Lect. (4)
Capacitance and Dielectrics
1. Capacitance
 Definition
 How to calculate the capacitance
2. Capacitor
3. Energy stored in a capacitor
4. Capacitor with dielectrics
5. Dielectrics explained in an atomic
view
1
Capacitance: the definition.

The capacitance, C, is defined as the ratio
of the amount of the charge Q on the
conductor to the potential increase ΔV of
the conductor because of the charge:
Q
C
V


Q
C=
V
This ratio is an indicator of the capability
that the object can hold charges. It is a
constant once the object is given,
regardless there is charge on the object or
not. This is like the capacitance of a mug
which does not depend on there being
water in it or not.
The SI unit of capacitance is the farad (F)
1F 
1C
1V
2
More About Capacitance




Capacitance will always be a positive quantity
The capacitance of a given capacitor is constant
The capacitance is a measure of the capacitor’s
ability to store charge
The farad is an extremely large unit, typically you
will see
microfarads (mF=10-6F),
nanofarads (nF=10-9F), and
picofarads (pF=10-12F)
3

Capacitors are devices that store electric charge



Any conductors can store electric charge, but
Capacitors are specially designed devices to store a lot of
charge
Examples of where
capacitors are used
include:




radio receivers
filters in power supplies
to eliminate sparking in
automobile ignition systems
energy-storing devices in
electronic flashes
4
How to Make a Capacitor?

Requirements:



Hold charges
The potential increase
does not appear outside of
the device, hence no
influence on other devices.


E 

2 0
E
For example, a `parallel plate’
capacitor, has capacitance
Q
A A
A
C


 0 ,
V Ed  d
d
0
A is the surface area
E0
E

0



2 0
2 0
E0
d
V  Ed
5
A Real Parallel Plate Capacitor
charged up with a battary

Each plate is connected to a terminal of the battery





If the capacitor is initially uncharged, the battery
establishes an electric field in the connecting wires
This field applies a force on electrons in the wire just
outside of the plates
The force causes the electrons to move onto the
negative plate
This continues until equilibrium is achieved





The battery is a source of potential difference
The plate, the wire and the terminal are all at the same
potential
At this point, there is no field present in the wire and
the movement of the electrons ceases
The plate is now negatively charged
A similar process occurs at the other plate, electrons
moving away from the plate and leaving it positively
charged
In its final configuration, the potential difference across
the capacitor plates is the same as that between the
terminals of the battery
V
6
Energy stored in a charged capacitor





Consider the circuit to be a
system
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
7
How Much Energy Stored in a
Capacitor?
To study this problem, recall that the work the field force
does equals the electric potential energy loss:
WE  U  QV
This also means that when the battery moves a charge
dq to charge the capacitor, the work the battery does
equals to the buildup of the electric potential energy:
q

E
-q
V
dq
WB  U
When the charge buildup is q, move a dq, the work is
q
dWB  Vdq  dq
C
We now have the answer to the final charge Q:
Q
Q
q
Q2
WB   dWB   dq 
 U
C
2C
0
0
8
Energy in a Capacitor, the formula

When a capacitor has charge stored in it, it also
stores electric potential energy that is
Q2 1
UE 
 C (V ) 2
2C 2



This applies to a capacitor of any geometry
The energy stored increases as the charge
increases and as the potential difference (voltage)
increases
In practice, there is a maximum voltage before
discharge occurs between the plates
9
Energy in a Capacitor, final
discussion



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
It can also be expressed in terms of the
energy density (energy per unit volume)
uE = ½ o E2
10
11
Circuit Symbols




A circuit diagram is a
simplified representation of
an actual circuit
Circuit symbols are used to
represent the various
elements
Lines are used to represent
wires
The battery’s positive
terminal is indicated by the
longer line
12
Capacitors are in Series:
When capacitors are in series,
the charge is the same on each capacitor.
Vt  V1  V2  V3
Qt Q1 Q2 Q3
 

Ct C1 C2 C3
Qt  Q1  Q2  Q3
1
1
1
1
 

Ct C1 C2 C3
Q
Q  CV  V 
C
Capacitors are in Parallel
When capacitors are in parallel ,
the total charge is the sum of that on each capacitor.
Qt  Q1  Q2  Q3
CtVt  C1V1  C2V2  C3V3
Vt V1  V2  V3
Ct  C1  C2  C3
Q  CV
Equivalent Capacitance,
Example



The 1.0-mF and 3.0-mF capacitors are in parallel as are the 6.0mF and 2.0-mF capacitors
These parallel combinations are in series with the capacitors
next to them
The series combinations are in parallel and the final equivalent
capacitance can be found
15
Dielectrics



A dielectric is a nonconducting material, an
electrical insulator, such as rubber, glass, or waxed
paper.
When a dielectric is inserted between the plates of a
capacitor, the capacitance increases by a dimensionless
factor , which is called the dielectric constant of the
material.
The dielectric constant is the ratio of the field without
the dielectric (Eo) to the net field (E) with the dielectric:
 = Eo /E
16
• Dielectrics
The voltages with and without the dielectric are related by
the factor  as follows:
• The dielectric constant  varies from one
material to another.
17
if E0 is the electric field without the dielectric, the field in the
presence of a dielectric is
18