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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