Section 28–1 Electromagnetism

 (Retarded Coulomb field / Correction term / Self-force)

 

The three interesting concepts discussed in this section are retarded coulomb field, its correction term, and self-force.

 

1. Retarded Coulomb field:

“Take the first term, E = −qe_r′/4πϵ₀r′². That, of course, is Coulomb’s law, which we already know: q is the charge that is producing the field (Feynman et al., 1963, section 28–1 Electromagnetism).”

 

Feynman says that the first term is Coulomb’s law in which q is the charge that is producing the field. In a sense, this is imprecise because the first term could be known as Retarded Coulomb field because it includes the retarded time, r′/c, or the time it takes for the influence to move from the charge to the point P at speed c. Note that Feynman uses the phrase delayed Coulomb field subsequently, but the notation of the term −qe_r′/4πϵ₀r′² should be improved to denote the delay effect. Some may prefer Jackson’s notation that expresses this term as E = (q/4pe₀)(Ȓ/R²)_ret. The idea of time delay may be illustrated by Feynman’s explanation of electric field using the analogy of two floating corks in water (Chapter 2) where water waves are analogous to electromagnetic waves.

Feynman explains that Coulomb’s law is wrong because influences cannot travel faster than the speed of light. Alternatively, one may clarify that Coulomb’s law is an idealization because we need to assume the “charges” are point objects and they are stationary instead of moving continuously. Furthermore, this law is not applicable for very short distances (less than 10⁻¹⁶ m) and we are unable to experimentally prove that it holds for very large (astronomical) distances. To give a better idea of field, Wilczek (2015) writes: “…in applications to fundamental physics, where quantum fluctuations are important, it becomes problematic because both forces and positions fluctuate. It can be salvaged as an approximate notion by doing some averaging over time and space (p. 355).”

2. Correction term:

“It suggests that we should calculate the delayed Coulomb field and add a correction to it, which is its rate of change times the time delay that we use. (Feynman et al., 1963, section 28–1 Electromagnetism).”

 

Feynman explains that the second term is as though nature was trying to allow for the fact that the effect is retarded. In a sense, this explanation is potentially misleading because the three terms of equation (28.3) have the same retarded effect. Many may prefer Jackson’s (1999) use of subscript “ret” in Feynman’s expression for the electric field is “E = (q/4pe₀) {[Ȓ/kR²]_ret + (¶/c¶t)[Ȓ/kR]_ret -(¶/c²¶t)[v/kR]_ret} (p. 284).” More important, Feynman adds that it is easy to show the first two terms vary inversely as the square of the distance in the next section. That is, we can use chain rule dy/dt = (dy/dx)(dx/dt) and in this case, d(1/R)/dt = d(1/R)/dR × dR/dt. It is simple to realize that d(1/R)/dR varies inversely as the square of the distance R.

In volume II, footnote 1 of chapter 21 states: “1. The formula was first published by Oliver Heaviside in 1902. It was independently discovered by R. P. Feynman, in about 1950, and given in some lectures as a good way of thinking about synchrotron radiation.” This footnote on Feynman’s expression for the electric field or equation (28.1) is incorrect because Heaviside’s formula is an expression for the magnetic field. However, Feynman’s expression for the electric field is equivalent to Heaviside’s expression for the magnetic field (Jackson’s 1999). Furthermore, footnote 2 of chapter 21 suggests readers not to derive Feynman’s expression for the electric field. On the contrary, this is a problem for students in Jackson’s (1999) Classical Electrodynamics by using Jefimenko’s equations. (Feynman’s expression for the electric field is a special case of Jefimenko’s equations.)

 

3. Self-force:

“…we want the field to act on, we get into trouble trying to find the distance, for example, of a charge from itself, and dividing something by that distance, which is zero (Feynman et al., 1963, section 28–1 Electromagnetism).”

 

According to Feynman, when we try to calculate the field from all the charges including the charge, we get into trouble, e.g., dividing something by the distance, which is zero. In other words, we have difficulties in calculating the self-force of a charge that seems to be infinity. Feynman adds that the problem of handling the electric field which is generated by the same charge was not yet solved. In volume II, Feynman elaborates that “…the infinity arises because of the force of one part of the electron on another—because we have allowed what is perhaps a silly thing, the possibility of the ‘point’ electron acting on itself (Feynman et al., 1964, section 28–4 The force of an electron on itself).” Recently, Gralla, Harte, and Wald (2009) determine the self-force by assuming the charge (q) and total mass (m) approach zero, and q/m reaches a well-defined limit.

 

Historically, Wheeler and Feynman propose that point charges interact only with other charges, but the interaction is half through the advanced waves and half through the retarded waves. In his Nobel lecture, Feynman (1965) says that “the idea seemed so obvious to me and so elegant that I fell deeply in love with it. And, like falling in love with a woman, it is only possible if you do not know much about her, so you cannot see her faults…” Interestingly, when Feynman was a graduate student in Princeton, he shared this idea in a technical talk and Einstein politely said: “I find only that it would be very difficult to make a corresponding theory for gravitational interaction … Since we have at this time not a great deal of experimental evidence, I am not absolutely sure of the correct gravitational theory (Feynman, 1985, p. 80).”

 

Review Questions:

1. Does the first term of the Heaviside-Feynman expression of electric field refers to Coulomb’s law or retarded Coulomb field?

2. How would you explain the correction term for the retarded Coulomb field (just like Feynman)?

3. Is the problem of self-force still not completely solved today?

 

The moral of the lesson: Coulomb’s law is an idealization not only because we need to assume the “charges” are point objects and they are stationary, but this law has not included the time-delay effect of influences.

 

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. Gralla, S. E., Harte, A. I., & Wald, R. M. (2009). Rigorous derivation of electromagnetic self-force. Physical Review D, 80(2), 024031.

3. Wilczek, F. (2015). A Beautiful Question: Finding Nature’s Deep Design. New York: Penguin Press.