It's important to realize that there is just one value of the E field
at a particular point in spce and time. The same is true for the
magnetic (B) field. So when you want to figure out the field somehwere,
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 enrgy 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.
Amother 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
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.
(published on 10/22/2007)