Calculating Energy in Fields

We 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 wave-like 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.

Getting killed by a solenoid

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

• + gravitational potential energy based on the distances between objects that interact gravitationally

• + electric potential energy based on the distances between objects that interact electrically

• + magnetic potential energy based on the distances between objects that interact magnetically

but in nonstatic situations we must use a different method:

kinetic energy

• + gravitational potential energy stored in gravitational fields

• + electric potential energy stored in electric fields

• + magnetic potential stored in magnetic fields

Surprisingly, the new method still gives the same answers for the static cases.

Energy stored in a capacitor

Potential energy of a pair of opposite charges

Energy in an electromagnetic wave

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.

Discussion Questions

A	The figure shows a positive charge in the gap between two capacitor plates. First make a large drawing of the field pattern that would be formed by the capacitor itself, without the extra charge in the middle. Next, show how the field pattern changes when you add the particle at these two positions. Compare the energy of the electric fields in the two cases. Does this agree with what you would have expected based on your knowledge of electrical forces?
B	Criticize the following statement: "A solenoid makes a charge in the space surrounding it, which dissipates when you release the energy."
C	In the example on the previous page, I argued that the fields surrounding a positive and negative charge contain less energy when the charges are closer together. Perhaps a simpler approach is to consider the two extreme possibilities: the case where the charges are infinitely far apart, and the one in which they are at zero distance from each other, i.e. right on top of each other. Carry out this reasoning for the case of (1) a positive charge and a negative charge of equal magnitude, (2) two positive charges of equal magnitude, (3) the gravitational energy of two equal masses.