In this post, we’ll do a mathy proof of the electrical energy density of an electric field. Here’s the statement:
For any continuous distribution of charge where the electric field vanishes at infinity, the total electrical energy of the system is given by
where is a small volume element, is the electric field magnitude at a particular point, and the integral is taken over all space. This allows us to interpret the quantity
as the density of electrical energy.
To prove this, we first need to make sure we understand what the electrical energy means.
what is electrical energy?
Here’s how we’ll define it.
The potential energy of a distribution of charge is the work that must be done to assemble it from a state where they are all infinitely far away from each other. The reason is simple: When they’re all infinitely far away, they’re not interacting at all!
For example, if I just had two charges, they’re infinitely far away from each other when one is at the origin and the other is at infinity. The potential energy is then the work I do to put the far away charge at a specified distance from the charge at the origin. Here’s another very important property. The path I take while bringing in the charge does not matter! This is because the electric field is conservative, which means that there is a well defined potential energy between two points.
Let’s say I have three charges as shown above. Then since the path you take to assemble the charge does not matter, you can have charge at the origin, bring in charge , and then finally bring in charge . Or you can fix charge and bring in and at the same time. It really doesn’t matter. For simplicity, we’ll go with the first option.
The work you do bringing beside is just the potential energy between and . And then the work you do brining in front of and is just the sum of the potential energies between and , and and . So the total potential energy is just the sum of the potential energies between and , and , and and .
our first formula for energy
In words, The total energy of a system of point charges is the sum of the potential energies between each pair of charges.
Converting this into math,
Here we have labeled each point charge with an index from to . The charge on the particle is and the distance between particle and particle is .The factor of is there because we are over counting by a factor of . For example, we will count both and .
Notice we can rearrange the formula to get
All we’ve done is factor out the from the sum over . This equation now has some physical meaning.
is the potential energy at particle due to all other charges other than particle . Let’s call this . Then
how about non-point charges?
If we aren’t dealing with point charges, we’re dealing with a continuos distribution of charge. In the continuum limit, the sum becomes an integral and becomes , where is a small volume element. So
Where just represents all of space.
By Gauss’s Law,
where is the electric field at a point not due to the infinitesimal charge located exactly at that point. If this is the definition, why should the gradient depend on the charge density at a certain point at all?
The answer is that Gauss’s Law takes the limit as we go infinitesimally close to our given point but never actually reach it. Therefore at these nearby points, always includes the contribution from the given point.
the rest is math
We can use this to simplify our integral further. Substituting
the electrical energy becomes
A neat trick
We can simplify it as follows
This is just a straightforward calculation. You can easily verify it by going through the product rule and the definition of the gradient and divergence. Using this expression, the electrical energy becomes
simplifying the integral
We can rewrite the first integral using the divergence theorem:
Here is the surface of region . For example, if were a sphere, then would be the surface of the sphere. is a small vectorial area element perpendicular to the surface.
It turns out this integral is zero. Here’s why. Let’s imagine as a sphere of radius of . Then we know falls off as . Since is just the electric field, it falls off as . The area grows as , so the integral falls off as . If we take , the integral vanishes.
This implies that
where is the magnitude of the electric field (remember that ).
We can interpret this as an energy density of
associated with each volume , which is what we set out to prove.