It’s important to realize that there is just one value of the E field at a particular point in space and time. The same is true for the magnetic (B) field. So when you want to figure out the field somewhere, you simply add contributions from all the sources.
If you were to multiply fields, the result would not be a field of the same type. It wouldn’t even have the right dimensions. Furthermore, since the fields are all vectors, not numbers, you have to figure out what type of product is relevant for a particular calculation.
Some important field products are E dot E and B dot B. These are NOT other E and B field vectors. They are not ordinary numbers, and their dimensions are those of energy per volume, in the simple cgs unit system. Multiplied by constants, these represent the energy present in the EM fields. Both terms are SQUARES so each is always positive.
Another important field product is E cross B. It also has units of energy density (in cgs units). However it is a vector, not a number. Multiplied by the speed of light (and another numerical constant) it represents the flow of energy in the EM fields. It’s called the Poynting vector.
Some of your remarks (’multiplication most naturally describes volume’) seem to assume that the things being multiplied represent ordinary lengths. That assumption doesn’t apply here.
(republished on 08/02/06)