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Capacitanc
e
and
Dielectrics
AP Physics C
Montwood High School
R. Casao
Definitions
Voltage—potential difference between two points in space
(or a circuit).
Capacitor—device to store energy as potential energy in
an E field.
Capacitance—the charge on the plates of a capacitor
divided by the potential difference of the plates: C = q/V
Farad—unit of capacitance, 1F = 1 C/V. This is a very
large unit of capacitance, in practice we use F (10-6) or
pF (10-12).
Electric circuit—a path through which charge can flow.
Battery—device maintaining a potential difference V
between its terminals by means of an internal
electrochemical reaction.
Terminals—points at which charge can enter or leave a
battery.
Definition of Capacitance
A capacitor consists of
two uncharged
conductors connected
across the terminals of a
voltage source (battery).
The battery will move
charge from one
conductor to another
until the voltage
produced by the charge
buildup is equal to the
battery voltage.
Definition of Capacitance
The charge on one conductor
will be charged to +Q and the
other conductor will be
charged to -Q.
An electric field E is
established between the
charged conductors.
A capacitor is a device whose
purpose is to store electrical
energy which can then be
released in a controlled
manner during a short period
of time.
When the switch is
closed, the battery
establishes an electric
field in the wire that
causes electrons to move
from the left plate into
the wire and into the
right plate from the wire.
As a result, a separation
of charge exists on the
plates, which represents
an increase in electric
potential energy of the
system of the circuit.
This energy in the
system has been
transformed from
chemical energy in the
battery.
Definition of Capacitance
Capacitors are used when a sudden release of energy
is needed.
Capacitance is the ratio of the charge Q on one
conductor of the capacitor to the potential difference
between the conductors, Vab.
Q
C
V
Capacitance is always a positive quantity.
The unit for capacitance is the Farad (F). 1 F = 1 C/V;
1 F = 1 x 10-6 F.
Capacitance of a Lone Spherical
Conductor
The capacitance of a device
depends on the geometric
arrangement of the
conductors.
For a spherical conductor
of radius R and charge +q
(the conductor with the –q
charge can be considered
as a hollow conducting
sphere of infinite radius):
– The voltage of the
sphere is
k Q
V
R
Capacitance of a Spherical
Conductor
The capacitance is given by:
Q
Q
R
C


 4  π  εo  R
V k Q k
R
eo is the permittivity of the insulating
material (called the dielectric) between the
plates.
ε o  8.854 x 10
12
C
2
Nm
2
Parallel Plate Capacitor
The charge on each plate is
distributed uniformly over
an area A;
Q
σ
A
Use Gauss's Law to
calculate the electric field E
between the plates of the
charged capacitor:
σ
Q
E

εo εo  A
A
+Q
d
-Q
A
Parallel Plate Capacitor
The potential difference between the plates is:
Vab
Qd

εo  A
Substitute into the capacitance equation to
determine the capacitance for a parallel plate
capacitor:
εo  A
Q
Q
C


Qd
V
d
εo  A
Parallel Plate Capacitor
The capacitance of a parallel plate capacitor is
directly proportional to the area of the plates and
inversely proportional to the distance between the
plates.
When air or a vacuum is the dielectric material
between the plates:
εo  A
C
d
When another material is present as the dielectric
material between the plates, the equation includes
a dielectric constant k:
k  εo  A
C
d
Cylindrical Capacitors
A cylindrical capacitor
consists of a cylindrical
conductor of radius a
and length l
surrounded by a
coaxial cylindrical shell
of radius b.
The electric field E is
perpendicular to the
axis of the cylinders
and is confined to the
region between the
cylinders.
Cylindrical Capacitors
Determine the voltage
between the two
cylinders:
Vb  Va  

b
a

E  dr
The electric field for a
cylinder is given by:
2k  λ
E
r
Cylindrical Capacitors
Vb  Va  

b2  k
λ
r
a
Vb  Va  2  k  λ 

b1
a
r

 dr
b1
a
r
 dr
 dr  lnr   ln b  ln a
b
a
b
ln b  ln a  ln
a
b
Vb  Va  2  k  λ  ln
a
Cylindrical Capacitors
The voltage V is
negative based on the
direction of the
integration.
Capacitance is a
positive quantity, so
use the magnitude of
the voltage. You can
also reverse the
direction of the
integration to Va – Vb.
Substitute into the
capacitance equation:
Q
Q
C

V 2  k  λ  ln b
a
Q
C
2kQ
b
 ln
L
a
L
C
b
2  k  ln
a
Cylindrical Capacitors
An example of this type of capacitor is a
coaxial cable.
The coaxial cable consists of two cylindrical
conductors of radii a and b separated by an
insulator. The cable carries currents in
opposite directions in the inner and outer
conductors.
The geometry is useful for shielding the
electrical signal from external influences.
Cylindrical Capacitor w/ Gauss’s Law
L
b
Qenclosed
 E   E dA 
eo
a
Q
E 2rL 
eo
+Q on center conducting cylinder
Q
E 
2e o rL
-Q on outer conducting cylinder
b
 Q 
Q
dr
Q
b
Vba   

ln  
  dr   

2eo rL 
2eo L a r
2eo L  a 
a
b
Cylindrical Capacitor w/ Gauss’s Law
L
b
a
Q  CVab
L
Q
C

Vab
2eo L

Q
b
b
ln   ln  
2eo L  a 
a
Q
Spherical Capacitors
The spheres have uniform
charge density and the electric
field is uniformly distributed
between the inner sphere of
radius a and the outer sphere of
radius b.
The electric field E between the
two spheres is given by:
 k Q
E 2
r
Spherical Capacitors
Determine the voltage between
the two spheres:
Vb  Va  

b
a

E  dr
k Q
 dr
a r2
b 1
Vb  Va  k  Q   2  dr
a r
Vb  Va   
b
b
1
1
1 1 1 1

dr





a r 2
r a
b
a
b a
b
b 
 1 1 
 a
Vb  Va  k  Q  
   k  Q  


 b a
a b a b
ba
Vb  Va  k  Q  

a

b


Spherical Capacitors
The voltage V is negative
based on the direction of
the integration.
Capacitance is a positive
quantity, so use the
magnitude of the
voltage. You can also
reverse the direction of
the integration to
Va – Vb.
Substitute into the
capacitance equation:
Q
Q
C

V k  Q  b  a 
a b
a b
C
k  b  a 
What is a lightning discharge ?
•Friction forces between the air
molecules within a cloud result in
positively charged molecules moving
to the lower surface, and the negative
charges moving to the upper surface .
•The lower surface induces a high
concentration of negative charges in
the earth beneath.
•The resultant electric field is very
strong, containing large amounts of
charge and energy.
•If the electric field is greater than than
breakdown strength of air, a lightning
discharge occurs, in which air
molecules are ripped apart, forming a
conducting path between the cloud
and the earth.