5.10 Energy Density
It is convenient to define a quantity called energy density, and we will denote this quantity by small u. It is defined as energy stored in the electric fields of the capacitor per unit volume. It is equal to u sub E divided by the volume of the region between the plates of the capacitor. If we consider a parallel plate capacitor, we know that such a capacitor consists of two parallel conducting plates separated by an insulating medium. And let’s say that the distance between the plates is equal to A, d is the separation distance, and let’s say A represents the surface area of the plate.
Therefore, for such a capacitor, we can express, let’s say that the upper plate is charged positively, the lower plate is charge negatively, and the electric field is filling the region between the plates originating from positive charged to negatively charged plate. The energy density, small u, is going to be equal to total energy stored in the electric field of this capacitor divided by the volume of the region between the plates. Since the surface plate area is A and the separation distance is d, that is going to be equal to A times d.
In explicit form, we can express the total energy stored between the plates of this capacitor as one-half capacitance of the plate times square of the potential difference between the plates divided by A times d.
Well, if we recall the capacitance of a parallel plate capacitor, the capacitance was equal to Epsilon zero times plate area divided by the distance between the plates of the capacitor. Therefore, if we substitute this for the capacitance here, then the energy density expression becomes one-half A times V squared divided by A times–here we have d from the equation and another d will come through substitution, d squared in the denominator. The plate areas will cancel in the numerator and the denominator, and also we have Epsilon zero. Let’s not forget that over here. Then we will end up with energy density is equal to one-half Epsilon zero V over d squared.
Let’s see what this ratio is equal to. If we recall the potential difference between the plates of a capacitor, V was equal to integral from positive to negative plate of E dot dl. So by choosing a straight line of path from positive to negative plate, then dl is the incremental displacement vector along that path. The angle between them, between an electric field vector and the incremental displacement vector in this case is zero, then this became equal to Edl cosine of zero. Cosine of zero is just 1.
For a parallel plate capacitor, we have seen that the electric field was constant. We found that by applying Gauss’s law. The result showed us that wherever we go between the plates of the parallel plate capacitor, the magnitude of the field was the same. Therefore we can take this outside of the integral and finally therefore the potential difference between the plates becomes equal to integral of dl integrated from positive to negative plate.
If we add all these incremental displacement vectors to one another along this distance, we are going to end up with the magnitude of that distance, which is equal to the separation distance d. So this expression therefore becomes equal to E times d. And solving for the electric field there, we will end up with E is equal to V over d.
Therefore this ratio is nothing but the electric field magnitude between the plates of this capacitor. Then we can replace that ratio by expressing the energy density, little u, is equal to one-half Epsilon zero times the magnitude of the electric field squared. Of course, the unit of energy density is going to be energy per unit volume, so the u will be equal to–the unit of energy in SI unit system is Joule–and the unit of volume is meter cubed, so Joule per meter cubed is the unit of energy density in SI unit system.
We will see the benefit of dealing with the energy density in the following example. This will enable us to be able to figure out the amount of energy stored in a specific region between the plates of the capacitor. Furthermore, although we obtained this expression for the parallel plate capacitor, of course this is going to be true also for spherical as well as the cylindrical capacitors. The only difference is going to be, of course, their associated electric fields.
For the parallel plate capacitor, electric field was constant between the plates all the time, therefore the energy density, energy per unit volume, is also constant. For the spherical as well as the cylindrical capacitors, the electric field is a function of the radial distance; therefore it will change point to point along the radial distance. As a result of that, the energy density will also not be a constant for those capacitors. It is going to change from point to point.