lecture 6 capacitance and capacitors electrostatic potential energy prof. viviana vladutescu
Post on 19-Dec-2015
221 views
TRANSCRIPT
![Page 1: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/1.jpg)
Lecture 6
Capacitance and CapacitorsElectrostatic Potential Energy
Prof. Viviana Vladutescu
![Page 2: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/2.jpg)
Capacitance
• A conductor in electrostatic field is equipotential and charges distribute themselves on the surface such way that E=0 inside the conductor Q on the surface is producing V
dsR
Vs
s
04
1
![Page 3: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/3.jpg)
k
QV
C
QV
linear dependence k=C
V
CFC SI
![Page 4: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/4.jpg)
Depends on –the geometry of the conductors -the dielectric constant of the medium between conductors
Capacitance (of the isolated conducting body) - is the electric charge that is added to the body per unit increase in its electric potential (is a constant of proportionality)
![Page 5: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/5.jpg)
Capacitors
![Page 6: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/6.jpg)
![Page 7: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/7.jpg)
![Page 8: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/8.jpg)
Electrolytic capacitors
![Page 9: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/9.jpg)
Determine capacitance
1- assume Vab Q (in terms of Vab)
use boundary conditions
2- assume Q Vab (in terms of Q)
![Page 10: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/10.jpg)
Q Vab
• Step 1Chose coordinate system for given geometry
• Step 2 Assume +Q and –Q on the conductors
• Step 3 Q E from D=εE=ρs or
• Step 4 E
• Step 5 C=Q/Vab
s
QdsE
a
b
ab ldEV
![Page 11: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/11.jpg)
Example
z
z
aS
QE
aD
E
S
QsQ ss
ED
S
QD s
Step 1
Step 2
Step 3
![Page 12: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/12.jpg)
d
z
a
b
zab S
Qddz
S
Qadza
S
QV
0
d
S
d
S
SQdQ
V
QC r
r
ab
0
0
Step 4
Step 5
![Page 13: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/13.jpg)
Vab Q
• Step 1Chose coordinate system for given geometry
• Step 2 Assume Vab between plates
• Step 3 Vab E D (from Laplace’s equation)
• Step 4 Boundary conditions at each plate
conductor –dielectric boundary:
ρs Q
. Step 5 C=Q/Vab
sND
![Page 14: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/14.jpg)
Example
z
Step 1
Step 2 Vab
Step 3 Laplace’s equation to find the potential everywhere in the dielectric
02 V
![Page 15: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/15.jpg)
There is no φ and z variation
BrArVr
A
r
VA
r
Vr
r
Vr
rr
Vr
rr
ln)(
0
001
![Page 16: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/16.jpg)
ba
VA
b
aAVbAaAV ab
abab
lnlnlnln
b
r
ab
V
b
r
ba
VbAr
ba
VrV ababab ln
lnln
lnlnln
ln)(
BbAbV
BaAVVaV abab
ln0)(br
ln)( arfor
![Page 17: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/17.jpg)
Step 4
rabab
r
a
ab
r
VE
br
b
ab
V
arVr
VE
ln1
1
ln
)( that know but we
ab
a
Va
ab
r
VED abr
arsrabr
r
lnln
000
![Page 18: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/18.jpg)
abVL
aL
ab
a
VSds abrabr
ln
2)2(
lnQ 00
s
s
s
Q on the inner conductor
Step 5
ab
L
VabVL
V
QC r
ab
abr
ab ln
2
ln
2 00
![Page 19: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/19.jpg)
Series Connected Capacitors
nsr CCCCC
1..........
1111
321
Parallel Connected Capacitors
nCCCCC ...........321
![Page 20: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/20.jpg)
Electrostatic Potential Energy
![Page 21: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/21.jpg)
Electric potential at a point in an electric field is the work required to bring a unit positive charge from infinity (at reference zero potential) to that point.
)(2
1
4
11222
11120
12222
VQVQW
VQR
QQVQW
![Page 22: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/22.jpg)
Now suppose we want to bring Q3 at R13 from Q1
and R23 from Q2
230
2
130
1333 44 R
Q
R
QQVQW
23
32
13
31
12
21
02 4
1
R
R
R
QQWWWtotal
)44
4444(
2
1
230
2
130
13
230
3
120
12
130
3
120
213
R
Q
R
R
Q
R
R
Q
R
QQWWtotal
![Page 23: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/23.jpg)
3322112
1VQVQVQWt
-can be negative -represents only interaction energy
![Page 24: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/24.jpg)
(eV)or )( 2
1
1N
kke JVQW
dvVWv
e 2
1
For a group of N discrete charges at rest
For a continuous charge distribution of density ρ
C
QQVCVWe 22
1
2
1 22
![Page 25: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/25.jpg)
Electrostatic energy in terms of field quantities
v
e VdvDW2
1
VDDVDV
VdvDdvDVWv v
e 2
1
2
1
dvEDdsaDVv
n
s
2
1
2
1
Substitute ρ
And by using
![Page 26: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/26.jpg)
Electrostatic Energy Density
v
e
v v
e
dvD
W
dvEdvEDW
(J) 2
1
2
1
2
1
2
2
3
22
mJ
22
1
2
1
DEEDwdvwW
v
eee
![Page 27: Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu](https://reader036.vdocuments.us/reader036/viewer/2022081514/56649d375503460f94a0f0df/html5/thumbnails/27.jpg)
Equipotential surfaces are at right angles to the electric field. Otherwise a force would act and work would be done on the path A to B.For a uniform electric field, equipotentials form planes perpendicular to the field.
Along AB, W = -q∆V = zero!
Example