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Capacitance and Capacitance and DielectricsDielectrics
CapacitanceCapacitors in combination #Series#Parallel
Energy stored in the electric field of capacitors and energy density
DielectricsDielectric Strength
Lesson 4
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Field Above ConductorField above surface of
charged conductor
Does not depend on thickness of conductor
E Q
A 0 0
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charge =
Area AE
conductor in electrostatic equilibrium
A 0
E dA EdA
A
closedcylinder
E dA
A
E
A
A 0
0
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Charged Plates
+ -
d
E
WFdQEd
UQEd U UVEdV V
+Q -Q
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Potential drops Ed in
going from + to -V- is Ed lower than V+
PD between Plates
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How does one make such a separation of charge? Must move positive chargeWork is done on positive charge in producing separation
Q -Q+QF
Work Done in Moving Charge
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What forms when we have separation of charge?An Electric Field
+Q -QE
Electric Field
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Capacitorb
The work done on separating charges to fixed positionsis stored as potential energy in this electric field, which can thus DO work
This arrangement is called a CAPACITORCAPACITOR
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How do we move charge?With an electric fieldalong a conduction pathconduction path
Moving Charge
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Picture
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The charge separation is
maintained by removing the conduction pathonce a charge separation has been
producedAn electric component that does
this is called A Capacitor
Charge Separation
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Capacitor Symbol
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+ -
Battery Symbol
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Charging CapacitorCan charge a capacitor by
connecting it to a battery
+
+ -
-
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CapacitancePlates are conductorsEquipotential surfacesLet V = P.D. (potential difference)
between platesQ (charge on plates) ~ V (why?)Thus Q = CVC is a constant called
CAPACITANCECAPACITANCE
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SI Units
FaradsVolts
Coulombs
V
C
V
QC
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Calculation of Capacitance
assume charge Q on platescalculate E between plates using Gauss’ LawFrom E calculate V Then use C = Q/V
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Capacitors
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Electric Field above Plates
0
00
plates to is
EAQ
A
QE
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Calculating Capacitance in General
going from positive to negative plate
V = Vf Vi E ds
i
f
0
E ds 0 choose path from + plate to - plateV = - V ( PD across plates )
Thus V = Eds+
-
( choose path | | to electric field )
C EA 0
Eds+
-
In order that
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for Parallel Plates Capacitor
-
+
CQ
V EA0
EdsEA0
EdA0
d
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C Q
V 2 0 L
lnb
a
•a = radius of inner cylinder•b = radius of outer cylinder•L = length of cylinder
for Cylindrical Capacitor
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Combination of Capacitors Parallel
Combinations of Capacitors in Combinations of Capacitors in equilibriumequilibrium
Parallelsame electric potential felt by
each elementSerieselectric potential felt by the
combination is the sum of the potentials across each element
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Picture
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Calculation of Effective Capacitance
V Q1
C1
Q2
C2
Total charge Q Q1 Q2
VC1 VC2 VCeq
Ceq C1 C2
In general
Ceq Ci
i
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Combination of Capacitors Series
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PictureNet charge zero
Net charge zero
Why are the charges on the plates of equal magnitude ?
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Calculation of Effective Capacitance I
If net charge inside these Gaussian surfaces is not zero
Field lines pass through the surfaces
and cause charge to flowThen we do have not equilibrium
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Calculation of Effective Capacitance II
Vtotal V1 V2 Q
C1
Q
C2
Q1
C1
1
C2
Q
1
Ceq
In general
1
Ceq
1
Cii
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Question I
Is this parallel or series?
=
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Question II
Is this parallel or series?
+
+ -
-
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Work Done in Charging Capacitor
Work done in charging capacitor
I +
+ -
-q
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CalculationV q q
Cif dq of charge is then transfered the work done is
dW V q dq
Thus total work done on charging is
W V q dq0
Q
1
Cqdq
0
Q
Q2
2C1
2CV 2
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Energy DensityThis work is stored as P.E.
EnergyDensity U
Volume
for parallel plate capacitor U
Ad
CV 2
2Ad1
2 0
V
d
2
1
20 E2
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DielectricsDielectrics
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Picture
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Picture
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Picture
Polarization
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Induced Electric Field
Polarization
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Charge Q stays the same, Total electric Field is less,
thus P.D. Veffective across plates is less
C Q
V C
Q
Veffective
C C
Dielectric Constant 1.00
Dielectric Constant
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C0 A
d
C C 0
A
d
C A
dwhere, PERMITTIVITY of the dielectric 0
Permitivity
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Permitivity in Dielectrics
For conductors ( not dielectrics )
For regions containing dielectrics all electrostatic equations containing
0 are replaced by e .g . Gauss ' Law
E dAQ
surface
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The Dielectric Strength Dielectric Strength of a non conducting material is the value of the Electric Field that causes it to be a conductor. When dielectric strength of air is surpassed we get lightning
Dielectric Strength