Section 23–3 Electrical resonance

(Circuit elements / Mechanical-electrical analogies / Electrical symbols)

In this section, Feynman discusses circuit elements, mechanical-electrical analogies, and electrical symbols that are related to the concept of electrical resonance.

1. Circuit elements:

“The three main kinds of circuit elements are the following. The first is called a capacitor… (Feynman et al., 1963, section 23–3 Electrical resonance).”

Feynman discusses an example of a capacitor that has two plane metallic plates spaced a very short distance apart by an insulating material. He says that all we need to know is that the potential difference across a capacitor is proportional to the charge: V = q/C; where C is the capacitance of the object. However, in The Art of Electronics, Horowitz and Hill (1989) write: “there’s a ‘memory’ effect (known as dielectric absorption), which is rarely discussed in polite society: if you charge a capacitor up to some voltage V₀ and hold it there for a while, and then discharge it to 0 V, then when you remove the short across its terminals it will tend to drift back a bit toward V₀ (p. 28).” In addition, real capacitors behave like they have resistance, inductance, and some frequency-dependent parallel resistance.

Feynman explains that an inductor is a circuit element that is analogous to the mass of an object. It is a coil which builds up a magnetic field within itself when there is a current in it and the coil’s changing magnetic field results in a voltage that is proportional to dI/dt. On the other hand, Horowitz and Hill (1989) explain that “inductors are magnetic devices, in which two things are going on: current flowing through the coil creates a magnetic field aligned along the coil’s axis; and then changes in that field produce a voltage (sometimes called ‘back EMF’) in a way that tries to cancel out those changes (an effect known as Lenz’s law) (p. 28).” Importantly, an ideal inductor has no resistance and no power is dissipated within the coil, whereas a real inductor has resistance and capacitance (due to a separation between the wires of the coil).

2. Mechanical-electrical analogies:

“If we think of the charge q on a capacitor as being analogous to the displacement x of a mechanical system, we see that the current, I = dq/dt, is analogous to velocity… (Feynman et al., 1963, 23–3 Electrical resonance).”

Feynman suggests that the charge q on a capacitor is analogous to the displacement x of a mechanical system and the current dq/dt is analogous to velocity. This is also known as Maxwell’s analogy or “force-voltage, velocity-current” analogy. In essence, the force on a mechanical element is analogous to the voltage (or electromotive force) on a circuit element, and the velocity of the mechanical element is analogous to the electric current through the circuit element. The analogy is derived from the similarity of the equations z = force/velocity and Z = voltage/current where z is the mechanical impedance and Z is the electrical impedance. However, Feynman could have clarified a limitation of this analogy: circuit elements connected in series are analogous to the corresponding mechanical elements connected in parallel, or vice versa. (Thus, he has inappropriately used the phrase exact analogy in chapter 25.)

Feynman mentions that the quantity R+iωL+1/iωC is a complex number, and it is called the complex impedance Z that is used in electrical engineering. One may explain that the concept of electrical impedance is seen as a ratio of the cause (voltage) to its effect (current). To have a better understanding of mechanical-electrical analogies, one should distinguish Maxwell’s analogy from Firestone’s analogy. Firestone (1933) suggests another analogy such that the force and velocity of a mechanical system are analogous to the Kirchhoff’s laws of current and voltage. The analogy is derived from the similarity of the equations ẑ = velocity/force and Z = voltage/current where ẑ is the reciprocal of the mechanical impedance.

3. Electrical symbols:

“The difficulties of science are to a large extent the difficulties of notations, the units, and all the other artificialities which are invented by man, not by nature (Feynman et al., 1963, section 23–3 Electrical resonance).”

In electrical engineering, the symbol j is commonly used to denote √−1 (instead of using i). Feynman elaborates that i must be the current for electrical engineers, but they get into trouble when they use j to denote current density. However, this is not true because electrical engineers use J (instead of j) to denote current density. In 1893, Charles Proteus Steinmetz introduces the concept of phasor and proposes the use of the symbol j because he found it useful in doing efficient calculations on alternating currents (Araújo & Tonidandel, 2013). One may add that the imaginary number j corresponds to a rotation of 90^o or a phase difference between the electric current and the electric potential difference.

Feynman explains that the difficulties of science are to a large extent the difficulties of notations, the units, and all the other artificialities, which are invented by man, not by nature. In his biography, Feynman (1997) writes that “I thought my symbols were just as good, if not better, than the regular symbols--it doesn’t make any difference what symbols you use-but I discovered later that it does make a difference. Once when I was explaining something to another kid in high school, without thinking I started to make these symbols, and he said, ‘What the hell are those?’ I realized then that if I’m going to talk to anybody else, I’ll have to use the standard symbols, so I eventually gave up my own symbols (p. 24).” In a sense, the difficulties of notations are worsened because Feynman also likes to invent his own symbols.

Questions for discussion:

1. How would you define a capacitor and an inductor?

2. What are the limitations of the Maxwell’s analogy and Firestone’s analogy?

3. Do electrical engineers really get into trouble because of the electrical symbol j?

The moral of the lesson: the difficulties of science are to a certain extent due to difficulties of notations, the units, and all the other artificialities, which are invented by man.

References:

1. Araújo, A. E. A. D., & Tonidandel, D. A. V. (2013). Steinmetz and the Concept of Phasor: A Forgotten Story. Journal of Control, Automation and Electrical Systems, 24(3), 388-395.

2. Feynman, R. P. (1997). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

3. 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.

4. Firestone, F. A. (1933). A new analogy between mechanical and electrical systems. The Journal of the Acoustical Society of America, 4(3), 249-267.

5. Horowitz, P. & Hill, W. (1989). The Art of Electronics (Second ed.). New York: Cambridge University Press.