The question itself is not quite nonsense. There is a perfectly reasonable answer of “everywhere”. It’s just not a particularly useful question, and this is because of the hidden assumptions behind it, which are wrong and can easily lead to nonsense questions. “What is the value of the electromagnetic field at X?” is a much more interesting question that can be asked once those incorrect assumptions are removed and replaced.
Eh. You can force an answer in English, sure, but it’s still not really the “right” answer. The electromagnetic field is a function from spacetime, to, uh, some sort of tangent bundle on it or something? My knowledge of how to formalize this sort of thing isn’t so great. My point is that it’s a function taking spacetime locations as inputs; it doesn’t really have a location itself any more than, say, the metric of spacetime does. When we say “it’s everywhere” what’s meant is something more like “it’s defined everywhere” or “at every location, it affects things”.
The EM field is used both for the function, and the values of that function. (I think it’s actually a skew-symmetric linear operator on the tangent space T_x M at a given point. This can be phrased in terms of a bivector at that point. A “bundle” TM = Union_x T_x M talks about an extended manifold connecting tangent spaces at a different points.) I think it’s entirely reasonable in common language to use “where” to mean “where it’s non-negligible”. Consider that physical objects are also fields. It’s entirely reasonable to ask “where an electron is” even though the electron field is a function of spacetime. Once we’re able to ask the right questions, this becomes a less-useful question, as it only applicable in cases where the field is concentrated. The EM field case just breaks down much sooner.
The question itself is not quite nonsense. There is a perfectly reasonable answer of “everywhere”. It’s just not a particularly useful question, and this is because of the hidden assumptions behind it, which are wrong and can easily lead to nonsense questions. “What is the value of the electromagnetic field at X?” is a much more interesting question that can be asked once those incorrect assumptions are removed and replaced.
Eh. You can force an answer in English, sure, but it’s still not really the “right” answer. The electromagnetic field is a function from spacetime, to, uh, some sort of tangent bundle on it or something? My knowledge of how to formalize this sort of thing isn’t so great. My point is that it’s a function taking spacetime locations as inputs; it doesn’t really have a location itself any more than, say, the metric of spacetime does. When we say “it’s everywhere” what’s meant is something more like “it’s defined everywhere” or “at every location, it affects things”.
The EM field is used both for the function, and the values of that function. (I think it’s actually a skew-symmetric linear operator on the tangent space T_x M at a given point. This can be phrased in terms of a bivector at that point. A “bundle” TM = Union_x T_x M talks about an extended manifold connecting tangent spaces at a different points.) I think it’s entirely reasonable in common language to use “where” to mean “where it’s non-negligible”. Consider that physical objects are also fields. It’s entirely reasonable to ask “where an electron is” even though the electron field is a function of spacetime. Once we’re able to ask the right questions, this becomes a less-useful question, as it only applicable in cases where the field is concentrated. The EM field case just breaks down much sooner.