Section 25–4 Analogs in physics

(Ohm’s law / Inductor and capacitor / Analog computer)

 

In this section, Feynman discusses Ohm’s law, “inductor and capacitor,” as well as the analog computer.

 

1. Ohm’s law:

“… in other words, proportional to how much voltage there is: V = IR = R(dq/dt). The coefficient R is called the resistance, and the equation is called Ohm’s Law (Feynman et al., 1963, section 25–4 Analogs in physics).”

 

Feynman states Ohm’s law as “if there is a current I, that is, so and so many charges per second tumbling down, the number per second that comes tumbling through the wire is proportional to how much voltage there is.” In short, the equation V = IR is called Ohm’s law by Feynman. However, one may distinguish two different versions of Ohm’s law: the law for a part of a circuit and the law for a whole circuit (Kipnis, 2009). The law for a part of a circuit is “electric current (I = DV/R) through a conductor is directly proportional to the potential difference at its ends (DV), and the resistance of the conductor (R) is constant.” The law for a whole circuit is “electric current (I = E/[R + r]) through a conductor is directly proportional to the potential difference at its ends and inversely proportional to its resistance (R + r).”

 

Feynman explains that the resistance obeys the Ohm’s law for almost all ordinary substances and says that “this law is extremely accurate for most metals.” In Volume II, he adds that “the relation between the current and the voltage for real conducting materials is only approximately linear (Feynman et al., 1964, section 22–1 Impedances).” We can specify three conditions of validity for Ohm’s law: (1) low constant voltage (assume ohmic devices), (2) constant temperature (assume no heating effect), and (3) constant size (assume no expansion). In addition, Feynman shows that the heating loss generated is equal to V(dq/dt) = VI = I²R. On the contrary, one may clarify that it can result in a heat gain in the resistor that increases the resistance such that the voltage-current relation is not strictly linear.

 

2. Inductor and capacitor:

“The equation is V = L(dI/dt) … is such that one volt applied to an inductance of one henry produces a change of one ampere per second in the current (Feynman et al., 1963, section 25–4 Analogs in physics).”

 

Feynman says that the current of an inductor does not want to stop after it is started. Based on the equation V = L(dI/dt), there is no voltage across the inductor if the current is constant. That is, we have idealized the inductor as a circuit element that has no resistance, and no power is dissipated by the current flowing through it. In Volume II, Feynman clarifies that we have to make four assumptions for an ideal inductor, e.g., “the magnetic field produced by currents in the coil does not spread out strongly all over space and interact with other parts of the circuit (Feynman et al., 1964, section 22–1 Impedances).” In a sense, the inductor has an inertial effect that resists a change in electric current, just like the inertial mass was explained as an inductive effect that is based on electrodynamics (Jammer, 1997).

 

Feynman mentions that the work done in moving a unit charge across the gap from one plate to the other is precisely proportional to the charge. He adds that we have V = q/C and the constant of proportionality is not called C, but 1/C for historical reasons. In Volume II, Feynman explains that “[t]his formula is not exact, because the field is not really uniform everywhere between the plates, as we assumed (Feynman et al., 1964, section 6–10 Condensers; parallel plates).” However, the dielectrics materials used in parallel plates capacitors are not perfect insulators. Historically, the formula S_Q = S_V/C does not mean Q = V/C and thus, the constant of proportionality is 1/C. In the context of an electroscope, S_Q is the deflection per unit charge (or charge sensitivity) and S_V is the deflection per unit potential-difference (or potential difference sensitivity).

 

3. Analog computer:

“This is called an analog computer. It is a device which imitates the problem that we want to solve by making another problem… (Feynman et al., 1963, section 25–4 Analogs in physics).”

 

According to Feynman, an analog computer is a device that imitates the problem that we want to solve by making another problem, which has the same equation, but in another circumstance of nature, and which is easier to build, to measure, and to adjust. Specifically, the analog computer has continuous quantities such as voltages or currents, instead of discrete states in a digital computer. An example of analog computer is the FERMIAC, a Monte Carlo mechanical device, that was used in the Manhattan (atomic bomb) Project to perform calculations for neutron diffusion. In his autobiography, Feynman (1997) mentions that he used IBM machines to find out what happened during the bomb’s implosion. In his words, “if you’ve ever worked with computers, you understand the disease--the delight in being able to see how much you can do (Feynman, 1997, p. 127).”

 

Feynman explains that the electrical circuit is the exact analog of the mechanical system, in the sense that whatever q does, in response to V (V is made to correspond to the forces that are acting), so the x would do in response to the force. However, it is inaccurate to use the phrase “exact analog” because circuit elements connected in series are analogous to the corresponding mechanical elements connected in parallel, and vice versa (Firestone, 1933). For example, a capacitor and an inductor connected in parallel have the same voltage V, but a spring and an object experience the same force when they are connected in series (horizontally). On the other hand, the same current passes through the capacitor and the inductor connected in series, but the object and the spring have the same velocity difference when they are connected in parallel. In short, the same voltage occurs in parallel, the same force occurs in series.

 

Questions for discussion:

1. How would you state Ohm’s law?

2. How would you explain the equation V = L(dI/dt) and Q = V/C? 

3. Does an electrical circuit in series have an exact analog of a mechanical system in series?

 

The moral of the lesson: we may replace the equation corresponding to the circuit L(d²q/dt²) + R(dq/dt) + q/C = V by the equation m(d²x/dt²) + γm(dx/dt) + kx = F in the sense that circuit elements connected in series are analogous to the corresponding mechanical elements connected in parallel.

 

References:

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

2. 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.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. 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. Jammer, M. (1997). Concepts of Mass in Classical and Modern Physics. Mineola, NY: Dover.

6. Kipnis, N. (2009). A law of physics in the classroom: The case of Ohm’s law. Science & Education, 18(3-4), 349-382.