lecture 9: capacitance and capacitors the interior of a sony walkman
TRANSCRIPT
Lecture 9: Capacitance and Capacitors
The interior of a Sony Walkman
Physics for Scientistsand Engineers
Chapter 24:Electrostatic Energy and Capacitance
Copyright © 2004 by W. H. Freeman & Company
Paul A. Tipler • Gene Mosca
Fifth Edition
Learning Objectives
•To introduce the concept of capacitance, C
•To provide a definition for the C of a conductor
•To calculate C of ideal capacitors
The Importance of Capacitors:(a) They store potential energy in the E-field
(b) Many applications
Examples include:
(i) use in electronics
(ii) flash units of cameras
(iii) pulsed lasers
(iv) computer keyboard
Capacitance for Charge
Q placed on a conductor changes the conductor’s potential by V
Define the capacitance C of the conductor by the equation
V
QC =
Unit of C - coulomb volt-1 - farad (F)
Q
Example: an isolated conducting sphere of radius a and carrying a charge Q
a
QV ⎟⎟
⎠
⎞⎜⎜⎝
⎛=
041πε
So aV
QC 04πε==
A sphere of radius 9 109 m (more than 103 that of the Earth) would have a capacitance of about 1 farad.
Common capacitors in use: pF-F
Earthing
aRadius b
Consider two conducting spheres
Charge q
The two spheres acquire the same potential
qa
qb
Earthing
€
Va =1
4πε0
⎛
⎝ ⎜
⎞
⎠ ⎟Qa
a
€
Vb =1
4πε0
⎛
⎝ ⎜
⎞
⎠ ⎟Qb
b
€
qa
a=
qb
bSame potential means
ba qqq +=Total charge is constant
) ( 1
baqb
a
ba
qqa >>≈
⎟⎠⎞
⎜⎝⎛ +
=Thus
€
qa
a=
qb
b=
q − qa
b
Important Conclusion
Earthed isolated charged bodies share their charge with the Earth - effectively lose that charge. The bodies and the Earth acquire a common potential - called zero of potential.
The potential of the earth does not change
€
Va =1
4πε0
⎛
⎝ ⎜
⎞
⎠ ⎟Qa
a
Because a is very Large
Similar to the use of sea level as a reference for height
Capacitors•It is a system of high capacitance, designed for the storage of separated positive and negative charges
•A capacitor is achieved by moving charge from one conductor to another
It is a measure of the ability of a capacitor to
store energy
V
QC ==
conductors e between th . .p dcharge either of magnitude
Capacitance of a Capacitor
Defined by:
€
x →8lim
1
x − 8= ∞
€
x →5lim
1
x − 5= 5
€
1
nsin x =
1
/ n si / n x = six = 6
Given that:
What is
Calculation of Capacitance
3 geometries - planar, cylindrical, and spherical
Procedure
•Calculate the change of V
•Apply C = Q/V
(a) The Parallel Plate Capacitor
We will ignore any fringing effects
d
Gauss’s Law: A
QE
00 εε
==
Potential difference between the two plates: Ed
dA
QEdVVV
0ε==−= −+
€
C =Q
V=
Aε0
d=
ε0A
d
The symbol of a capacitor in electric circuits
If the voltage across a parallel plate capacitor is doubled, its capacitance:
(a)Doubles
(b)Drops by Half
(c)Remains the same
€
C =ε0A
d
If the charge on an isolated spherical conductor is doubled, its capacitance:
(a)Doubles
(b)Drops by Half
(c)Remains the same
aV
QC 04πε==
(b) A Long Cylindrical Capacitor (co-axial cable)
(b) A Long Cylindrical Capacitor (co-axial cable)
What is the capacitance per unit length?
This is important in determining the transmission characteristics of the cable.
Choose a cylindrical Gaussian surface
rE
02πελ
=
⎟⎟⎠
⎞⎜⎜⎝
⎛=−=−= ∫
a
b
r
r
rr r
rlnEdrVVV
a
b
ba
02πε
λ
⎟⎟⎠
⎞⎜⎜⎝
⎛==
a
b
rr
lnV
C 02πελ
rb
ra
(c) A Spherical Capacitor
ra
rb
Appropriate Gaussian surface is a sphere concentric with, and between, the conducting spheres+Q
-Q 204 r
QE
πε=
∫−=−=a
b
ba
r
r
rr EdrVVV
⎟⎟⎠
⎞⎜⎜⎝
⎛−=−= ∫
ba0
r
r2
0 r
1
r
1
4
Q
r
dr
4
QV
a
bπεπε
Hence
or( )
ba
ab
rr
rrQV
04πε
−=
ab
ba
rr
rr
V
QC
−== 04πεTherefore
The symbol for spherical capacitor
Review and Summary
Procedure for calculating the capacitance of a capacitor
•Calculate the E-field (e.g. use Gauss’s Law)
∫−= ldEV .•Use
V
QC =•Followed by
€
ε0A
d
ab
ba
rr
rr
V
QC
−== 04πε
⎟⎟⎠
⎞⎜⎜⎝
⎛==
a
b
rr
lnV
C 02πελ
aV
QC 04πε==
Parallel plates capacitor
Cylindrical capacitor
Spherical capacitor
An isolated sphere
Some CapacitancesCapacitor Type CapacitanceParallel Plate
Cylindrical
Spherical
Isolated Sphere
Capacitors
