Download Lecture #12, October 21

Lecture #3-5
Capacitors
Today we shall continue our discussion of electrostatics and, in particular, the
concept of electrostatic potential energy and electric potential.
The main example which we have been considering so far was the example of the
charged point-like object inside of the uniform electric field of the parallel-plate
capacitor. In this situation there is electrostatic force acting on the object from the field.
So, if one releases this charge, it will start moving and the electric field of the capacitor
will perform work on the charge transferring potential energy of electric field into kinetic
energy of the moving charge. So, the capacitor is capable of storing potential energy PE,
which can later be used for accelerating of a charge. This potential energy depends on
configuration (or arrangement) of the system which means that it depends on the design
of the capacitor.
We have also introduced the concept of electric potential. The change of electric
potential is defined as the change of electric potential energy per unit positive charge. It is
similar to the concept of electric field, where the field itself does not depend on the
testing charge, the electric potential depends on configuration of the system but it does
not depend on the absolute value of the testing charge. For instance, we have proved the
relationship between the electric field (which only depends on the charge density of
capacitor’s plates) of the parallel-plate capacitor and the potential difference between its
plates. If we have a testing charge q moving in the direction of the electric field along the
x-axis perpendicular to the pates of the capacitor then
W
qEx

  Ex,
q
q
V
E
x
V  
(3.5.1)
As we said, this means that electric field is the rate of change of electric potential. We
know that the constant value of the electric field inside of the plane capacitor is
E

,
0
(3.5.2)
If the distance between the plates of the parallel-plate capacitor is d, which is a fixed
value for any given capacitor then we can calculate the potential difference between the
plates combining equations 3.5.1 and 3.5.2 as
V

 E 
,
d
0
d
Qd
V  Ed 

0
A 0
(3.5.3)
So, we can see that potential difference between the plates of the parallel-plate capacitor
is only dependent on its characteristics, such as the distance between the plates, the area
of each plate and charge placed on the pates. For a given capacitor the values of d and A
are fixed. So, if you provide a certain potential difference across the plates of the
capacitor (for instance by connecting it to the battery) then the capacitor will be charged.
It will hold the charge of absolute value Q (positive on one plate and negative on the
other plate), which is
Q 
A 0
V
d
(3.5.4)
Or if the given value of electric charge is placed on the plates of the parallel-plate
capacitor, then there will be a certain value of potential difference across the plates
defined by the equation 3.5.3.
A capacitor is the electric device which can store electric energy or electric charge.
So, the better the capacitor is the more charge you can store in it. But for any given
capacitor and given potential difference this amount of charge is limited by the equation
3.5.4. Capacitor gets its name because of the capacity to store energy and charge. One
can introduce a special physical quantity, which shows this capacity for a given potential
difference, which is
C
Q
,
V
(3.5.5)
The quantity C is called capacitance of the capacitor and it is measured in special units
called Farad (F), 1F=1C/1V. Even though the example, which we just have considered,
was the example of the parallel-plate capacitor, but the fact that the charge is proportional
to the potential difference between the plates (equation 3.5.4) is general. Therefore, one
can introduce capacitance for any type of the capacitor by means of its definition 3.5.5.
In the case of the parallel-plate capacitor we just showed that
C
0 A
,
d
(3.5.6)
It is obvious that the larger area of the plates allows more space for the charge, so C gets
bigger. Increase of the distance between the plates causes the decrease of the potential
change rate and as a result the decrease of the electric field, so the smaller charge is
needed to provide this smaller field.
More complicated shapes of the capacitors, such as cylindrical capacitor or
spherical capacitor, are often used in practice. In the case of those capacitors C depends
on A and d in more complicated way than the one described by equation 3.5.6, but the
general conclusion that it depends on area of the plates and separation distance between
the plates is still true.
As we just have seen the capacitance depends on the shape and the size of the
capacitor, however, we have only limited our attention by the case when the space
between the plates is empty. In reality it will always be filled with some substance.
The purpose of the capacitor is to hold as much charge as possible. To achieve
this goal we can insert dielectric material between the plates of the capacitor. Every time
when this dielectric material is inside of the external electric field of the capacitor it will
be polarized. This means that electric dipoles (molecules of dielectric) will be oriented in
such a way that their negative sides will be directed towards the positive plate of the
capacitor and their positives sides will be directed towards negative plate of the capacitor.
Thus the total electric field of all the diploes is oriented in such a way that it is directed
opposite to the external field of the capacitor which has caused that orientation. So, the
total field inside becomes less compared to what it was before the dielectric was inserted.
One can introduce a special quantity known as dielectric constant  (or electric
permittivity of dielectric), which shows by how much electric field in dielectric is smaller
compared to the electric field in vacuum, so that

E0
,
E
(3.5.7)
where E stands for the electric field in dielectric and E 0 for the electric field in vacuum.
Value of dielectric constant depends on material. Those values are listed in table 19.1 on
page 584 in the book. This phenomenon can be explained, by appearance of additional
induced charge inside of the dielectric which has the opposite sign to the original charge,
so we can also say that the effective charge enclosed between the plates of the capacitor
is now qeff 
q

.
As a result of having dielectric inside of the capacitor the effective electric field
reduces causing the effective reduce of the potential difference by the same factor of  ,
so foe the parallel-plate capacitor we have
C
 A
Q
 0
V / 
d
(3.5.8)
This means that capacitance gets higher if some dielectric is inserted inside of the
capacitor. However, this does not set the limit on how large the charge stored can be. If
you continue to increase potential difference between the plates of the capacitor, the
molecules of dielectric can be broken apart by strong enough external electric field. Then
the dielectric becomes a conductor and electric charge flows from one plate to another
plate of the capacitor. So, we say that dielectric break down has occurred.
Let us now find the energy stored inside of the capacitor. This energy storage in the
capacitor takes place in the process of its charging, when the charge is transferred from
one of the capacitor’s plates to another plate until the potential difference between the
plates reaches a given value (for instance one provided by the battery). During this
process both charges on the plates and potential difference between the plates are
changing. The charge built up on the plates is related to the current value of the potential
difference. Since C is constant for a given capacitor then charge increases as a linear
function of increasing potential difference. For every small charge transferred from one
plate to another plate the potential energy of the system increases by a small amount. To
find the total energy stored in the capacitor, one has to perform a summation of all these
small energies which means to find the area under V , q  graph, which is
1
1
1 Q2
U  QV  CV 2 
.
2
2
2 C
(3.5.9)