Transcript A e - Personal.psu.edu

Lesson 4
Capacitance and
Dielectrics
Capacitance
Capacitors in combination
#Series
#Parallel
Energy stored in the electric field
of capacitors and energy density
Dielectrics
Dielectric Strength
Field above surface of
charged conductor
Field Above Conductor
Q
s
E = Ae = e
0
0
Does not depend on thickness of conductor
conductor in electrostatic
equilibrium
sA
=
e0
=E


E  dA =
closed
cylinder

EdA
Area A
E
A
dA
A
sA
s
\E =
=
Ae 0 e 0
charge = s
E
Charged
Plates
d
+Q
+
-Q
-
W = Fd =QEd
DU = -QEd = U- - U+
\DV = -Ed =V- - V+
PD between Plates
Potential drops Ed in
going from + to V- is Ed lower than V+
Work Done in Moving
Charge
How does one make such a
separation of charge?
Must move positive charge
Work is done on positive charge
in producing separation
+Q
F
Q
-Q
Electric Field
What forms when we have separation
of charge?
An Electric Field
+Q
E
-Q
The work done on separating
charges to fixed positions
is stored as potential energy
in this electric field, which
can thus DO work
This arrangement is called a
Capacitorb
CAPACITOR
Moving Charge
How do we move charge?
With an electric field
along a conduction path
Picture
Charge Separation
The charge separation is
maintained
by removing the conduction path
once a charge separation has been
produced
An electric component that does
this is called A Capacitor
Capacitor Symbol
Battery Symbol
+
-
Can
charge
a
capacitor
by
Charging itCapacitor
connecting
to a battery
+
+
-
-
Plates are conductors
Capacitance
Equipotential surfaces
Let V = P.D. (potential difference)
between plates
Q (charge on plates) ~ V (why?)
Thus Q = CV
C is a constant called
CAPACITANCE

Q C
C  = =
V  V
SI Units
Coulombs
= Farads
Volts
Calculation of Capacitance
assume charge Q on plates
calculate E between plates
using Gauss’ Law
From E calculate V
Then use C = Q/V
Capacitors
Electric Field above Plates
s
Q
E= =
e0 Ae0
is  to plates 
Q = EAe0
going from positive to negative plate
Calculating
Capacitance in
General
DV = V V = - E  ds  0
f
f

i
In order that
i
E  d s  0 choose path from
+ plate to - plate
D V = - V ( PD across plates )
Thus V =

-
Eds
(choose path
+
e0
EA
C = -

+
Eds
|| to electric field
)
for Parallel Plates Capacitor
Q EAe0 EA e0 Ae0
=
=
C= =
V
Ed
d
Eds

-
+
for Cylindrical Capacitor
Q 2pe0 L
=
C =
b 
V

ln 
a 
•a = radius of inner cylinder
•b = radius of outer cylinder
•L = length of cylinder
Combination of Capacitors
Combinations of Capacitors in
Parallel
equilibrium
Parallel
same electric potential felt by
each element
Series
electric potential felt by the
combination is the sum of the
potentials across each element
Picture
Q
Q
1
V =Calculation
= 2 of Effective
C1 Capacitance
C2
Total charge = Q = Q1 + Q2
= VC1 + VC2 = VCeq
\ Ceq = C1 + C2
In general
Ceq =

Ci
Combination of Capacitors
Series
Net charge zero
Net charge zero
Picture
Why are the charges on the plates of
equal magnitude ?
If net charge inside these
Gaussian surfaces is not zero
Field lines pass through the
surfaces
and cause charge to flow
Then we do have not
equilibrium
Calculation of Effective
Capacitance I
Q
Q
V total
= V1 + Vof
=
+
Calculation
2 Effective
C1
C2
Capacitance II
 1
1 
1
= Q 
+
 = Q
C2 
Ceq
C1
In general
1
1
=
Ceq
Ci

i
Is this parallel or series?
=
Question I
Is this parallel or series?
+
-
Question II
+
-
Work done in charging
capacitor
+ - in Charging
WorkI Done
q
Capacitor
+
-
q
Calculation
V q  =
C
if dq of charge is then transfered the work done is
dW = V q dq
Thus total work done on charging is

Q
1
W = V q dq =
C
0

Q
Q2 1
qdq =
= CV 2
2C 2
0
This work is stored as P. E.
Energy Density
U
EnergyDensity=
Volume
U
for parallel plate capacitor=
Ad
2
2
CV
1 V  1
=
= e 0   = e 0 E 2
2 Ad 2 d  2
Dielectrics
Picture
Picture
Picture
Polarization
Polarization
Induced Electric Field
Dielectric Constant
Charge Q stays the same, Total electric Field is less,
thus P. D. V effective across plates is less
Q
Q
\C =  C =
V
V effective
C =  C
Dielectric Constant   1. 00
Permitivity
C=
e0 A
d
A
\ C =  C = e 0
d
A
 C  = e
d
where, PERMITTIVITY of the dielectric e = e 0
Permitivity in Dielectrics
For conductors (not dielectrics )
 =e =
For regions containing dielectrics
all electrostatic equations containing
e 0 are replaced by e
e . g . Gauss ' Law
F=

E  dA =
surface
Q
e
Dielectric Strength
The 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