Download Chapter 24 - Capacitance

Chapter 24 - Capacitance
I.
Introduction: Consider two conductors connected to the terminals of a battery. The battery will
supply an equal amount of charge to each of the conductors, but the charge on each of the
conductors will have opposite signs. The equation that gives the potential difference in terms of the

b
electric field Va  Vb  E  d l  can be used to derive expressions that relate the potential difference
a


between two conductors to the charge on the conductors.

A.
Two conducting parallel plates separated by a distance d with charges +Q and -Q. The potential
difference between the plates (from one plate to the other) is
area, A
+Q

Qd
Va  Vb  V  Ed   d 

A
o
 o
B.
d
-Q
Two conducting concentric spheres of radii a and b. The potential difference between the spheres is
Va  Vb  V 
Q 1 1
  
4 o  a b 
b
+Q
a
-Q
C.
Coaxial cable (two concentric conducting cylinders) of length L. The inside conductor has a radius a
with charge + and the inside surface of the outside conductor is b with charge -.
Va  Vb  V 

b
Q
b
ln 
ln
2 o a 2 oL a
b
+Q
a
-Q
24-1
D.
Note that in all cases that the charge Q is proportional to the potential difference or voltage, V, and
that the proportionality constant only depends on constants and the geometry (arrangement and
size) of the charged conductors, that is, Q  V. Because of this, a constant C, can be introduced such
that Q = CV.
E.
This constant C is defined as the capacitance of the system and depends on the geometry of the
system. Electrically, the capacitance, C, represents
F.
For units: C  
Q  Coulomb
V 
volt
 farad F  . Common values of capacitance vary from microfarads,
F (10-6 farads) to picofarads, pF (10-12 farads).
II.
What are the capacitances for the above arrangements of conductors?
A.
Two parallel plates:
Example: There are electrical devices that are designed to store energy in this fashion. These devices
are referred to a "capacitors." To get an idea of the magnitude of the unit Farad, find how large a
parallel plate capacitor must be in order to have a capacitance of one Farad (1.0 F). Let the distance
between the plates to be 0.1 mm – approximately the thickness of a sheet of paper.
B.
Concentric spheres of radii a and b:
C.
Coaxial cable – In this case, the coaxial cable can have different lengths. Because of this we find the
capacitance per unit length:
24-2
III. Capacitors in Electrical Circuits
A.
Real capacitors – some examples
B.
The circuit diagram of a capacitor
C.
You can "charge" a capacitor by connecting the capacitor to a battery (power supply). Remember
that in the electrostatic situation the wires (conductors) are equipotentials.
D.
Combinations of Capacitors
1.
Capacitors in Parallel: Find one capacitor, Ceq , that is equivalent to the capacitors in parallel
(does the same job as the capacitors in
+
parallel).
"top to top, bottom to bottom"
"left to left, right to right"
voltage
V
"The voltage is the same across all capacitors in
parallel."
24-3
_
C1
C2
C3
2.
Capacitors in Series: Find the equivalent capacitance, Ceq.
+
"one after another" - similar to a train engine pulling its cars
C1
"The charge is the same for all capacitors connected in series.
voltage
V
C2
C3
_
Special case: Two capacitors in series:
3.
Parallel and series combinations:
Example 1: Find the charge on each capacitor and the voltage across each capacitor.
2 F
3 F
4 F
12 v
+
_
24-4
Example 2: Find the equivalent capacitance between points A and B.
4 F
6 F
3 F
8 F
A
B
2 F
Example 3: Find the equivalent capacitance
between points A and B.
3 F
6 F
A
2 F
B
24-5
8 F
4 F
Example 4: A 3 F and a 6 F capacitor are connected in parallel and are charged by a 12 volt
battery, as shown. After the capacitors are charged, the battery is then disconnected from the
circuit. The capacitors are then disconnected from each other and reconnected after the 6 F
capacitor is inverted. Find the charge on each capacitor and the voltage across each.
+
A
B
12 v
3 F
C
D
6 F
A
B
D
C
_
Example 5: Place the two capacitors in example 4 in series and connect the 12 v power supply.
After the capacitors are charged, disconnect the capacitors and reconnect them so that the A
and C leads are together and the B and D leads are together. Find the charge on each capacitor
and the voltage across each.
+
A
B
3 F
C
D
6 F
12 v
_
24-6
IV. Energy stored in the capacitor.
A.
When a capacitor is being "charged" by a battery (or power supply), work is done by the battery to
move charge from one plate of the capacitor to the other plate. As the capacitor is being charged, we
can say that the capacitor is storing energy (What kind of energy?). Find the stored energy.
Consider a capacitor being charged by a battery. After a time t elapses, the voltage across the
capacitor is V and an amount of charge q has accumulated (so far) on the plates of the capacitor. To
move an additional amount of charge dq from one plate to the other, the battery must do an amount
of work dW, where dW = (dq)V. (Remember from before that Wab  Ua  Ub  qVa  Vb  , or W = qV
is the work done moving a charge q through a voltage V.)
dq
+
+q
V
-q
-
V
at time t, before
being fully charged
B.
Where is the energy stored? (What was different after the capacitor was charged?)
C.
What is the density of the energy stored in the electric field, or what is the energy density, uE , in
Joules/m3 ?
24-7
V.
How will dielectrics (insulators) affect a capacitor?
A.
mechanically - simpler to construct capacitor
B.
dielectrics change the capacitance
1.
C.
Polar and nonpolar atoms and molecules in electric fields
a.
polar
b.
nonpolar
Compare parallel plate capacitors - one without a dielectric and one with a dielectric. Assume that
the plates of the capacitors have the same amount of charge. (The diagram is not to scale. The area
of the plates is very large and the distance between the plates is small.) Compare charges Q, electric
fields E, voltages V, and capacitances C.
without dielectric
+Qo
with dielectric
+ + + + + + + + + + + +o
+Qo
+ + + + + + + + + + + +o
-Qi
_
_
_
_
_
_
-i
d
-Qo
_
_
_
_
_
_
_
_
_
_
_
-o
+
+Qi
-Qo
24-8
_
+
_
_
+
_
_
+
_
_
+
_
_
+
_
_
-o
+i
Without Dielectric
With Dielectric
Relative Values
Electric Field
Potential
Difference
Capacitance
1.
To relate the effectiveness of a dielectric, we define a constant called the dielectric constant, 
(kappa) as the ratio of the capacitance with the dielectric to the capacitance without the
dielectric:
C
,
or
C = Co

Co
Some values -- vacuum:  = 1, glass:  = 5 to 10, mica:  = 3 to 6, strontium titinate =233
Related quantity: dielectric strength - for air = 3 x 106 V/m, for strontium titinate = 14 x 106 V/m
2.
For a parallel plate capacitor:
3.
Note how the capacitance, voltage, and electric field are related to each other in a capacitor with
a dielectric and without a dielectric:
4.
How is the charge on the conducting plates of the capacitor, Qo , related to the induced charge
on the surfaces of the dielectric, Qi ?
24-9
D.
Examples:
For examples 1 and 2 below, assume that you have a parallel capacitor whose plates have an area of
10 cm2 and are separated by a distance of 0.05 mm. In the first case nothing is between the plates
and in the second case a dielectric with dielectric constant  = 5 completely fills the space between
the plates of the capacitor.
1.
Suppose that the capacitor is first connected to a battery, receives a charge of 2.00 x 10-9 C. and
is then disconnected from the battery maintaining the charge. Find the following:
with nothing between the plates
the capacitance
of the capacitor
the voltage
across the
capacitor
the electric
field between
the plates
the energy
stored in the
capacitor
the energy
density
between the
plates
24-10
with the dielectric between the plates
2.
Suppose the capacitor is connected to a 6.00 volt battery, and the battery remains connected.
Find the following:
with nothing between the plates
the capacitance
of the capacitor
the charge on
the plates of
the capacitor
the electric
field between
the plates
the energy
stored in the
capacitor
the energy
density
between the
plates
24-11
with the dielectric between the plates
3.
4.
A parallel plate capacitor is constructed so that two
dielectrics fill the space between the plates. The area
of the plates is A, the dielectric constants are and
 and the thickness of the dielectrics are d1 and d2.
Find the capacitance of this capacitor.
area A



Find the capacitance of the capacitor shown. The total area of
the plate is A. The area covered by each of the dielectrics A1 and
A2.
A1
d1
d2
A2
1
5.
Find the capacitance of the capacitor shown.
A/2
d
A/2
2
1
3
24-12
2
d/2
d/2