Ch 26 – Capacitance and Dielectrics
The capacitor is the first
major circuit component
we’ll study…
Ch 26.1 – Capacitance
All conductors display some degree of capacitance.
Usually, the term “capacitor” refers to two separate pieces of
metal acting together.
Each piece of metal is referred to as a plate.
Ch 26.1 – Capacitance
Electric field lines generated by a real parallel
plate capacitor. Notice, the field is essentially
uniform between the plates.
Charging a capacitor:
Fig 26-4b, p.800
Charging a capacitor:
Chemical potential energy maintains
charge separation in the battery 
the battery generates an E-field.
Fig 26-4b, p.800
Charging a capacitor:
Chemical potential energy maintains
charge separation in the battery 
the battery generates an E-field.
The battery’s E-field accelerates
charges in the wires.
Electrons flow off the orange plate
and toward the blue plate.
Fig 26-4b, p.800
Charging a capacitor:
Chemical potential energy maintains
charge separation in the battery 
the battery generates an E-field.
The battery’s E-field accelerates
charges in the wires.
Electrons flow off the orange plate
and toward the blue plate.
Now, there is a charge imbalance
across the capacitor’s plates. So…
what must exist in between the plates
of the capacitor?
… an electric field.
Fig 26-4b, p.800
Charging a capacitor:
Chemical potential energy maintains
charge separation in the battery 
the battery generates an E-field.
The battery’s E-field accelerates
charges in the wires.
Electrons flow off the orange plate
and toward the blue plate.
Now, there is a charge imbalance
across the capacitor’s plates. So…
what must exist in between the plates
of the capacitor?
… an electric field.
Charge continues to flow until the Efield in the capacitor is strong enough
to cancel the E-field in the battery.
Fig 26-4b, p.800
Ch 26.1 – Capacitance
A capacitor stores electrical potential energy by virtue of
separating charges.
The stored energy is “in” the capacitor’s E-field.
Based on geometry and materials, some capacitors are better
at storing energy than others.
Ch 26.1 – Capacitance
Called a capacitor because the device has some “capacity” to store
electrical charge, given a particular applied potential difference
(voltage).
The ability of a capacitor to store charge given a certain applied voltage is
called its “capacitance.”
Ch 26.1 – Capacitance – factors affecting capacitance
• Size of the capacitor (A, d)
• Geometric arrangement
– Plates
– Cylinders
• Material between conductors
– Air
– Paper
– Wax
Ch 26.1 – Capacitance
The “capacitance,” C, of a capacitor is the ratio of
the charge on either conductor to the potential
difference between the conductors:
C 
Q
V
Ch 26.1 – Capacitance
Units: 1 F = 1 C/V
The “farad”
C 
Q
V
magnitude of charge
on one plate
voltage across the
capacitor
EG – Definition of Capacitance
(a) How much charge is on each plate of a 4.00μF capacitor
when it is connected to a 12.0-V battery?
(b) If this same capacitor is connected to a 1.50-V battery,
what charge is stored?
Ch 26.2 – Parallel Plate Capacitors
- one plate has +Q, the other -Q
- for each plate σ = Q/A
Ch 26.2 – Parallel Plate Capacitors
- one plate has +Q, the other -Q
- for each plate σ = Q/A
- Gauss’s Law 
E 

0

Q
0A
the E-field just outside one of the plates
E 

0

Q
0A
Uniform
E-field
Ch 26.2 – Parallel Plate Capacitors
E 

0

Q
0A
Working backwards from the uniform E-field, the
magnitude of the voltage between the plates is
V  

b
a
V 
Qd
0A


E  d s  Ed
Ch 26.2 – Parallel Plate Capacitors
E 

0

Q
0A
Working backwards from the uniform E-field, the
magnitude of the voltage between the plates is
V  

b


E  d s  Ed
a
V 
Capacitance of parallel
plate capacitor
Qd
0A
But: C

Q
V

C 
Q
Qd
0A

C 
0A
d
Ch 26.2 – Parallel Plate Capacitors
C 
 0 A Capacitance of parallel
d
plate capacitor
Capacitance of a parallel-plate capacitor is
directly proportional to the area of the plates and
inversely proportional to the distance between
the plates.
Think about:


V    E  ds
EG 26.1 – Cylindrical Capacitor
A solid, cylindrical conductor of
radius a and charge Q is coaxial
with a cylindrical shell of
negligible thickness, radius
b>a, and charge –Q.
Find the capacitance of this
capacitor if its length is l.
EG 26.2 – Spherical Capacitor
A spherical capacitor consists of a
spherical conducting shell of
radius b and charge –Q
concentric with a smaller
conducting sphere of radius a
and charge Q.
Find the capacitance of this device.
Ch 26.3 – Combinations of Capacitors
Capacitors are intentionally used in circuits to alter the rates of change of
voltages.
Capacitors can be hooked up in two ways:
- networked in parallel
- networked in series
Ch 26.3 – Combinations of Capacitors – parallel network
•
•
•
•
Connecting wires are conductors in
electrostatic equilibrium  Ein =0
Left plates at same electric potential
as the positive terminal of the
battery.
Right plates at same electric
potential as the negative terminal of
the battery.
Therefore, all capacitors in a
parallel network experience the
same potential difference, in this
case, ΔV.
ΔV1=ΔV2=ΔV
C2
Q2
C1

Q1
ΔV

Ch 26.3 – Combinations of Capacitors – parallel network
ΔV1=ΔV2=ΔV
The individual voltages across parallel
capacitors are equal, and they are equal to
the voltage applied across the network.
C2
Q2
C1

Q1
ΔV

Ch 26.3 – Combinations of Capacitors – parallel network
•
•
When battery is attached to circuit
 capacitors quickly reach
maximum charge, Q1 and Q2.
ΔV1=ΔV2=ΔV
C2
 total charge stored by the circuit
is Qtot = Q1 + Q2.
•
So, for a given applied voltage, this
network has some “capacity” to
store charge.
•
In other words, the network itself
can be thought of as a single
capacitor, even though it has many
components.
Q2
C1

Q1
ΔV

Ch 26.3 – Combinations of Capacitors – parallel network
•
•
Let’s replace the two-capacitor
network with a single equivalent
capacitor that has capacitance Ceq.
ΔV1=ΔV2=ΔV
C2
Based on the definition of
capacitance, Ceq = Qtot/ΔV.
•
 Qtot= CeqΔV
•
But, Qtot = Q1 + Q2, so
Q2
C1

Q1
 CeqΔV = C1ΔV1+ C2ΔV2
•
ΔV
In conclusion:
 Ceq= C1+ C2 (parallel network)

Ch 26.3 – Combinations of Capacitors – parallel network
•
Ceq
In general,
 Ceq= C1+ C2+… (parallel network)
The equivalent capacitance of a parallel network is:
Qtot


-the algebraic sum of the individual capacitances
-greater than any of the individual capacitances
composing the network
ΔV
Ch 26.3 – Combinations of Capacitors – series network
•
Left plate of capacitor 1 is at same
potential as positive terminal of the
battery.
•
Right plate of capacitor 2 is at same
potential as negative terminal of
battery.
•
“Middle leg” has no net charge.
C1
C2
Q1
Q2
ΔV
Ch 26.3 – Combinations of Capacitors – series network
•
When battery is connected,
electrons flow off the left plate of C1
and onto the right plate of C2.
•
Electrons accumulate on right plate
of C2, establishing an electric field.
•
E-field forces electrons off the left
plate of C2 and onto right plate of
C1.
•
All right plates end up with –Q, and
all left plates end up with +Q.
C1
C2
Q1
Q2
ΔV
Ch 26.3 – Combinations of Capacitors – series network
Q1 = Q2 = Q
•
Additionally, once the circuit
reaches electrostatic equilibrium,
the voltage across the network must
cancel the battery’s voltage.
•
ΔVtot = ΔV1 + ΔV2
C1
C2
Q1
Q2
{
•
In other words, the magnitude of
charge on all the plates is equal.
ΔV2
{
•
ΔV1
ΔV
Ch 26.3 – Combinations of Capacitors – series network
•
In other words, even though the
network has multiple components, it
can be modeled using a single
equivalent capacitor.
•
Lets build the same circuit using a
single equivalent capacitor.
C1
C2
Q1
Q2
{
This series network has some ability
to store charge given an applied
voltage, ie., it has some capacitance
ΔV2
{
•
ΔV1
ΔV
Ch 26.3 – Combinations of Capacitors – series network
C1
C2
Q1
Q2
{
From the definition of capacitance,
ΔVtot = Q/Ceq
ΔV2
{
Q1 = Q2 = Q
(previous result)
ΔVtot = ΔV1 + ΔV2 (previous result)
ΔV1
Substituting for Δ Vtot,
Q/Ceq = Q1/C1 + Q2/C2
Cancelling Q,
1/Ceq = 1/C1 + 1/C2 (series combination)
ΔV
Ch 26.3 – Combinations of Capacitors – series network
From the definition of capacitance,
ΔVtot = Q/Ceq
ΔV
{
Q1 = Q2 = Q
(previous result)
ΔVtot = ΔV1 + ΔV2 (previous result)
Ceq
Q
Substituting for Δ Vtot,
Q/Ceq = Q1/C1 + Q2/C2
Cancelling Q,
1/Ceq = 1/C1 + 1/C2 (series combination)
ΔV
Ch 26.3 – Combinations of Capacitors – series network
• The inverse of the equivalent
capacitance is the algebraic sum of the
inverses of the individual capacitances.
• The equivalent capacitance is always
less than any individual capacitances in
the network.
{
• In general,
• 1/Ceq = 1/C1 + 1/C2 +… (series combination)
ΔV
Ceq
Q
ΔV
EG 26.3 – Equivalent Capacitance
Find the equivalent capacitance between a and b for the combination of
capacitors shown. All capacitances are in microfarads.