Figure 4.1: Two thin plate conductors with fixed distance d, plate area A, and charges +Q on one plate and -Q on the other form a parallel-plate capacitor.
Methods of storing energy for the purpose of doing work and directly studying electric fields and their effect on matter was an important focus of research in the mid-1700's. Methods of storing electrical charges for long periods of time were developed, the most important being the creation of the Leydan jar (so-called because it was first created at the University of Leiden in 1746 by one of its professors, Pieter van Musschenbroek). Methods of improving the Leydan jar by simplifying its construction and increasing its capacity for storing charge soon resulted in devices consisting of just two flat metal sheets separated by air. One of the sheets can be charged, then disconnected from all electrically conducting material. The charged sheet induces an equal charge on the second sheet. In its simplest model, any two oppositely charged metal objects separated by a distance forms a "condenser" of charge or a capacitor, as we now say. In order to provide a "figure of merit" for how effective any type of capacitor is at holding charge, we define the capacitance through the relationship
| (4.1.1.1) |
The higher the capacitance, the more effective the capacitor is in the following sense: if we look at the relationship between C, Q, and V as defined, we note that the more charge we store per unit of V, the higher the value of C. Since V relates to the amount of work needed to produce the charge Q on the plates of the capacitor, having a high capacitance means we store more charge for less work done in charging. As we will show, capacitance also is a measure of how effective the capacitor is at storing energy, and hence how effective the capacitor can be at doing work after it is charged.
First, let's consider the parallel-plate capacitor as shown in figure 4.1.
The two conducting plates each have an area A and are separated by a distance d. By using a source of EMF or ElectroMotive Force (a device which moves charges from one point to another), we can drive electrons onto the bottom plate while removing them from the top plate. After the plates are charged, we can, if we wish, remove the EMF source and the plates, if isolated from any other conducting material will hold their charges for a considerable period of time; years in some cases.
We can evaluate the capacitance for the parallel-plate case by noting that the electric field between the plates is, according to Gauss's Law as applied in previous lectures, constant in magnitude and direction.
Hence, we know that E = Q/e0A. The voltage
between the plates increases in going from the negative
to the positive plate and has a magnitude V = Ed.
Therefore,
| (4.1.1.2) |
We note that we are more familiar with the concept of storing electrical energy in batteries. Hence, it makes some sense to distinguish between batteries and capacitors. The purpose of a battery is to use chemical (or other) processes to produce the EMF needed to move charges around. This is done by using the chemical or other process to maintain a constant potential between the poles (i.e. the positive and negative parts of the battery). This constant potential provides the EMF for moving charges from one conductor of a capacitor to another, for example. Hence, this is the principal distinguishing feature of a battery and a capacitor: a battery maintains a constant potential difference between it's poles whereas a capacitor can maintain a potential difference between its positive and negatively charged conductors, but there is no need for that potential difference to be constant - the potential difference depends on the magnitude of the charge on the conductors! Thus, while batteries store energy, capacitors store both energy and charge. The motion of charges can be quite rapid and readily controlled hence capacitors can be used for electrical energy storage that requires timing as part of its function. Batteries are generally not good for this purpose.
We can now talk about the energy stored in the electric field of or, equivalently, stored in the arrangement of electric charges within the capacitor. Again, we use the parallel-plate capacitor because it is reasonably simple. First, note that we will use schematic representations of circuit elements to save drawing time. For a source of EMF, or voltage that can move charges, we have the symbol in figure .
To calculate the energy stored, we hook the EMF source to a capacitor, e.g. by closing a switch that allows conducting lines to link the EMF source and a capacitor together.
As soon as the EMF source is able to push charge onto the capacitor plates, it does so by stealing electrons from the top plate of the capacitor and pushing them onto the bottom plate. As soon as the first infinitesimal bit of charge hits the capacitor, an electric field is created between the plates. Although the field is also infinitesimally small, it still produces a voltage (again, infinitesimally small). This voltage operates opposite to the direction that charge is flowing, i.e. the capacitor voltage tends to push positive charge in the counter-clockwise direction while the EMF source pushes it in the clockwise direction. Since the EMF source has a much higher voltage, it succeeds in pushing more charge onto the plates of the capacitor. As the charge continues to build though, the electric field strength, and hence the voltage across the plates of the capacitor, continues to build. When the capacitor voltage equals that of the EMF source, charge flow stops and the situation is static. At this point, the voltage across C is just V0, the same as that across the EMF source.
To find the amount of work done by the battery to bring this
situation about, let's translate the above description into
calc-speak, i.e say it mathematically. The first bit of charge
to arrives takes no work on the part of the EMF source assuming
that C is initially uncharged. This bit of charge on C causes
a potential V¢ to appear across C. The next
infinitesimal bit of charge, dq, coming through the EMF source
must have work done on it to push it against the potential from
C. This work is just dW = dq V¢. Now we can integrate
to see what the total work is to get V¢ up to
V0.
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| (4.1.3.4) |
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We originally started the discussion of capacitors by noting their importance as a means for storing charge (and hence electrical energy) that could be used to do work or make studies of the electric field. As such, there was a continual push to develop capacitors of larger charge-storing capability. There are practical limitations to storing charge in that the electric field between the parallel plates of a capacitor, and hence the electric potential, goes up as the charge goes up. At a certain point, the air gap between the plates breaks down, i.e. the nitrogen in the air is ionized, becomes conducting, and therefore provides a path that enables the plates to short out. One can also decrease the distance between the plates. Since the potential across the plates goes as Ed = Qd/e0A, reducing d decreases V for a given value of Q on the plates. This has practical limitations in keeping the distance between the plates uniform over the area. To see why keeping the plates apart is a problem, let's calculate the force between the plates. First, look at figure below showing a parallel-plate capacitor with a separation distance x between the plates. To pull the plates apart by an additional infinitesimal distance dx, we need to exert a force because the positive and negative charge plates attract one another.
To be formal about this, note that the potential energy
of the original capacitor is
| (4.1.4.6) |
| (4.1.4.7) |
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So, how do we get around the essential constraints on capacitance? First, note what happens if we take any single, charged parallel-plate capacitor and insert a slab of conducting material into the space between the plates, but do it such that the conducting material does not touch either plate.
The conducting material assumes the charge separation as
shown in figure 4.6. The electric
field above and below the conductor remains the same as
before the conducting material was introduced,
E = q/e0A because the electric field in the
vicinity of the charged plates is unaffected by the presence
of the conducting material. The electric field in the
conducting material is zero, so the potential from the
bottom to the top of the conducting material is the same.
Therefore, the potential difference between the bottom and
top plates are (note that the spaces between the conductor
and the plates are both (d - b)/2)
| (4.1.4.9) |
| (4.1.4.10) |
This still doesn't help, technologically speaking, with making better capacitances, because we still have to keep the conductor from touching the plates of the capacitor. To get around this, we need something that reduces the electric field magnitude in the volume of space between the plates without allowing for conduction between the plates. The solution is to fill the space between the plates with a dielectric material. Dielectrics are non-conductors that are formulated from molecules which either have permanent electric dipole moments or can be induced to have dipole moments in the presence of an external electric field.
These dielectric molecules align partially with any external
electric field. They generate a field of their own within
the material. This field opposes that of the external field
and therefore reduces the magnitude of E
within the material. While not quite as effective as a
conductor, this material does not conduct charge between the
plates of a conductor and still reduces the voltage as a function
of charge on the plates. The reduction factor is referred to
as the dielectric constant of
the material and is usually represented by the letter K. The
electric field produced by the charge on the plates of the
capacitor remains E0 = q/e0A in any region between the plates which is
not occupied by dielectric. In the region which is occupied by
dielectric material, the electric field is E¢ = E0/K
with K > 1, i.e. the electric field magnitude is reduced.
Since the potential difference between the plates goes as the
electric field times the distance, the lower E¢ (and
therefore the higher K is), the more charge can be stored
on the plates for a given potential difference.
One way to think about the proper use of K is to note that
any formula derived for capacitance can be adopted to the case
in which a dielectric completely fills the space between the
plates of the capacitor by simply replacing every instance of
the permittivity, e0, with the value Ke0.
Hence, the voltage between the plates of a parallel-plate
capacitor in which a material with dielectric constant K
fills the volume between the plates with separation d is
| (4.1.4.11) |
Before and after the dielectric is inserted, we have, if
we let e = Ke0,
| Before insertion | After insertion |
| C = (e0A)/d | C¢ = (eA)/d |
| V = qd/(e0A) | V¢ = qd/(eA) |
| U = 1/2CV2 = (q2d)/(e0A) | U¢ = (q2d)/(eA) |
Notice that the energy stored has decreased by a factor of K! Where did this energy go? To understand, we again have to turn to forces exerted by the capacitor plates. For any real capacitor we must have fields which bow out at the edges, the so-called fringe fields.
As the dielectric is introduced into the fringe field, the dielectric molecules start to align with the external field. This creates an attractive force as the plus charges of the dielectrics are aligned so that they are closest to the negative plate of the capacitor (and of course the same process occurs for the negative dielectric charges and positive plate of the capacitor). The horizontal components of the fringe field yield a horizontal, attractive force which pulls the dielectric into the space between the capacitor plates. The work done by this force in accelerating the dielectric into the capacitor volume decreases the energy stored in the capacitor, thereby lowering the energy stored in the capacitor. Some energy is also converted into electromagnetic radiation and into heat.
Send comments to larryg@upenn5.hep.upenn.edu.
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