Section 29–2 Energy of radiation

(Energy content / Energy flux / Energy loss)

 

The three interesting concepts discussed in this section are the energy content of a wave, energy flux within a conical angle, and energy loss by an oscillating charge.

 

1. Energy content:

“Now we must point out that the energy content of a wave, or the energy effects that such an electric field can have, are proportional to the square of the field… (Feynman et al., 1963, section 29–2 Energy of radiation).”

 

Feynman points out that the energy content of a wave, or the energy effects that an electric field can have, is proportional to the square of the electric field. He explains that it is because the electric field can act on a charge or oscillator and makes it move. However, this does not explain why the energy content of a wave is proportional to the square of the field. In Volume II, section 8–5 Energy in the electrostatic field, Feynman shows that the field energy is equal to U = ϵ₀/2∫E.EdV. In his words, “when an electric field is present, there is located in space an energy whose density (energy per unit volume) is u = ϵ₀(E.E)/2 = ϵ₀E²/2 (Feynman et al., 1964).” Thus, the field energy (potential energy) can be converted into kinetic energy of the charge that is proportional to the square of the electric field.

 

At the beginning of this section, Feynman says that at any particular moment or in any particular place, the strength of the electric field varies inversely as the distance r, as it was mentioned previously. In the last sentence of the previous section, he mentions: “Now, ignoring the angle θ and the constant factors, let us see what that looks like as a function of position or as a function of time (Feynman et al., 1963, section 29-1).” Based on this assumption, the oscillation of a charge clearly causes the electric field to be maximum at the horizontal plane (q = 90°) for any distance r (thus, inversely proportional to the distance r) as shown in Fig 29-1. On the other hand, the electric field is zero at any point on the vertical axis (q = 0°) through the charge. Perhaps the last sentence of section 29-1 on ignoring the angle q should be shifted to the beginning of section 29-2.

 

2. Energy flux:

“So the fact that the amplitude of E varies as 1/r is the same as saying that there is an energy flux which is never lost, an energy which goes on and on, spreading over a greater and greater effective area (Feynman et al., 1963, section 29–2 Energy of radiation).”

 

According to Feynman, the energy that a source can deliver decreases as it is farther away and it varies inversely as the square of the distance. He adds that the energy we can take out of a wave within a conical angle is independent of the distance from where we are. Strictly speaking, the energy flux or energy flow through a cone may vary with the angle θ unless the energy flux through the cone has cylindrical symmetry (through the axis of motion). Thus, the energy flux through any conical angle is not definitely the same and this property allows us to choose the direction of maximum energy flow and help to save the energy needed for an antenna. In section 29-4, Feynman will discuss how multiple antennas will result in a stronger beam (energy flow) in a preferred direction using interference (the title of this chapter).

 

Feynman explains that all the energy we could pick up from a wave in a certain cone at a distance r₁ and at another distance r₂ are the same because the area of the surface intercepted by the cone varies directly as the square of the distance r. It may be worthwhile to mention that the energy flux through the charge is different from its electric flux. More importantly, we may define energy flux using Poynting vector (S) that is mathematically represented by S = (1/m₀)E ´ B in which E is the electric field vector and B is the magnetic field vector. Feynman opines that “[t]here are, in fact, an infinite number of different possibilities for u and S, and so far no one has thought of an experimental way to tell which one is right! (Feynman et al., 1964, section 27-4.)” In other words, energy flux is a calculational device and it is not an observable.

 

3. Energy loss:

“So if we are far enough away that our basic approximation is good enough, the charge cannot recover the energy which has been, as we say, radiated away… We shall study this energy loss further in Chapter 32. (Feynman et al., 1963, section 29–2 Energy of radiation).”

 

Feynman initially says that an oscillating charge would lose some energy which it can never recover. Subsequently, he explains that the charge cannot recover the energy because it has been radiated away. However, he elaborates that the energy is not really lost because it still exists somewhere, and it is available to be picked up by other systems (such as a radio). Perhaps he could have clarified that energy loss does not mean that energy can be destroyed, but it can be transformed into another form of energy. For example, in Volume II, chapter 22, Feynman mentions: “[w]hat about the energy loss when a generator is connected to an arbitrary impedance z? (By ‘loss’ we mean, of course, conversion of electrical energy into thermal energy.)”

 

Feynman ends the section by saying this energy loss will be discussed further in Chapter 32. In Chapter 32, he suggests that “to the driving circuit the antenna acts like a resistance, or a place where energy can be lost (the energy is not really lost, it is really radiated out, but so far as the circuit is concerned, the energy is lost). In an ordinary resistance, the energy which is lost passes into heat; in this case the energy which is lost goes out into space.” In this context, when Feynman uses the phrase energy loss, it may refer to the conversion of electrical energy of a system (antenna) into field energy or the spreading of energy into space. In a sense, the idea of energy loss is dependent on the definition of a system such as an antenna, or the system may include the space surrounding the antenna.

 

Review Questions:

1. Is the strength of an oscillating charge’s electric field really vary inversely as the distance r?

2. Is the energy flux through any conical angle independent of the angle q?

3. Does an oscillating charge lose some energy which it can never recover?

 

The moral of the lesson: the energy content of a wave of an oscillating charge is related to the energy flux through a cone and the energy “loss” by the oscillating charge.

 

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.