PHYSICS 1B – Fall 2009courses.physics.ucsd.edu/.../Chapter_16part3.pdf · capacitance of 1m2 of...
Transcript of PHYSICS 1B – Fall 2009courses.physics.ucsd.edu/.../Chapter_16part3.pdf · capacitance of 1m2 of...
PHYSICS 1B – Fall 2009
Electricity&
Magnetism
Professor Brian Keating
SERF Building. Room 333
Capacitor combinations
Capacitors connected in seriesand parallel
Electrical circuit elements
Circuit diagramVoltage source resistor capacitor
Conductor
VR
C
+-
one capacitor
+q
-q+
-
CΔV Vc=ΔV
qCV
=!
Two Capacitors in Parallel
equivalentcapacitance
=!eqqCV
C1 C2ΔV
q
q1q2 Ceq
q
qΔV
What single capacitor has the same properties as two capacitors in parallel?
=
PHYSICS 1B – Fall 2009
Electricity&
Magnetism
Professor Brian Keating
SERF Building. Room 333
Capacitors
Two Parallel Capacitors
C1 C2ΔV
1 2V V V! = ! = !
1 2eq
C V C VqCV V
! + != =! !
equivalentcapacitance
=!eqqCV
= + = ! + !1 2 1 2q q q C V C V
q
q1 q2
1 2eqC C C= +
Parallel Capacitors
= + +1 2 ........eq NC C C C
For N capacitors in parallel
Ceq is the sum of capacitances
Like a larger capacitor, larger area
5µF 3µFΔV
10µF
Find the equivalent capacitance
µ
= + +
= + + =1 2 3
5 3 10 18eq
eq
C C C C
C F
A. 15 uF
B. 17 uF
C.18 uF
D.20 uF
5µF 3µFΔV
10µF
Find the equivalent capacitance
µ
= + +
= + + =1 2 3
5 3 10 18eq
eq
C C C C
C F
Two Capacitors in Series
C1
C2
ΔV
q +q-q
+q-q
q
q=!eqqCV
What is the equivalentcapacitance?
ΔV1
ΔV2
the charge on bothcapacitors in seriesis q
Two Capacitors in Series
C1
C2
ΔV
q +q-q
+q-q
q
q
=!eqqCV
q=q1=q2
1 2
1 1 ( )eq
V q qC q q C C
!= = +
! = ! + ! = +1 21 2
q qV V VC C
1 2
1 1 1
eqC C C= +
For N capacitors in series
= + +1 2
1 1 1 1.........eq NC C C C
Capacitors in series
Ceq is smaller than the smallest capacitance.You store less charge on series capacitorsthan you would on either one of them alonewith the same voltage!
+ + + + + + + + + + + + +
- - - - - - - - - - -
+ + + + + + + + + + + +
- - - - - - - - - - -
- - - - - - - - - - - + + + + + + + + + + + + +
C
ΔV ΔV/2
ΔV/2
C1=C2=Ceq =
The equivalent capacitance is less than thecomponent capacitances
C1
C2
2CC
Ceq <C1 or C2
Physical ArgumentTake a parallel plate capacitor and place a thin metal plate with the same area in the middle of the gap.
the component capacitancesare larger than the total
34. Find the equivalent capacitance.
X
X
Ceq= 4.00+2.00+6.00=12.00 µF
= + = =1 1 1 4 1
24 8 24 6seriesC
Cseries=6µF
34. Find the charge on each capacitor.
X
XCseries =6µF
= !q C V
C1
C4
C3
C2
q1=C1ΔV=4x10-6(36)=1.44x10-4 C
q2 =C1ΔV=2x10-6(36)=0.72x10-4 C
q3=q4=CseriesΔV=6x10-6(36)=2.16x10-4C
34.Find the voltage drop across eachcapacitor.
ΔV1=ΔV2=36V
ΔV1 ΔV2
ΔV3
ΔV4
! =qVC !
!" = = =
4
3 63
2.16 10 9.024 10
q xV V
C x!
!" = = =
4
4 64
2.16 10 278 10
q xV V
C x
series capacitorsThe larger C has the smaller voltage drop
16.9 Dielectrics, Energy
Dielectric constant-effect on capacitanceEnergy stored in a capacitorEnergy density (depends on E2)Biological Membranes
Dielectric material – insulators such as paper, glassplastic, ceramic. Used in the gap in capacitors.
“Dielectric Strength” - is the electric field at which conduction occurs through the material
dielectric material
-q
+q
Electric Fields in Dielectric FilledCapacitors
Vacuum
dielectric material
ΔVo
oVV!"
" =
Effects of a dielectric material inserted into a capacitor,with charge q
-q
-q
+q
+q
Potential due tocharge qdecreases by κ
ΔV
κ= dielectric constant (dimensionless)
Dielectric Properties of Selected Materials
Dielectric Strength Material dielectric constant, κ (Volt/m)
Vacuum 1.000000 -----Air 1.00059 2x106
Polystyrene 2.3 24x106
Paper 3.4 16x106
Pyrex 5.6 14x106
Water 80 -------
ΔVo
-q
-q
+q
+q
ΔV
How does the capacitance change?
Co
C=o
q qV V
!=
" "
oC C!=Capacitance increases
Originally
Add dielectric
ΔVo
-q
-q
+q
+q
ΔV
How does the E field change?
E= oVVd d!
""=
oEE!
=
Electric field decreases (when not connected to a battery)
Permittivity is increased
Compared to vacuum
o! "!=
o
q qEA A!" "
= =
ΔVo
-q
-q
+q
+q
ΔV
Example: A parallel plate capacitor consists of metal sheets(A= 1.0m2) separated by a Teflon sheet (κ=2.1) with a thicknessof 0.005 mm. (a) find the capacitance. (b) Find the maximumvoltage. The maximum electric field across Teflon is 60x106
V/m. – this is its dielectric strength.
12
3
6
2.1(8.8 10 )(1.0)0.005 10
3.7 10
oA xCd x
C x F
!" #
#
#
= =
=
(a)Κ=2.1
A=0.25m2
d=0.005m
A parallel plate capacitor consists of metal sheets(A= 0.25m2)separated by a Teflon sheet (κ=2.1) with a thickness of 0.005mm. (a) find the capacitance. (b) Find the maximum voltage.The maximum electric field across Teflon is 60x106
V/m.(dielectric strength)
6 3max 60 10 (0.005 10 ) 300dsV E d x x V!" = = =
(b)
Molecular basis for dielectric constant
Dipolar molecules.
e.g. water
+
-
Eare oriented inan E field
Oriented molecules decrease the net charge nearthe plates
The E field in the Capacitor is reduced
!= oEE
Polarization of Dielectric
Dielectric Screening
High dielectric constant of water allows ions to dissociate
Na+ Cl-
+
+-
+- +-+-+ -
+-
+-
+-
+-
+ --
!= =
80o oV V
V
Find the potential energy in electron volts for theinteraction of Na+ and Cl- separated by 0.5 nm in water.
19 2
12 9
2119
(1.6 10 )4 4 (80)(8.8 10 )(0.5 10 )
15.8 101.6 10
Na cl
o
q q xPE
r x x
eVPE x Jx
x J
!"# !
$
$ $
$$
$= =
% &= $ ' (
) *
PE= -0.036 electron Volts
comparable to thermal energies (kinetic energy ofthe ion) about 0.025 eV at room temperature
Energy stored in a capacitor.
q=0
q=0
Workdone tocharge acapacitor
ΔV=0
+q
-q
ΔVq
221 1
2 2= = !
qW C VC
So work depends on the square of q or ΔV
A parallel plate capacitor consists of metal sheets (A= 1m2)separated by a teflon sheet (κ=2.1) with a thickness of 0.005mm. Find the maximum energy that can be stored.
C
C=3.7 x10-6F ΔVmax=300V
2 6 2
1
1 1(3.7 10 )(300)2 21.7 10
Energy C V x
Energy x J
!
!
= " =
=
Insert the dielectric material with dielectric constant κ into the capacitor keeping the voltage source connected. Find C,q,E, PE
-qo
+qoΔVo
ΔVo
q
-q
Co
C=
E=
q=
oC!
o oCV C V q! != =
oo
VV Ed d
!!= =
PE= 2 21 12 2 o oC V C V PE! !" = " =
(quick quiz 16.6)
Energy Density in a Capacitor
Suppose you wanted to store a large amount of energyin a capacitor with a given volume of 1m3 using TeflonAs the dielectric (dielectric strength of 60x106 V/m).What is the maximum energy that could be stored?
212
Energy CV=
oAC
d!"
=
22 2
2
1 1 1 ( )2 2 2
o oo
A Ad VEnergy V E volume
d d!" !"
!"= = =
212 o
Energy Evolume
!"= The energy density dependsonly on the E field squared.
2
12 6 2
4
1 ( )21(2.1)(8.8 10 )(60 10 ) (1)23.4 10
oEnergy E volume
Energy x x
Energy x J
!"
#
=
=
=
For comparison the energy content of burning
1 gallon of gasoline is 1.3x108 J
Chemical energy has a higher energy density.
For a 1 m3 capacitor atthe maximum voltage.
For the maximum electric field = dielectric strength
Capacitance of Biological Membranes
Axon - Nerve cells
+
+
+ + ++ ++
-
- ----
Potential difference across the membrane
membrane
Nerve transmission – involves a discharge of membrane potential
The low dielectric portion of a biological membranehas a thickness of 2.0 nm. Assume that it has adielectric constant of 2.5 (silicone oil) find thecapacitance of 1m2 of membrane.
12
9
2
2.5(8.8 10 )(1)2.0 10
1.1 10
xC
xC x F
!
!
!
=
=
High Capacitance
Compare to 3.7x10-6 F for1m2 the Teflon capacitor.
Biological membranes –Capacitance
oAAC
d d!""
= =
lowdielectric
high dielectric
A nerve cell has a potential across it of 60 mV. Find the density of charges on the membrane that can give rise to this potential
ΔV=60 mV
+ + + + + +- - - - - -
37
960 10 3 10 /2 10
V xE x V m
d x
!
!
"= = =
This is close to the dielectric strength
12 7
4 2
2.5(8.8 10 )(3 10 )
6.6 10 /
o o
o
qE
A
E x x
x C m
!"# "#
! "#
!
$
$
= =
= =
=
This corresponds to an ion density of 4.1x10-3 ions /nm2
or a distance between ions of about 16 nm. A small numberof excess charges.