Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information.... 
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Calculating Energy in FieldsWe have seen that the energy stored in a wave (actually the energy density) is typically proportional to the square of the wave's amplitude. Fields of force can make wave patterns, for which we might expect the same to be true. This turns out to be true not only for wavelike field patterns but for all fields:
Although funny factors of 8π and the plus and minus signs may have initially caught your eye, they are not the main point. The important idea is that the energy density is proportional to the square of the field strength in all three cases. We first give a simple numerical example and work a little on the concepts and then turn our attention to the factors out in front.
In chapter 5 when we discussed the original reason for introducing the concept of a field of force, a prime motivation was that otherwise there was no way to account for the energy transfers involved when forces were delayed by an intervening distance. We used to think of the universe's energy as consisting of kinetic energy
but in nonstatic situations we must use a different method: kinetic energy
Surprisingly, the new method still gives the same answers for the static cases.
Now let's give at least some justification for the other features of the three expressions for energy density,
besides the proportionality to the square of the field strength. First, why the different plus and minus signs? The basic idea is that the signs have to be opposite in the gravitational and electric cases because there is an attraction between two positive masses (which are the only kind that exist), but two positive charges would repel. Since we've already seen examples where the positive sign in the electric energy makes sense, the gravitational energy equation must be the one with the minus sign. It may also seem strange that the constants G, k, and μ_{o} are in the denominator. They tell us how strong the three different forces are, so shouldn't they be on top? No. Consider, for instance, an alternative universe in which gravity is twice as strong as in ours. The numerical value of G is doubled. Because G is doubled, all the gravitational field strengths are doubled as well, which quadruples the quantity g^{2} . In the expression
we have quadrupled something on top and doubled something 8πG on the bottom, which makes the energy twice as big. That makes perfect sense.
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