23. electrostatic energy and capacitors. 2 topics capacitors electrostatic energy using capacitors
Post on 18-Dec-2015
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TRANSCRIPT
4
Capacitors
A capacitor is a device that stores charge –Q on one conductor and charge +Q on the other conductor
The stored chargecreates an electricfield, and therefore,a potential difference between the conductors
–Q+Q
5
Capacitance
QC
V
The capacitance of a device is defined by
The unit is the farad (F) = Coulomb/VoltBut since the farad is such a huge unit, the morecommonly used units are F = 10-6 F or pF = 10-12 F
Q is the charge stored on a conductorV is the potential difference betweenthe conductors
6
CapacitanceSpherical Conductor
QV k
R
The potential on the surface of aspherical conductor of radius R is
04Q R
C RV k
Therefore, its capacitance is
7
CapacitanceParallel Plate Capacitors
If two conducting plates ofarea A are separated by a small distance d the electric field between them will beapproximately constant and of magnitude
0 0/ /( )E Q A
8
CapacitanceParallel Plate Capacitors
0 0( / ) /( )
V E
Q
d
d Ad
Since the electric field isconstant, the potential difference between the plates is simply
so the capacitance is
0Cd
A
9
CapacitanceCylindrical Capacitors
0
12
2
ln( / )RC
R
L
A coaxial cable of length L is an example of a cylindrical capacitor
R2
R1
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Electrostatic Energy
q1
q2
q3
Total work done1 3 2 31 2
2 3
kq q kq qkq qW W W
a a a
2 2 1 21W q q
kqV
a a
1 23 3 1 2 3( ) ( )W q q
kq kqV V
a a
aa
Work is required to assemble a charge distribution
12
Electrostatic Energy
dW dqV
W dW Vdq
dq
The work dW required to add an element of charge dqto an existing charge distribution is
where V is the potential at the finallocation of the charge element. The
total work required is thereforeSince the electric is conservative, the work is stored as electrostatic energy,U.
13
Storage of Electrostatic Energy
Work must be done to move positive chargefrom a negatively charged conductor to onethat is positively charged. Or to move negative charge in the reverse direction.
14
Storage of Electrostatic Energy
In moving charge dq, the electrostaticenergy of the capacitor is increased by dU Vdq
2
0
1
21
2
Q q QU dq
C C
QV
Therefore,
21
2U CV
15
Energy Density of Electric Field
1
2U VQ
0/( )E Q A
20 0
1 1( )( ) ( )
2 2EdE dA AU E
V Ed
Potential energy
Electric field
Electric potential
16
/( )E
energyu U
vold
umeA
The energy density uE
20
1
2Eu EThis expressionholds true forany electric field
Energy Density of Electric Field
17
Example – A Thunderstorm
How much electrical energy is stored in a typical thundercloud?
Assume a cloud of
height h = 10 km,
radius r = 10 km,
with a uniform
electric field
E = 105 V/m.
http://redcrossggr.files.wordpress.com/2008/06/thunderstorm.jpg
18
Example – A Thunderstorm
Narrative
The problem is about stored electric energy.
Since the electric field, E, is uniform, so to is the energy density uE = ½ 0 E2 in the cloud. Therefore, the electric energy stored in the cloud is just the electric energy density times the volume of the cloud.
20
Example – A Thunderstorm
Calculation
The electric energy density in the cloud is
uE = ½ 0 E2 = 4.4 x 10-2 J/m3.
The volume of the cloud is, v = r2 h, that is,
v = 3.1 x 1012 m3.
Therefore, the total electric energy stored in the cloud is U = uEv = 140 GJ.
22
The Effect of Dielectrics
Michael Faraday1791 – 1867
wikimedia
Michael Faraday discovered that the capacitance increases when the space between conductors is
replaced by a dielectric.
Today, we understand this to be a consequence of the polarization of molecules.
23
The Effect of Dielectrics
The polarized molecules of the dielectric tend toalign themselves parallel to the electric field createdby the charges on the conductors
b b
--------
++++++++
24
The Effect of Dielectrics
The bound charge b induced on the surface of the dielectric creates an electric field opposed to the electric field of the free chargef on the conductors,thereby reducing thefield between them.
25
The Effect of Dielectrics
The reduction in electric field strength from theinitial field E0 to the reduced field E is quantifiedby the dielectric constant (kappa)
0EE
The dielectric increases the capacitance by the same factor .
26
The Effect of Dielectrics
For a parallel plate capacitor, with a dielectricbetween the plates, the electric field is
0 is called thepermittivity
The product of the dielectric constant and thepermittivity of free space 0
0/( )E Q A
27
Capacitors in Parallel
At equilibrium, the potential across each capacitor is the same, namely, 12 V
same potential
same potential
30
The potential V1
across C1 plus thepotential V2 across C2 is equal to thepotential difference V between points a and b:V = V1 + V2
1 21/ 1/ 1/C C C