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Chapter 23
Capacitance
Capacitance
• The capacitance, C, is a measure of the amount of
electric charge stored (or separated) for a given electric
potential
Q
C
V
• SI unit – Farad (F): 1 F = 1 C / V
Michael Faraday
1791 – 1867
• A 1 Farad capacitance is very large – µF or pF
capacitances are more common
Capacitor
• A capacitor is a device used in a variety of electric
circuits to store electric charge
• Capacitance of a capacitor is the ratio of the
magnitude of the charge on either conductor (plate)
to the magnitude of the potential difference between
the conductors (plates)
• This capacitance of a device depends on the
geometric arrangement of the conductors
Parallel-Plate Capacitor
• This capacitor consists of two parallel
plates (each of area A) separated by a
distance d each carrying equal and
opposite charges
• Each plate is connected to a terminal of
the battery
• The battery is a source of potential
difference
• If the capacitor is initially uncharged, the
battery establishes an electric field in the
connecting wires
Parallel-Plate Capacitor
• The field applies a force on electrons in
the wire just outside of the plates, causing
the electrons to move onto the negative
plate until equilibrium is achieved
• The plate, the wire and the terminal are all
at the same potential, and there is no field
present in the wire and the movement of
the electrons ceases
• At the other plate, electrons are moving
away from the plate and leaving it
positively charged
Parallel-Plate Capacitor
• The electric field due to one
plate
σ
E
2ε o
• The total electric field
between the plates is given
by
σ
σ
E2

2ε o ε o
• The field outside the plates
is zero
Parallel-Plate Capacitor
• In the final configuration, the
potential difference across the
capacitor plates is the same as that
between the terminals of the battery
• For a parallel-plate capacitor whose
plates are separated by air:
A
Q A A

C


V V Ed ( /  o ) d
A
C  o
d
Electric Field in a Parallel-Plate Capacitor
• The electric field between the plates is
uniform near the center and
nonuniform near the edges
• The field may be taken as constant
throughout the region between the
plates
Spherical Capacitor
b
 
V  Vb  Va    E  ds    Er dr
b
a
b
a
Q
1 1 
keQ
  ke 2 dr 
 keQ   
r
r a
b a 
a
a b
ab
 keQ
Q

C
ab
k e ( a  b)
V
b
Chapter 23
Problem 26
A parallel-plate capacitor with 1.1-mm plate spacing has ±2.3 µC on its
plates when charged to 150 V. What’s the plate area?
Electric Circuits
• A circuit is a collection of objects
usually containing a source of
electrical energy (such as a battery)
connected to elements that convert
electrical energy to other forms
• A circuit diagram – a simplified
representation of an actual circuit – is
used to show the path of the real
circuit
• Circuit symbols are used to represent
various elements (e.g., lines are used
to represent wires, battery’s positive
terminal is indicated by a longer line)
Capacitors in Parallel
• When capacitors are first connected in
parallel in the circuit, electrons are
transferred from the left plates through the
battery to the right plates, leaving the left
plates positively charged and the right
plates negatively charged
• The flow of charges ceases when the
voltage across the capacitors equals that of
the battery
• The capacitors reach their maximum charge
when the flow of charge ceases
Capacitors in Parallel
• The total charge is equal to the sum of the
charges on the capacitors: Qtotal = Q1 + Q2
• The potential differences across the
capacitors is the same and each is equal to
the voltage of the battery
• A circuit diagram for two
capacitors in parallel
Capacitors in Parallel
• The capacitors can be replaced with one capacitor with
a equivalent capacitance Ceq – the equivalent capacitor
must have exactly the same external effect on the
circuit as the original capacitors
Q  Q1  Q 2
Q1  C1 V
Q2  C 2 V
Q  (C1  C 2 )  V  C eq  V
C eq  C1  C 2
Capacitors in Parallel
• For more than two capacitors in parallel:
C eq  C 1  C 2  C 3  ...
• The equivalent capacitance of a parallel
combination of capacitors is greater
than any of the individual capacitors
Capacitors in Series
• When a battery is connected to the
circuit, electrons are transferred
from the left plate of C1 to the right
plate of C2 through the battery
• As this negative charge
accumulates on the right plate of C2,
an equivalent amount of negative
charge is removed from the left plate
of C2, leaving it with an excess
positive charge
• All of the right plates gain charges
of –Q and all the left plates have
charges of +Q
Capacitors in Series
• An equivalent capacitor can be
found that performs the same
function as the series combination
• The potential differences add up to
the battery voltage  V   V   V
Q
Q
V1 
 V2 
C1
C2
Q Q
Q

V 

C1 C 2 Ceq
1
2
1
1
1


Ceq C1 C 2
Capacitors in Series
• For more than two capacitors in series:
1
1
1
1



 ...
C eq C1 C 2 C3
• The equivalent capacitance is always
less than any individual capacitor in the
combination
Problem-Solving Strategy
• Be careful with the choice of units
• Combine capacitors:
• When two or more unequal capacitors are connected
in series, they carry the same charge, but the potential
differences across them are not the same
• The capacitances add as reciprocals and the
equivalent capacitance is always less than the
smallest individual capacitor
Problem-Solving Strategy
• Be careful with the choice of units
• Combine capacitors:
• When two or more capacitors are connected in parallel,
the potential differences across them are the same
• The charge on each capacitor is proportional to its
capacitance
• The capacitors add directly to give the equivalent
capacitance
Problem-Solving Strategy
• Redraw the circuit and continue
• Repeat the process until there is only one single
equivalent capacitor
• To find the charge on, or the potential difference
across, one of the capacitors, start with your final
equivalent capacitor and work back through the circuit
reductions
Equivalent Capacitance
Chapter 23
Problem 51
What’s the equivalent capacitance measured between A and B in the
figure?
Energy Stored in a Capacitor
• Before the switch is closed, the energy
is stored as chemical energy in the
battery
• When the switch is closed, the energy
is transformed from chemical to
electric potential energy
• The electric potential energy is related
to the separation of the positive and
negative charges on the plates
• A capacitor can be described as a
device that stores energy as well as
charge
Energy Stored in a Capacitor
• Assume the capacitor is being charged and, at some
point, has a charge q on it
• The work needed to transfer a charge from one plate to
the other:
q
dW  Vdq 
• The total work required:
W 
Q
0
C
dq
q
Q2
dq 
C
2C
• The work done in charging the capacitor appears as
electric potential energy U:
Q2 1
1
U
 QV  C(V )2
2C 2
2
Energy Stored in a Capacitor
• This applies to a capacitor of any geometry
• The energy stored increases as the charge increases
and as the potential difference increases
• The energy can be considered to be stored in the
electric field
• For a parallel-plate capacitor, the energy can be
expressed in terms of the field as U = ½ (εoAd)E2
Q2 1
1
U
 QV  C(V )2
2C 2
2
Capacitors with Dielectrics
• A dielectric is an insulating material (e.g., rubber,
plastic, etc.)
• When placed between the plates of a capacitor, it
increases the capacitance: C
• κ - dielectric constant
• The capacitance is
multiplied by the factor κ
when the dielectric
completely fills the region
between the plates
= κ Co = κ εo (A/d)
Capacitors with Dielectrics
• Tubular – metallic foil interlaced
with thin sheets of paraffinimpregnated paper rolled into a
cylinder
• Oil filled (for high-V capacitors)
– interwoven metallic plates are
immersed in silicon oil
• Electrolytic (to store large
amounts of charge at relatively
low voltages) – electrolyte is a
solution that conducts
electricity via motion of ions in
the solution
Dielectric Strength
• For any given plate separation, there is a maximum
electric field that can be produced in the dielectric
before it breaks down and begins to conduct
• This maximum electric field is called the dielectric
strength (breakdown field)
Chapter 23
Problem 56
A 470-pF capacitor consists of two 15-cm-radius circular plates,
insulated with polystyrene. Find (a) the thickness of the polystyrene
and (b) the capacitor’s working voltage.
An Atomic Description of Dielectrics
• The molecules that make up the dielectric are modeled
as dipoles; in the absence of an electric field they are
randomly oriented
• An external electric field produces a torque on the
molecules partially aligning them with the electric field
An Atomic Description of Dielectrics
• The presence of the positive (negative) charge on
the dielectric effectively induces some of the
negative (positive) charge on the metal
• This allows more charge on the plates for a given
applied voltage and the capacitance increases
Answers to Even Numbered Problems
Chapter 23:
Problem 22
14 C
Answers to Even Numbered Problems
Chapter 23:
Problem 32
(a) 6.0 μF
(b) 0.55 μF