V = Vi #Vo = #v E $dv r

ro

ri%

!

"V = #$

2%&0rro

ri' dr

!

="2#$0

ln ro( )% ln ri( )( )

!

"V =#2$%0

ln rori

&

' (

)

* +

!

C =Q"V

!

=2"#0L

ln rori

$

% &

'

( )

!

v E "d

v A # =

qenc$0

!

E2"rl =#l

$0

ri

#

-#ro

!

=Q

2"#0Lln ro

ri

$

% &

'

( )

Cylindrical Capacitors

Capacitance and Dielectrics 8

Energy and Capacitance

Area = Energy

!

Check units : C( ) "JC# $ %

& ' ( =[ ] J

!

Q = C"VQ

!V

Capacitance and Dielectrics 9

Energy and Capacitance

The electric potential energy stored in a chargedcapacitor is:

!

UC =12Q"V

!

=12C "V( )2

!

=12Q 2

C

Area = Energy!

Q = C"VQ

!V

Capacitance and Dielectrics 10

Capacitors with Dielectric Fillers

When the space between the conductors is filledwith a dielectric (nonconducting) material otherthan air the capacitance increases by a factor $called the dielectric constant of the material.

!

="Ad

!

="#oAd

!

C = "Co

For a parallel-plate capacitor, the capacitancebecomes:

!

" =CCo

!

C ="#oAd

Capacitance and Dielectrics 11

Capacitors with Dielectric Fillers

Dielectrics serve three functions:1.) Provide the mechanical support between the

plates.2.) Increase the maximum possible potential

difference between the plates. (Air ionizesat 3 x 106 V/m.)

3.) Increase the capacitance of a capacitor withgiven dimensions.

Capacitance and Dielectrics 12

More on Dielectrics

Inserting a dielectric material decreases the electricfield between the plates.

!

E =Eo"

Surface charge on the conducting plates does notchange, but an induced charge of the opposite signappears on each surface of dielectric.

Capacitance and Dielectrics 13

Capacitors with Dielectric Fillers

!

E < E0

!

"V < "V0

!

C =Q"V

!

C > C0

" "i-"i "i -"" "i-"i "i -"

!

E0 ="#0

!

E =" #" i

$0

!

"V = Ed

Capacitance and Dielectrics 14

Still More on Dielectrics

The induced charge can be found by comparing theelectric field with and without the dielectric.

!

E =" #" i$o

!

Eo ="#o

,

The induced surface charge density is:

!

" i = " 1# 1$

% & '

( ) *

!

=Eo"

!

="#$o

!

and " =#

# $# i

Capacitance and Dielectrics 15

Capacitors in Series

• Charge is the same on each capacitor• Voltage drop across each capacitor is different unless

the capacitance is the same

+Q -Q+Q -Q +Q -Q

_+

!V

C2C1 C3

!

1Cs

=1Cii

"

!

"V = "V1 + "V2 + "V3

!

QCs

=QC1

+QC2

+QC3

Capacitance and Dielectrics 16

Capacitors in Parallel• Voltage drop is the same across each capacitor• Charge is different on each capacitor, the higher the

capacitance the higher the charge

+Q1 -Q1

+Q2 -Q2

+Q3 -Q3

_+

!V

C1

C2

C3

!

Q = Q1 + Q2 + Q3

!

Q = C1 + C2 + C3( ) " #V

!

"V =Q

C1 + C2 + C3( )

!

Cp = Cii"

!

= C1"V + C2"V + C3"V

!

=QCp