When we introduced fields we discussed the "direct method" of calculating forces and the "field method" for calculating forces. At the time it seemed like the field was little more than a tool to help us consider the problem. If were willing we could use the direct method between all the particles of interest. When we introduced magnetic forces, we never even discussed a "direct method" because the field method was so much easier, but it seemed possible to construct a direct method with enough patience. Nothing we discussed so far seems to require fields; fields just made our job easier. One might be tempted to ask if fields actually exist, or if they're just convenient mathematical constructions.
To show why fields (or something like them) have to exist, consider moving a charge around for a short time. Energy is required to move the charge around, and the charge releases that energy to electromagnetic waves. Sometime later the wave hits some other charges, and starts them oscillating. At all stages energy is conserved.
Suppose we carried this out without knowledge of fields; we would provide energy to the charges which would slow down and eventually stop. Sometime later the other set of charges would start to oscillate all by themselves! It would appear as if we lost some energy for a short time, and then gained it back later. This would lead us to one of three possibilities:
- Maybe the conservation of energy is not valid.
This is wrong.
- Maybe the energy went into an energy system (such as a field) that we haven't considered, and everything is really okay.
We advocate this second choice, but some may still object. There is a third possibility that allows us to conserve energy without fields:
Maybe the energy of a system does not only depend on what it is doing right now, but also what happened to it in the past. In other words, maybe energy is not a state function.
Energy is conserved because energy is a state function. The third possibility is a very clever attempt to get around having to accept fields, but unfortunately it fails. If it were true, we would have to know the entire history of an object (in principle) to figure out what its energy should be now. However we've seen that, with regard to energy, you only need to consider initial and final conditions of an object. What happens in between isn't as important, so long as it doesn't lose energy. Energy must be a state function.
In order to save the principle of energy conservation in a useful manner, we are forced into accepting that there is an additional system containing energy – the field. This means when we choose our initial and final conditions for a system, we must also consider how the energy in the fields changes in the interim.
Authors of Phys7C (UC Davis Physics Department)