The earth exerts a gravitational force on the moon (Fe) while the moon exerts a gravitational force on the earth (Fm). By Newton's third law of motion, these forces are of equal strength and in opposite directions. However, many people (including Newton) have been reluctant to accept this idea of "action at a distance." Such people are bothered by the idea that the earth could directly exert a force on an object over 200,000 miles away. (The distance is irrelevant, really, but the point is, the earth isn't touching the moon.) An alternate way that doesn't rely so much on action at a distance is to think of a force in terms of "fields."
The moon (and any other object with mass) is also surrounded by a gravitational field, and any object, such as the earth, in that field experiences a force as well. (I've left this out of the picture above, though. Too much clutter, otherwise.)
We can look at electric forces the same way. Consider, for example, the electric force between a proton and an electron. We can say that the proton is surrounded by an electric field, and this electric field exerts a force on the electron. and the electron is surrounded by an electric field, which exerts a force on the proton.
Either way, the result is the same. The moon experiences the same force, regardless of what we consider to be the immediate source of that force, the earth, or a gravitational field. So do these fields really exist? Well, they're not tangible, certainly, but they do seem to have some important features. For example, a fluctuating electric field creates a magnetic field, and a fluctuating magnetic field creates an electric field. So these fields sort of have a life of their own. This is especially true when we consider the consequences of quantum physics.
The Quantum Picture of Forces
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