Section 26–3 Fermat’s principle of least time

(Law of reflection / Law of refraction / Infinitesimal change)

 

In this section, Feynman explains the laws of reflection and refraction through Fermat’s principle of least time, showing that the correct path is the one for which the first-order infinitesimal change in travel time vanishes. A more accurate title could be the principle of stationary time, since the path light follows is one where the travel time is stationary — meaning that small variations give no first-order change. This stationary value could correspond to a minimum, a maximum, or even a saddle point, though in optics it usually manifests as a minimum.

 

1. Law of reflection:

“…the angle of incidence equals the angle of reflection is equivalent to the statement that the light goes to the mirror in such a way that it comes back to the point B in the least possible time (Feynman et al., 1963, section 26–3 Fermat’s principle of least time).”

 

According to Feynman, Fermat formulated the principle of least time around 1650 as: “that out of all possible paths that it might take to get from one point to another, light takes the path which requires the shortest time.” In a 1657 letter to Cureau de la Chambre, Fermat suggested that the law of refraction could be deduced from such a minimum principle. The principle can be expressed as: a light ray traveling between two points follows the path of shortest optical path length. The optical path length, nd, is the distance in vacuum that light would cover in the same time it traverses a distance d in a medium of refractive index n. In other words, the path of least time corresponds to the path with the minimum number of wavelengths.

 

“The first way of thinking that made the law about the behavior of light evident was discovered by Fermat in about 1650, and it is called the principle of least time, or Fermat’s principle (Feynman et al., 1963, section 26–3 Fermat’s principle of least time).”

 

Feynman states that the first way of thinking which made the laws of light evident was discovered by Fermat. However, as early as ~100 AD, Hero of Alexandria proposed that light reflects from a plane mirror by taking the path of shortest distance. Later, around ~200 AD, Damianus suggested an idea closer to the principle of least time. As Mach (1953) notes: “Damianus says ‘… the nature of our ray of vision did not permit of aimless wandering, the ray would be bent (reflected) at equal angles.’ As already mentioned, it is not quite clear whether the teleological conception refers to a minimum of time, or space, or perhaps both” (p. 28). Ultimately, Fermat’s contribution stands out because he formulated the principle in a mathematically precise way and extended it to refraction, where the shortest path is no longer valid.

 

2. Law of refraction:

“…let us demonstrate that the principle of least time will give Snell’s law of refraction (Feynman et al., 1963, section 26–3 Fermat’s principle of least time).”

 

Using the principle of least time to explain Snell’s law, Feynman (1985) illustrates the problem with a vivid analogy: a beautiful girl has fallen out of a boat and is crying for help in the water. In his lecture on QED for the public, he rephrases the problem as: “Suppose you’re the lifeguard, sitting at S, and the beautiful girl is drowning at D” (p. 51). He then distinguishes four kinds of path: (1) the path of least time, (2) the path of least water, (3) the path of least distance, and (4) the path certainly not of least time. To clarify why the least-time path is optimal, Feynman adds that “light seems to go slower in water than it does in air, which makes the distance through water more ‘costly,’ so to speak, than the distance through air” (p. 51). The word costly is fitting, since Fermat described his law as the principle of natural economy.

Source: (Feynman, 1985)

Recently, researchers have discovered light behaviors that go beyond Fermat’s principle. For example, a new class of laser beams called spacetime wave packets that do not follow the traditional law of refraction. As Bhaduri et al. (2020) observe, “space-time refraction defies our expectations derived from Fermat’s principle.” This reminds us that Fermat’s principle should not be treated as fundamental, nor should we assume that all light beams necessarily follow the shortest-time path. In fact, light can obey different rules depending on the experiment, and its behavior is more comprehensively explained by the broader framework of the principle of stationary action. Strictly speaking, the phrase “least action” is a misnomer: the true path corresponds to a stationary value of the action, which may represent a minimum, a maximum, or even a saddle point (Zengel, 2024).

 

3. Infinitesimal change:

“This means that if we move the point X to points near C, in the first approximation there is essentially no change in time because the slope is zero at the bottom of the curve. So our way of finding the law will be to consider that we move the place by a very small amount, and to demand that there be essentially no change in time (Feynman et al., 1963, section 26–3 Fermat’s principle of least time).”

Feynman explains that if we compare the travel times at points near C, there is essentially no change in time to first order, because the slope is zero at the bottom of the curve (Fig. 26–5). In other words, optical laws can be deduced by comparing the times at neighboring points that differ only by an infinitesimal displacement. To see what “first approximation” means more precisely, we can use Taylor’s theorem: f(x) = f(a) + (x−a)f′(a)/1! + (x–a)²f′′(a)/2! + … (x – a)ⁿfⁿ(a)/n! + … The zeroth-order approximation is simply the constant term f(a). The first-order approximation adds the linear term (x−a)f′(a), which locally replaces the curve with its tangent line at x=a. In other words, the curve is approximated by using a straight line in the immediate neighborhood of the point.

“Of course, there is an infinitesimal change of a second order; we ought to have a positive increase for displacements in either direction from C (Feynman et al., 1963, section 26–3 Fermat’s principle of least time).”

 

Feynman remarks that the infinitesimal change near point C is of second order, meaning that for small displacements to either side of C, the travel time increases only quadratically (Fig. 26–5). To clarify this idea, we can turn to the second-order term of Taylor’s expansion, (x–a)²f′′(a)/2!, which represents a quadratic approximation to the curve. In other words, while the first-order approximation replaces the curve with a straight line, the second-order approximation uses a parabola. As Feynman later emphasizes in The Feynman Lectures on Physics, Vol. II (1964), “one of the properties of the minimum is that if we go away from the minimum in the first order, the deviation of the function from its minimum value is only second order.” Put differently, the second-order effect, by contrast, is governed by the second derivative of the function, which reflects the curvature of the curve at the minimum.

 

Review Questions:

1. Feynman credits Fermat with the first clear way of thinking about the behavior of light. Do you agree, or do earlier thinkers like Hero of Alexandria and Damianus deserve more recognition?

2. How would you explain the law of refraction using Fermat’s principle of least time?

3. The idea of an infinitesimal change of first order plays a key role in Fermat’s principle. How would you interpret this mathematically and conceptually?

 

Key Takeaway: We can derive optical laws (such as the laws of reflection and refraction) by analyzing how the first-order change in travel time vanishes (stationary point) for different possible paths of light.

 

The Moral of the Lesson:

The Taoist concept of Wúwéi, often translated as "non-action" or "effortless action," does not imply passivity or doing nothing. Rather, it is the art of achieving maximum effect with minimal, economical, or natural effort—of moving in harmony with the inherent patterns of nature, like following the path of least resistance. A wise president who practices Wúwéi governs not through forceful control, but by aligning with the natural order—just as a physical system follows the principle of stationary action without the burden of unnecessary constraints.

 

References:

Bhaduri, B., Yessenov, M., & Abouraddy, A. F. (2020). Anomalous refraction of optical spacetime wave packets. Nature Photonics, 14, 416-421.

Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

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.

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

Mach, E. (1953). The Principles of Physical Optics: An Historical and Philosophical Treatment. New York: Dover.

Zengel, K. (2024). Why the action?. American Journal of Physics, 92(11), 885-888.

Section 26–2 Reflection and refraction

 (Law of reflection / Law of refraction / Ptolemy’s experimental results)

 

In this section, Feynman discusses the law of reflection, the law of refraction, and Ptolemy’s experimental results.

 

1. Law of reflection:

“For some reason it is customary to measure the angles from the normal to the mirror surface. Thus the so-called law of reflection is θ_i = θ_r (Feynman et al., 1963, section 26–2 Reflection and refraction).”

 

Feynman states the law of reflection simply using the equation θ_i = θ_r. In Volume II, Feynman adds that “[t]he angle of reflection is equal to the angle of incidence. With the angles defined as shown in Fig. 33-1, θ_r = θ_i (Feynman et al., 1964, section 33–1 Reflection and refraction of light).” Some authors prefer the term “plane mirror reflection” and include the second law of reflection: “the incident ray, the reflected ray, and the normal at the point of incidence all lie in the same plane.” To be precise, we can have a vectorial law of reflection using the cross product and dot product. For example, the incident ray, reflected ray, and normal can be represented by i = u_ix_i + u_iy_j + u_iz_k, r = u_ix_i + u_iy_j − u_iz_k and k respectively. The triple product of these three vectors (i × r).k is zero because they all lie in the same plane (Tkaczyk, 2012).

 

Feynman explains that when the light hits the mirror, it does not continue in a straight line, but bounces off the mirror into a new straight line. It is analogous to the collision of an elastic ball on a hard surface (neglecting the influence of gravity). One may clarify that the law of reflection is an empirical law, but it can be proved by using Maxwell’s equations. Historically, the first law of reflection, “the angle of incidence is equal to the angle of reflection,” appeared in Euclid’s Catoptrics. Euclid is also one of the very intelligent men that has proposed “something comes out of the eye” (mentioned in the last section).

 

2. Law of refraction:

“…θ_i is the angle in air and θ_r is the angle in the water, then it turns out that the sine of θ_i is equal to some constant multiple of the sine of θ_r: sin θ_i = n sin θ_r ... Equation (26.2) is called Snell’s law (Feynman et al., 1963, section 26–2 Reflection and refraction).”

 

Feynman states Snell’s law as “if θ_i is the angle in air and θ_r is the angle in the water, then it turns out that the sine of θ_i is equal to some constant multiple of the sine of θ_r: sin θ_i = n sin θ_r. In Volume II, he makes a revision of the law to “[t]he product n sin θ is the same for the incident and transmitted beams (Snell’s law): n₁sin θ_i = n₂sin θ_t (Feynman et al., 1964, section 33–1 Reflection and refraction of light).” Some authors prefer to include the second law of refraction: “the incident ray, refracted ray, and the normal to the interface of two media at the point of incidence all lie on the same plane.” Better still, one may express a vector form of Snell’s law that encompasses the two laws of refraction. For small values of i (in radians), Snell’s law can be approximated by using the equation n i = n¢ i¢.

 

Feynman says that the rule (Snell’s law) was found by Willebrord Snell, a Dutch mathematician. Specifically, Snell used the cosecant of the angles instead of the sine of the angles in his law of refraction (Smith, 1987). Historically, Descartes was the first to publish the law of refraction using the sine of the angles in 1637 and he was initially regarded as the originator of this sine law. On the other hand, some have argued that the first discovery of the sine law was made by the sixteenth-century English scientist Thomas Harriot (Dudley & Kwan, 1997). In addition, the Persian scientist Ibn Sahl applied his law of refraction to demonstrate the optical properties of hyperbolic lenses (Mark, 1999). Thus, it is debatable who was the first to discover the law of refraction.

 

3. Ptolemy’s experimental results:

“Claudius Ptolemy made a list of the angle in water for each of a number of different angles in air (Feynman et al., 1963, section 26–2 Reflection and refraction).”

 

Feynman presents Table 26–1 to show the angles in the air, the corresponding angle measured in the water, and concludes that there is a remarkable agreement with Snell’s law. Based on Ptolemy’s experimental results, one may deduce Snell’s law to be governed by the quadratic equation r = ai – bi² or r = R – (n²d₂ – nd₂)/2 where i is the angle of incidence, r is the angle of refraction, and n is the refractive index (Smith, 1996). That is, it is more difficult to deduce Snell’s sine law of refraction using Ptolemy’s results that fit a parabolic curve. Furthermore, it was also difficult for Ptolemy to formulate the quadratic equation because of the limitations of the Euclidean theory of proportionality (Smith, 1996). Thus, we should not conclude that the numbers based on Snell’s law are in a remarkable agreement with Ptolemy’s results.

 

Surprisingly, Feynman includes the following statement: “it should be noted, however, that these do not represent independent careful measurements for each angle but only some numbers interpolated from a few measurements, for they all fit perfectly on a parabola.” This statement presents Ptolemy’s results in Table 26–1 positively, but the numbers were manipulated to fit the parabolic curve. Importantly, Feynman criticizes the manipulation of experimental results in a talk titled Cargo Cult Science. In Feynman’s words, “[w]hen they got a number closer to Millikan's value they didn’t look so hard. And so they eliminated the numbers that were too far off, and did other things like that (p. 342).” Perhaps Feynman should explain how the manipulation of numbers could make it impossible to discover Snell’s law.

Review Questions:

1. How would you state the law(s) of reflection?

2. Was the law of refraction first discovered by Willebrord Snell?

3. Should Feynman conclude that the numbers based on Snell’s law are in a remarkable agreement with Ptolemy’s list?

 

The moral of the lesson: the adjusted Ptolemy’s experimental results (Table 26–1) are slightly different from the numbers based on Snell’s law (Table 26–2), and thus, it could have posed difficulties to discover Snell’s sine law of refraction.

 

References:

1. Dudley, J. M., & Kwan, A. M. (1997). Snell’s law or Harriot's?. Physics Teacher, 35(3), 158-159.

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. Smith, A. M. (1987). Descartes's theory of light and refraction: a discourse on method. Transactions of the American Philosophical Society, 77(3), 1-92.

5. Smith, A. M. (1996). Ptolemy's theory of visual perception: an English translation of the Optics (Vol. 82). Philadelphia: American Philosophical Society.

6. Smith, A. M. (1999). Ptolemy and the foundations of ancient mathematical optics: A source based guided study. Transactions of the American Philosophical Society, 89(3), 1-172.

7. Tkaczyk, E. R. (2012). Vectorial laws of refraction and reflection using the cross product and dot product. Optics Letters, 37(5), 972-974.

Section 26–1 Light

 (Geometrical optics / Wave theory / Quantum theory)

 

In this section, Feynman discusses the properties of light that are from the perspectives of geometrical optics, wave theory, and quantum theory.

 

1. Geometrical optics:

“… a condition exists in which the wavelengths involved are very small compared with the dimensions of the equipment … we can make a rough first approximation by a method called geometrical optics (Feynman et al., 1963, section 26–1 Light).”

 

According to Feynman, we can apply geometrical optics if the wavelengths are very small compared with the dimensions of an equipment and the photon energies are small compared with the energy sensitivity of the equipment. That is, light is idealized as a stream of particles (or rays) in which the wavelength l ® 0 and one may clarify that the equipment could be a curved mirror or a lens. In Chapter 27, Feynman mentions that “[t]he most advanced and abstract theory of geometrical optics was worked out by Hamilton … (Feynman et al., 1963, section 27–1 Introduction).” In short, Hamilton intended to show that all problems of geometric optics can be solved by only one method. In a sense, Hamilton has also contributed to wave optics and quantum optics because Schrödinger applied the Hamiltonian principle by passing from geometrical optics to wave optics, and developed the Schrödinger’s equation.

 

Feynman says that he does not even bother to say what the light is, but just find out how it behaves on a large scale compared with the dimensions of interest. In general, one may describe light as a photon, electromagnetic waves, or quantum field. Perhaps Feynman could have stated the behavior of light using three empirical observations (or idealized rules): 1. Straight lines: Light rays travel in straight lines in free space. 2. Law of reflection: the angle of incidence is equal to the angle of reflection. 3. Law of refraction: the ratio of sine of angle of incidence to the sine of angle of refraction for two refractive media is a constant. 

 

2. Wave theory:

“… the wavelengths are comparable to the dimensions of the equipment, which is difficult to arrange with visible light but easier with radiowaves … This method is based on the classical theory of electromagnetic radiation (Feynman et al., 1963, section 26–1 Light).”

 

In the second example, Feynman says that the classical theory of electromagnetic radiation is used to study the wavelengths (e.g., radio waves) that are comparable to the dimensions of the equipment. In this case, he disregards quantum mechanics because the photon energies are negligibly small. As a suggestion, one should explain that the equipment may be a narrow slit that is shorter than the wavelengths of radio waves. Importantly, the diffraction of light through the narrow slit cannot be explained by geometrical optics. For relatively longer wavelengths, we can apply wave optics to study wave properties of light in phenomena such as diffraction, interference, and polarization of light for which the ray approximation is not valid. 

According to Feynman, the light goes from one place to another in straight lines are based on observations, and the rays do not seem to interfere with one another. In other words, light is crisscrossing in all directions, but the light that is passing across our line of vision does not affect the light that comes to us from some objects. Specifically, Huygens (1690) uses his wave theory of light to explain how it is possible for two persons mutually seeing one another’s eyes. He did not only use this argument to refute the corpuscular theory of light, but shows that the speed of light is slower in a denser medium. However, it is incorrect to say that the light does not interfere with each other because we can see 3D hologram that is due to the interference of light rays (or lasers).

 

3. Quantum theory:

“… furthermore, the photon energies, using the quantum theory, are small compared with the energy sensitivity of the equipment (Feynman et al., 1963, section 26–1 Light).”

In the third example, Feynman discusses very short wavelengths, where we can disregard the wave properties of photons. We can use the quantum theory when the photons have very high energy compared with the energy sensitivity of the equipment. This condition is unclear because he did not state a possible range of wavelengths and the equipment. Feynman could have specified the type of electromagnetic wave such as ultraviolet radiation or x-rays that has very short wavelengths as compared to the dimensions of the equipment. Furthermore, one may explain how the photon energies are higher as compared with the energy sensitivity of the equipment using the formula E = hc/l. It is related to Quantum Optics that uses quantum physics to study submicroscopic phenomena involving light.

 

Feynman did not elaborate on the nature of photons, but only gives a rough picture. During a lecture titled QED, Feynman (1985) says that “[t]he first important feature about light is that it appears to be particles… (p. 36).” Currently, the word photon has different meanings, for example, Hentschel (2018) identifies six photon models: 1. the corpuscular model, 2. the singularity model, 3. the binary model of photons, 4. wave packet model, 5. the semiclassical model and, 6. QED model. According to QED, “photons are quantized states of the electromagnetic field whose energy generally belongs to the whole region of space occupied by the radiation field (Hentschel, 2018, p. 174).” Interestingly, Glauber (his Nobel prize is related to quantum optics) says that “I don’t know anything about photons, but I know one when I see one.”

 

Review Questions:

1. How would you explain the condition in which the wavelengths involved are very small compared with the dimensions of the equipment?

2. How would you explain the condition in which the wavelengths are comparable to the dimensions of the equipment and the photon energies are still negligibly small?

3. How would you explain the condition in which the wavelengths are very short and the photons have a very high energy compared with the sensitivity of the equipment?

 

The moral of the lesson: we may apply geometrical optics, waves optics, or quantum optics depending on the wavelengths of electromagnetic waves and their energy.

 

References:

1. Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

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. Hentschel, K. (2018). Photons: The History and Mental Models of Light Quanta. Cham: Springer.

4. Huygens, C. (1690). Traité de la Lumière. Leiden: Pieter van der Aa.

Section 25–5 Series and parallel impedances

 (Series impedance / Parallel impedance / Kirchhoff’s laws)

 

In this section, Feynman discusses how to calculate the series impedance and parallel impedance of alternating current (AC) circuits as well as Kirchhoff’s circuital laws.

 

1. Series impedance:

“This means that the voltage on the complete circuit can be written V = IZ_s, where the Z_s of the combined system in series is the sum of the two Z’s of the separate pieces: Zs = Z₁ + Z₂ (Feynman et al., 1963, section 25–5 Series and parallel impedances).”

 

Feynman asks what happens if we put two circuit elements in series that have impedance Z₁ and Z₂ by applying a voltage. Then, he shows that the total voltage is the sum of the voltages across the two circuit elements and it is equal to V = V₁ + V₂ = (Z₁ + Z₂)I. However, there is a problem of sequence because Feynman could have first proved the Kirchhoff’s circuital laws. Alternatively, he could first explain why the current is the same in the circuit elements and apply the law of conservation of energy for circuit elements that are connected in series. To have a better understanding, one may distinguish reactance and impedance as follows: reactance of a capacitor or inductor is like the resistance that opposes the AC current, whereas impedance of an AC circuit is the vector sum of the net reactance and the resistance in the circuit.

 

Feynman explains that the voltage on the complete circuit can be written as V = IZ_s, where Z_s is the effective impedance of the combined system in series. That is, the effective impedance of two circuit elements is the sum of the two Z’s: Z_s = Z₁ + Z₂. Perhaps one may provide a definition of the impedance of an AC circuit as a measure of the total opposition to the current that is analogous to the resistance by using the equation Z_s = V/I. It is beneficial to clarify that although impedance is a complex number, but it is not the same as V or I that is a phasor. One may represent the phasor using a complex number and explain how it behaves like a vector that can rotate continuously. In short, all phasors are complex numbers (or a kind of sinusoidal vectors), but not all complex numbers are phasors.

 

2. Parallel impedance:

“This can be written as V = I/[(1/Z₁) + (1/Z₂)] = I/Z_p. Thus, 1/Z_p = 1/Z₁ + 1/Z₂ (Feynman et al., 1963, section 25–5 Series and parallel impedances).”

 

According to Feynman, if the connecting wires are perfect conductors and a constant voltage is effectively applied to both of the impedances, it will cause currents in each independently. He shows that the current in Z₁ is equal to I₁ = V/Z₁ and the current in Z₂ is I₂ = V/Z₂. Alternatively, one may explain why the voltage is the same and apply the law of conservation of charge for circuit elements that are connected in parallel. Note that Kirchhoff’s current law is not really a fundamental principle, but it is a consequence of the law of conservation of charge. When we apply this law in DC circuit, we assume the ideal condition of steady-state current (or constant current).

 

Feynman states that the total current which is supplied to the terminals is the sum of the currents in the two sections: I = V/Z₁ + V/Z₂. In addition, he says that “it is the same voltage” without further explanations. The phrase “same voltage” could be unclear because it is about the “same AC voltage” that is varying sinusoidally. Kirchhoff’s circuit laws are intuitively correct and applicable to DC voltage, but it can be generalized to AC voltage provided both AC voltage and AC current are of the same frequency. Essentially, we have idealized the circuit elements only interact with each other through connecting wires. Specifically, we have assumed that there is no induced current and the wires are not capacitively coupled when the circuit devices are connected in parallel.

 

3. Kirchhoff’s laws:

“These rules are called Kirchhoff’s laws for electrical circuits. Their systematic application to complicated circuits often simplifies the analysis of such circuits (Feynman et al., 1963, section 25–5 Series and parallel impedances).”

 

Feynman ends the chapter by stating Kirchhoff’s laws for electrical circuits. Kirchhoff’s current law and voltage law are also known as Kirchhoff’s junction rule (or nodal rule) and loop rule (or mesh rule) respectively. That is, they are rules that involve idealizations and approximations. In Volume II, Feynman elaborates that “one of our idealizations has been that negligible electrical charges accumulate on the terminals of the impedances. We now assume further that any electrical charges on the wires joining terminals can also be neglected… (Feynman et al., 1964, section 22–3 Networks of ideal elements; Kirchhoff’s rules).”

 

Feynman states a condition for Kirchhoff’s law: the voltages in little generators have no impedance. This implies that the voltage of a generator is approximately equal to the electromotive force provided its internal resistance is negligible and thus, there is no loss of electrical energy in the generator. Feynman adds that the systematic application of Kirchhoff’s laws to complicated circuits can simplify the analysis of such circuits. Interestingly, we can apply Kirchhoff’s laws for mechanical elements: 1. Kirchhoff’s force law means that “the algebraic sum of all the forces acting on any junction of mechanical elements is zero (Firestone, 1933, pp. 254-255).” 2. Kirchhoff’s velocity law means that “the algebraic sum of the velocity differences around any closed mechanical circuit is zero (Firestone, 1933, p. 255).”

 

Questions for discussion:

1. Would you derive the formula for two circuit elements in series without using Kirchhoff’s voltage law?

2. Would you derive the formula for two circuit elements in parallel without using Kirchhoff’s current law? 

3. How would you state Kirchhoff’s laws for electrical circuits?

 

The moral of the lesson: we can apply Kirchhoff’s circuital laws to determine the effective impedance of any circuit, however, we have essentially used Kirchhoff’s laws to derive the formula for series impedance and parallel impedance.

 

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.

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