Mass of a photon (zero or non-zero?)

Question: What is the mass of a photon?

The answer to this question is dependent on your definition of mass. If your definition of mass is based on the concept of rest mass or invariant mass, then the mass of a photon is zero. Hence, photons are sometimes said to be massless. However, there is no experiment that can establish the photon’s rest mass to be exactly zero. Experimental physicists can only place limits on it. On the other hand, if your definition of mass is based on the concept of relativistic mass or effective mass, then the mass of a photon is dependent on the energy it possesses. According to Einstein’s principle of mass-energy equivalence, the mass of the photon is equivalent to its energy. Thus, its mass can be calculated by using the equation, m = E/c². If the energy of the photon is hf, then its mass is hf/c².

As another alternative, one may include the concept of Meissner mass (non-zero photon mass). This is related to the Meissner effect in which magnetic fields penetrate a finite distance into a superconductor. For example, Wilczek (2005) explains that “[a]n unusual but valid way of speaking about the phenomenon of superconductivity is to say that within a superconductor the photon acquires a mass. The Meissner effect follows from this. Indeed, to say that the photon acquires a mass is to say that the electromagnetic field becomes a massive field. Because the energetic cost of supporting massive fields over an extended volume is prohibitive, a superconducting material finds ways to expel magnetic fields (p. 241).”

How would Feynman answer?

Feynman may answer this question from the perspectives of rest mass and relativistic mass, as well as discuss problems of defining photon’s mass as shown below.

1. Rest mass of a photon:

Based on the concept of rest mass, Feynman mentions that “The masses given here are the masses of the particles at rest. The fact that a particle has zero mass means, in a way, that it cannot be at rest. A photon is never at rest, it is always moving at 186,000 miles a second (Feynman et al. 1963, section 2–4 Nuclei and particles).” This concept of mass can be mathematically represented by m₀, and its value is Lorentz invariant. In other words, the rest mass of a photon does not change with the inertial frame of reference of an observer.

2. Relativistic mass of a photon:

Feynman may also provide an answer based on the concept of relativistic mass or Einstein’s principle of mass-energy equivalence. In Volume I of The Feynman Lectures on Physics, he explains that “[i]n the Einstein relativity theory, anything which has energy has mass — mass in the sense that it is attracted gravitationally. Even light, which has an energy, has a “mass.” When a light beam, which has energy in it, comes past the sun there is an attraction on it by the sun. Thus the light does not go straight, but is deflected. During the eclipse of the sun, for example, the stars which are around the sun should appear displaced from where they would be if the sun were not there, and this has been observed (Feynman et al. 1963, section 7–8 Gravity and relativity).” However, the relativistic mass is dependent on the inertial frame of reference of an observer.

In Volume II of The Feynman Lectures on Physics, Feynman elaborates that “[a] photon of frequency ω₀ has the energy E₀ = ℏω₀. Since the energy E₀ has the relativistic mass E₀/c² the photon has a mass (not rest mass) ℏω₀/c², and is ‘attracted’ by the earth. (Feynman et al. 1964, section 42–6 The speed of clocks in a gravitational field).” Although the speed of a photon is constant, the photon’s frequency may vary with the inertial frame of reference of the observer. That is, the relativistic mass of the photon may be increased or decreased and this can be explained by using Doppler’s effect.

3. Problems of defining a photon’s mass:

Feynman might discuss problems of defining (or determining) a photon’s mass by sharing his discussion with a physicist in Paris: “In this connection, I would like to relate an anecdote, something from a conversation after a cocktail party in Paris some years ago. There was a time at which all the ladies mysteriously disappeared, and I was left facing a famous professor, solemnly seated in an armchair, surrounded by his students. He asked, ‘Tell me, Professor Feynman, how sure are you that the photon has no rest mass?’ I answered ‘Well, it depends on the mass; evidently if the mass is infinitesimally small, so that it would have no effect whatsoever, I could not disprove its existence, but I would be glad to discuss the possibility that the mass is not of a certain definite size. The condition is that after I give you arguments against such mass, it should be against the rules to change the mass.’ The professor then chose a mass of 10^-6 of an electron mass.

My answer was that, if we agreed that the mass of the photon was related to the frequency as ω = (k² + m²)^1/2, photons of different wavelengths would travel with different velocities. Then in observing an eclipsing double star, which was sufficiently far away, we would observe the eclipse in blue light and red light at different times. Since nothing like this is observed, we can put an upper limit on the mass, which, if you do the numbers, turns out to be of the order of 10^-9 electron masses. The answer was translated to the professor. Then he wanted to know what I would have said if he had said 10^-12 electron masses. The translating student was embarrassed by the question, and I protested that this was against the rules, but I agreed to try again.

If the photons have a small mass, equal for all photons, larger fractional differences from the massless behavior are expected as the wavelength gets longer. So that from the sharpness of the known reflection of pulses in radar, we can put an upper limit to the photon mass which is somewhat better than from an eclipsing double star argument. It turns out that the mass had to be smaller than 10^-15 electron masses. After this, the professor wanted to change the mass again, and make it 10^-18 electron masses. The students all became rather uneasy at this question, and I protested that, if he kept breaking the rules, and making the mass smaller and smaller, evidently I would be unable to make an argument at some point (Feynman et al., 1995, pp. 22-23).”

       To conclude, the rest mass of a photon is zero, whereas its relativistic mass is dependent on the photon’s frequency and the observer’s inertial frame of reference.

References:

1. Feynman, R. P., Morinigo, F. B., & Wagner, W. G. (1995). Feynman Lectures on gravitation (B. Hatfield, ed.). Reading, MA: Addison-Wesley.

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. Wilczek, F. (2005). In search of symmetry lost. Nature, 433(7023), 239-247.

About the author

I am a fan of Feynman, with over ten years of experience teaching introductory physics. I don’t really like travelling, but I visited the following countries/places: Australia (Perth), Austria (Salzburg, Vienna), Bosnia, Brunei, China (Guangzhou, Hong Kong, Macau), Croatia (Split), Egypt (Cairo, Mount Sinai), France (Toulouse, Marseille, Nice, Paris), Germany (Frankfurt), Indonesia (Batam, Jakarta, Pulau Bintan), Israel (Mount Carmel, Golan Heights, Jerusalem), Italy (Assisi, Milan, Rome, Venice), Japan (Tokyo), Malaysia (Kenyir Lake, Kuala Lumpur, Malacca, Pulau Redang, Pulau Pemanggil), Portugal, South Korea (Gwangju, Mokpo, Seoul), Spain (Barcelona), Switzerland (Bern, Interlaken, Jungfrau), Taiwan (Taipei, Mount Alishan, Kaohsiung), Thailand, United Kingdom (London), United States (Hawaii, Pittsburgh), Vatican, and Vietnam (Ho Chi Minh, Quảng Trị).

My blogs:
Meditating Feynman’s Lectures
How would Feynman answer examination questions?

Problems of physics assessment

Selected Publications:

1. Wong, C. L., Chu, H. E., & Yap, K. C. (2016). Are Alternative Conceptions dependent on Researchers’ Methodology and Definition? : A Review of Empirical Studies related to Concepts of Heat. International Journal of Science and Mathematics Education. 14(3), 499-526.

2. Wong, C. L., Chu, H. E., & Yap, K. C. (2014). Developing a Framework for analyzing definitions: A Study of The Feynman Lectures. International Journal of Science Education. 36(15), 2481-2513.

3. Wong, C. L. (2013). Is there a best definition of weight? The Physics Teacher, 51(9), 516-7.

4. Lim, K. M., Lee, G. C. F., Sheppard, C. J. R., Phang, J. C. H., Wong, C. L., & Chen, X. (2011). Effect of polarization on a solid immersion lens of arbitrary thickness. Journal of the Optical Society of America A, 28(5), 903-911.

5. Yap, K. C., & Wong, C. L. (2007). Assessing conceptual learning from quantitative problem solving of a plane mirror problem. Physics Education, 42(1), 50-55.

Speed and velocity

Question: What is the difference between speed and velocity?

In some physics textbooks, speed is defined as the rate of change of distance with time, whereas velocity is the rate of change of displacement with time. In addition, speed is a scalar quantity and velocity is a vector quantity. Simply put, the difference between these two physical quantities is that speed does not have a direction, whereas velocity has a direction. However, the difference is also related to the concepts of distance and displacement.

To be more precise, a theoretical definition of speed is the rate of change of distance traveled by an object with respect to time in an inertial frame of reference. Similarly, velocity is the rate of change of displacement traveled by an object with respect to time in an inertial frame of reference. Essentially, the speed and velocity of the object are dependent on an observer’s reference frame. Moreover, an operational definition of speed is “what the speedometer measure.” In other words, the measured speed and velocity of the object are dependent on the type of speedometer used.

How would Feynman answer?

Feynman would provide a definition of speed, a definition of velocity, and explain the difference between speed and velocity as shown below.

1. A definition of speed:

In Feynman’s own words, “[m]any physicists think that measurement is the only definition of anything. Obviously, then, we should use the instrument that measures the speed -- the speedometer (Feynman et al., 1963, section 8–2 Speed).” In short, the measured speed of an object is dependent on the speedometer used as well as the experimental operations. This is related to operationalism (a kind of philosophy) which means that “the concept is synonymous with the corresponding set of operations’ (Bridgman 1927, p. 5).” Thus, a theoretical concept may be considered meaningless if it cannot be measured. Based on the same philosophy, some physicists prefer to define weight as “what the weighing scale measure” instead of “gravitational force on an object.”

In addition, Feynman elaborates that “we can define the speed in this way: We ask, how far do we go in a very short time? We divide the distance by the time, and that gives the speed. But the time should be made as short as possible, the shorter the better, because some change could take place during that time (Feynman et al., 1963, section 8–2 Speed).” That is, the speed of an object is a ratio of distance moved to the time interval measured. Importantly, the speed of the object is dependent on the measurement procedure such as how the time interval is measured. For example, an experimenter may measure the total distance traveled by the object and the time elapsed in one hour or in one second. Interestingly, Feynman also distinguishes the meaning of 88 feet per second and 60 miles per hour.

2. A definition of velocity:

Feynman also provides a mathematical definition of velocity: “[l]et us try to define velocity a little better. Suppose that in a short time, ϵ, the car or other body goes a short distance x; then the velocity, v, is defined as v = x/ϵ, an approximation that becomes better and better as the ϵ is taken smaller and smaller (Feynman et al., 1963, section 8–2 Speed). This definition of velocity is based on the ratio of an infinitesimal distance to the corresponding infinitesimal time. Theoretically speaking, we imagine what happens to that ratio as the time we use is shorter and shorter. In other words, we take a limit of the distance traveled divided by the time elapsed, as the time taken is assumed to be shorter and shorter, ad infinitum. This idea was independently invented by Newton and Leibnitz and it is now known as calculus.

Alternatively, Feynman mentions that “we have another law that the velocity is equal to the integral of the acceleration. This is just the opposite of a = dv/dt; we have already seen that distance is the integral of the velocity, so distance can be found by twice integrating the acceleration (Feynman et al., 1963, section 8–5 Acceleration).” That is, the velocity of an object can be determined not only by differentiation, but it can be calculated by using integration or summing the total area under a curve. Moreover, the velocity of the object can be mathematically represented as a two-dimensional quantity or three-dimensional quantity. For instance, we can represent the velocity as v = ds/dt = √(v_x² + v_y²).

3. The Difference between speed and velocity:

Feynman clarifies that “[o]rdinarily we think of speed and velocity as being the same, and in ordinary language they are the same. But in physics, we have taken advantage of the fact that there are two words and have chosen to use them to distinguish two ideas. We carefully distinguish velocity, which has both magnitude and direction, from speed, which we choose to mean the magnitude of the velocity, but which does not include the direction (Feynman et al., 1963, section 9–2 Speed and velocity).” In short, velocity is speed in a specified direction. Thus, we can also define velocity by describing how the x-, y-, and z-coordinates of an object change with time, as well as write v_x = Δx/Δt, v_y = Δy/Δt and v_z = Δz/Δt. (Mathematicians may disagree with physicists’ interpretation of notations such as v = dx/dt or v = Δx/Δt.)

On the other hand, there are problems in defining speed as well as determining speed accurately. For example, Feynman mentions that a speedometer may be spoilt, however, the speedometer has inherent uncertainty depending on the technologies used. In general, there are different kinds of speedometer such as an electronic speedometer, Doppler radar, and Global Positioning System (GPS) device. The measurement uncertainty of a car’s electronic speedometer is dependent on the interaction between a precision watch mechanism and a mechanical pulsator driven by the car’s wheel. The uncertainty of a Doppler traffic radar is dependent on a car’s direction in moving and the wavelength of the radar waves generated. The positional accuracy of a GPS device is dependent on the satellite signal quality and the position averaging software used by GPS to reduce errors.

References:

1. Bridgman, P. W. (1927). The Logic of Modern Physics. New York: Macmillan.

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.

Why does a wave on a string invert after reflection?

Question: Why does a wave pulse on a string invert after it is reflected from a rigid boundary?

Below is a conventional explanation on this question: We may assume an end of a string is rigidly clamped to a wall and a wave pulse on a string moves from left to right towards this end which is fixed. As the wave pulse approaches the fixed end, the string exerts an upward force on the wall or rigid boundary. According to Newton's third law of motion, the wall exerts an equal downward (or restoring) force on the end of the string. This restoring force generates an inverted wave pulse that propagates from right to left, having the same wave speed and amplitude as the incident wave. In addition, the displacement of the wave remains zero at the rigid boundary and the there is a phase change of 180^o. (To have a more complete understanding of the phenomenon, some textbook authors include explanations on a string that has a “free end.”) 

In general, we can derive a wave equation of a vibrating string that requires a small segment of the string obeying Newton’s second law of motion. However, we need more information to specify the complete movement of the vibrating string. That is, it is necessary to know initial conditions (position and velocity) of the string at a particular time and boundary conditions at a particular location on the string. The boundary condition may refer to the displacement of the string that is zero at the rigid boundary or the slope of the string that is zero if the end of the string is attached to a frictionless boundary. 

Alternatively, Pierce (2006) explains the inverted wave pulse by using a voltage wave. Imagine a voltage wave is traveling down a transmission line. If the transmission line is open at one end, the reflected voltage wave will be the negative of the incident voltage wave. If the transmission line is shorted at the point of reflection, the voltage is zero at that point instead. In other words, during the process of reflection, the incident voltage wave plus the reflected voltage wave must always be zero. Hence, we may infer that the reflected voltage wave must be the negative of the incident voltage wave during the process of reflection. 

How would Feynman answer?: 

Feynman did not provide an intuitive explanation by using Newton’s Third law of motion. In The Feynman Lectures on Physics, he has provided a mathematical explanation:  “Suppose that the string is held at one end, for example by fastening it to an “infinitely solid” wall. This can be expressed mathematically by saying that the displacement y of the string at the position x = 0 must be zero because the end does not move… we know that the general solution for the motion is the sum of two functions, F(x − ct) and G(x + ct)… (Feynman et al., 1963, section 49–1 The reflection of waves).” Note that the mathematical expression F(x − ct) represents a wave traveling in the string to the right at the speed c and G(x + ct) represents another wave traveling to the left at the speed c. Thus, the displacement y of the string can be expressed as F(x − ct) + G(x + ct) by using the principle of superposition. This physical principle is applicable to the vibrating string because it is a linear system.

We should not assume that Feynman has the attitude of “shut up and calculate” or he simply believes in the unreasonable effectiveness of mathematics. More important, Feynman would have a concern on the different interpretations of mathematics. According to Feynman, “[t]he next great awakening of human intellect may well produce a method of understanding the qualitative content of equations. Today we cannot. Today, we cannot see whether Schrödinger’s equation contains frogs, musical composers, or morality - or whether it does not. We cannot say whether something beyond it like God is needed, or not. And so we can all hold strong opinions either way (Feynman et al., 1964, p. 41-12).” It is possible to have different philosophical perspectives for a mathematical equation that is applied in the physical world. The “shut up and calculate” attitude could be attributed to Mermin (2004) instead of Feynman.

Interestingly, Feynman elaborates that we can imagine a hypothetical wave traveling in the opposite direction that is behind the wall. In his own words, “We say hypothetical because, of course, there is no string to vibrate on that side of the origin. The total motion of the string is to be regarded as the sum of these two waves in the region of positive x. As they reach the origin, they will always cancel at x = 0, and finally, the second (reflected) wave will be the only one to exist for positive x and it will, of course, be traveling in the opposite direction… (Feynman et al., 1963, section 49–1 The reflection of waves).” That is, we can understand the reflected wave by imagining an inverted wave that comes out from behind the wall. In short, we may assume that the string is connected to an infinitely massive string at x = 0. This explains the boundary condition in which the displacement of the string at x = 0 must always be zero.

Note: 

Feynman might mention problems of defining waves. During a BBC interview, Feynman (1994) explains that “[i]f I'm sitting next to a swimming pool, and somebody dives in, and she's not too pretty, then I can think about something else. I like to think about the waves that are formed in the water, and when lots of people have dived into the pool, there's a very great choppiness of all these waves all over the surface. Now to think that it's possible, maybe, that in those waves there's a clue as to what's happening in the pool: that an insect of sufficient cleverness could sit in the corner of the pool, and just by being disturbed by the waves and by the nature of the irregularities, the insect could figure out who jumped in where, and when, and what's happening all over the pool. It seems incredible, but that’s what we're doing when we looking at something… (p. 130).” 

References: 
1. Feynman, R. P. (1994). No Ordinary Genius: The Illustrated Richard Feynman. New York: W. W. Norton & Company. 

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. Mermin, N. D. (2004). Could Feynman have said this. Physics Today, 57(5), 10.

5. Pierce, J. R. (2006). Almost all about waves. New York: Dover.

Refractive index (Absolute speed or effective speed?)

Question: What is the refractive index of a medium?

A student in Singapore complained the following examination question: 

What is the refractive index of a medium? 

A. the ratio of the speed of light in air to the speed of light in the medium. 

B. the ratio of the speed of light in the medium to the speed of light in air. 

C. the ratio of the speed of light in the medium to the speed of light in vacuum. 

D. the ratio of the speed of light in vacuum to the speed of light in the medium. 

The student chose the answer “A,” but his physics teacher told him that the answer should be “D.” However, this does not seem reasonable to him because the recommended textbook states that “the refractive index is a ratio between the speed of light in air or vacuum and the speed of light in a medium (Chew & Chow, 2007, p. 231).” Subsequently, this led to complaints that the textbook has errors and the ministry of education has recommended the wrong physics textbook. Alternatively, we can define absolute index of refraction as a ratio of the speed of light in a vacuum to the speed of light in a medium. Thus, this is also a problem of terminology and thus, the examination question should be phrased more carefully.

How would Feynman answer?

An ability to state the refractive index as a ratio of the speed of light in vacuum and the speed of light in a medium does not imply a genuine understanding of the concept. In short, Feynman would explain that the speed of light waves in a medium remains constant, but it is the effective speed of the light waves that is decreased. We should have some knowledge of Feynman’s answer from the perspective of wave mechanics, electromagnetism, quantum mechanics, and problems of defining index of refraction as shown below.

1. Effective speed of light waves: Strictly speaking, it is imprecise and incomplete to state that the speed of light is reduced in a transparent material or medium. In general, there are different kinds of speed (or velocity) of light waves such as phase velocity, group velocity, and effective speed. In Feynman’s words, “[i]t is approximately true that light or any electrical wave does appear to travel at the speed c/n through a material whose index of refraction is n, but the fields are still produced by the motions of all the charges — including the charges moving in the material — and with these basic contributions of the field travelling at the ultimate velocity c. Our problem is to understand how the apparently slower velocity comes about (Feynman et al., 1963, section 31–1 The index of refraction).” Simply phrased, light or light waves still travel at the same ultimate speed of light, c, instead of c/n. However, the constant speed of light waves appears to slow down as a result of moving charges in the material. 

Importantly, there should be a deeper understanding of the meaning of effective speed (or apparent speed) of light. We should distinguish the term absolute speed and effective speed for the speed of light in the material. Feynman explains that “[b]efore we proceed with our study of how the index of refraction comes about, we should understand that all that is required to understand refraction is to understand why the apparent wave velocity is different in different materials. The bending of light rays comes about just because the effective speed of the waves is different in the materials (Feynman et al., 1963, section 31–1 The index of refraction).” Although the bending of light is sometimes explained by the principle of least time, it is also important to understand the concept of absolute speed of light in different materials. Better still, we should emphasize that the refractive index is related to the effective speed of light in the material or medium.

Furthermore, the effective or apparent speed of light can be explained by the phase shift of light waves. Feynman elaborates that “[i]n spite of the fact that it is said that you cannot send signals any faster than the speed of light, it is nevertheless true that the index of refraction of materials at a particular frequency can be either greater or less than 1. This just means that the phase shift which is produced by the scattered light can be either positive or negative (Feynman et al., 1963, section 31–3 Dispersion).” Note that the refractive index of a material is related to the phase velocity or speed of nodes of the waves. Moreover, there is a phase difference between an incident light wave and the emitted light wave generated by an atom. If the phase of the emitted light wave is delayed, the effective speed of light is slowed down.

2. Theories of refractive index:

The concept of refractive index can be explained by wave mechanics, electromagnetism, and quantum mechanics. Generally speaking, the oscillation of atoms in a medium can be modeled by using wave mechanics. Feynman mentions that “[y]ou may think that this is a funny model of an atom if you have heard about electrons whirling around in orbits. But that is just an oversimplified picture. The correct picture of an atom, which is given by the theory of wave mechanics, says that, so far as problems involving light are concerned, the electrons behave as though they were held by springs (Feynman et al., 1963, section 31–2 The field due to the material).” In a sense, we can idealize the electrons as tiny oscillators with a resonant frequency and having a linear restoring force. Therefore, the driven motion of the electrons can re-emit light waves through the material.

In addition, the physical principles behind the oscillation of molecules are based on electromagnetism. Feynman clarifies that “[t]he electric field of the light wave polarizes the molecules of the gas, producing oscillating dipole moments. The acceleration of the oscillating charges radiates new waves of the field. This new field, interfering with the old field, produces a changed field which is equivalent to a phase shift of the original wave. Because this phase shift is proportional to the thickness of the material, the effect is equivalent to having a different phase velocity in the material (Feynman et al., 1964, section 32–1 Polarization of matter).” Essentially, Feynman has simplified the discussion by excluding complications that arise from the effects of light waves changing the electric fields at the oscillating charges. He has assumed the forces on the charges in the atoms came only from the incoming wave, and did not delve deeper in the re-emitted waves from all other atoms.

Fundamentally speaking, we can have a quantum interpretation of the equation derived by wave mechanics on the refractive index. In a footnote of his lecture, Feynman states that “[i]n quantum mechanics even an atom with one electron, like hydrogen, has several resonant frequencies. Therefore N_k is not really the number of electrons having the frequency ω_k, but is replaced instead by Nf_k, where N is the number of atoms per unit volume and f_k (called the oscillator strength) is a factor that tells how strongly the atom exhibits each of its resonant frequencies ω_k (Feynman et al., 1963, p. 31-8).” In short, the interactions of light with a material can be visualized as the absorptions and re-emissions of photons. As a result, the absorptions and emissions of light waves in the material cause the phase shift and the reduction of (effective) speed of light.

3. Problems of defining refractive index:

The refractive index is not a simply a constant and it is dependent on the frequency (or wavelength) of light waves. According to Feynman, “we have also learned how the index of refraction should vary with the frequency ω of the light. This is something we would never understand from the simple statement that ‘light travels slower in a transparent material’ (Feynman et al., 1963, section 31–3 Dispersion).” In essence, the speed of light is dependent on the color and the refractive index is not exactly defined by the ratio of the speed of light in vacuum to the speed of light in the medium. Mathematically, the index of refraction can be modeled by the equation n = 1 + Nq_e²/2ϵ₀m(ω₀² − ω²) in which N is the number of atoms per unit volume in a plate. We can use this equation to explain the phenomenon of dispersion.

Interestingly, the refractive index can be represented by a complex number. Feynman explains that “the index of refraction is now a complex number! What does that mean? By working out what the real and imaginary parts of n are we could write n = n′ − in′′, where n′ and n″ are real numbers (Feynman et al., 1963, section 31–4 Absorption).” The imaginary part of refractive index means that some energy of light waves can be absorbed by the material or medium. This implies a decrease in the amplitude of the light waves that is proportional to the thickness of the medium such as a piece of glass plate. In other words, the light waves that come out from the other side of the glass plate have lesser energy. Therefore, n″ is sometimes known as the “absorption index.”

On the other hand, there is still a problem of incomplete knowledge in the refractive index. In Feynman’s words, “We still have the problem, of course, of knowing how many atoms per unit volume there are, and what is their natural frequency ω₀. We do not know this just yet, because it is different for every different material, and we cannot get a general theory of that now. Formulation of a general theory of the properties of different substances — their natural frequencies, and so on — is possible only with quantum atomic mechanics (Feynman et al., 1963, section 31–3 Dispersion).” Therefore, it is challenging to have a general mathematical equation for all refractive indices that is applicable to all materials. Currently, there are meta-materials in which the refractive index can even be negative.

       To conclude, the knowledge of speed of light is reduced in a transparent material does not mean a good understanding of the concept of refractive index. Nevertheless, Feynman would explain that it is the effective speed of light that is reduced and this is related to the phase shift of light waves in the material. Importantly, the concept of refractive index can be modeled by using wave mechanics, electromagnetism, and quantum mechanics. Furthermore, we should be cognizant of problems of defining refractive index such as it is depending on the wavelengths of light and it is possible to have an imaginary part in the refractive index.

Note: Feynman has a short and interesting explanation on the bending of light in a medium, “[f]inding the path of least time for light is like finding the path of least time for a lifeguard running and then swimming to rescue a drowning victim: the path of least distance has too much water in it; the path of least water has too much land in it; the path of least time is a compromise between the two (Feynman, 1985, p. 51).”

References: 

1. Chew, C. & Chow, S. F. (2007). GCE ‘O’ Level Physics Matters. Singapore: Marshall Cavendish. 

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

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

Newton’s Second Law (law or definition?)

Question: State Newton’s second law of motion in words. Explain the meaning of Newton’s second law.

In Principia, Newton states the second law as “[t]he alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.” Simply phrased, this means that the force is proportional to the rate of change of momentum, but we do not find the equation F = ma in the original Newton’s second law. Depending on the grading criteria, students could be penalized when they state Newton’s second law that is related to F = ma instead of F = dp/dt. 

On the other hand, one may explain that Newton’s second law is a definition of force. Currently, there is no agreement on whether Newton’s second law is simply an empirical law or merely a definition. Furthermore, one may debate to what degree Newton’s second law is a definition or a law. However, physicists could be more precise by specifying their definition of definition. For instance, what they have in mind may be a theoretical definition or an operational definition. Alternatively, Newton’s second law can be considered to be a physical model or simply a mathematical relationship between force and motion.

Below are examples of Newton’s second law of motion that are stated in textbooks from different countries. A Russian textbook: A force acting on a body is equal to the product of the mass of the body and the acceleration produced by this force, the directions of the force and the accelerations coinciding. (Landsberg, 1971). A UK textbook: The rate of change of momentum of an object is proportional to the resultant force which acts on the object (Breithaupt, 2000). A US textbook: An object of mass m subjected to forces F1, F2, F3, … will undergo an acceleration a given by a = Fnet/m where the net force Fnet = F1 + F2 + F3 +…  is the vector sum of the individual forces. The acceleration vector a points in the same direction as the net force vector Fnet (Knight, 2004).

How would Feynman answer?

In The Feynman Lectures on Physics, we can find Newton’s second law of motion that is based on the equation F = ma and F = dp/dt. Additionally, Feynman disagrees that Newton’s second law is simply a definition. We will discuss possible answers of Feynman from the perspective of F = ma, F = dp/dt, and problems of defining force.

1. F = ma 

Although Feynman often makes fun of philosophers, he is interested in the meaning of knowledge, and thus opines that it is always important to ask, “What does it mean?” In Feynman’s words, “‘What is the meaning of the physical laws of Newton, which we write as F = ma? What is the meaning of force, mass, and acceleration?’ Well, we can intuitively sense the meaning of mass, and we can define acceleration if we know the meaning of position and time. We shall not discuss those meanings, but shall concentrate on the new concept of force. The answer is equally simple: ‘If a body is accelerating, then there is a force on it.’ That is what Newton’s laws say, so the most precise and beautiful definition of force imaginable might simply be to say that force is the mass of an object times the acceleration (Feynman et al., 1963, section 12–1 What is a force?) Importantly, he emphasizes that the force is supposed to have some independent properties, for example, it has a material origin, and thus, it is not just a definition.

In addition, Feynman explains that “the acceleration a is the rate of change of the velocity, and Newton’s Second Law says more than that the effect of a given force varies inversely as the mass; it says also that the direction of the change in the velocity and the direction of the force are the same (Feynman et al., 1963, section 9–1 Momentum and force).” Similarly, according to Feynman, “we see that Newton’s Second Law, in saying that the force is in the same direction as the acceleration, is really three laws, in the sense that the component of the force in the x-, y-, or z-direction is equal to the mass times the rate of change of the corresponding component of velocity: F_x = m(dv_x/dt) = m(d²x/dt²) =ma_x, F_y = m(dv_y/dt) = m(d²y/dt²) = ma_y, F_z = m(dv_z/dt) = m(d²z/dt²) = ma_z (Feynman et al., 1963, section 9–3 Components of velocity, acceleration, and force).” Essentially, physicists define the force in terms of F = ma in Euclidean space. Thus, we can visualize the force as a vector such that the directions of the force and acceleration are the same.

Interestingly, Feynman disagrees that F = ma is a definition because it is not exactly true. Firstly, Feynman mentions that “[o]ne might sit in an armchair all day long and define words at will, but to find out what happens when two balls push against each other, or when a weight is hung on a spring, is another matter altogether, because the way the bodies behave is something completely outside any choice of definitions (Feynman et al., 1963, section 12–1 What is a force?).” Note that we have idealized the equation F = ma and prediction cannot be simply made from a mathematical definition. Secondly, Feynman clarifies that “[t]he forces on a single thing already involve approximation, and if we have a system of discourse about the real world, then that system, at least for the present day, must involve approximations of some kind. (Feynman et al., 1963, section 12–1 What is a force?) That is, Newton’s second law is not exact and it is important to understand that this physical law involves idealizations and approximations.

2. F = dp/dt

Historically speaking, Newton proposes the second law of motion as the rate of change of motion instead of the product of a mass of an object and its acceleration. However, Feynman states that “the motion of an object is changed by forces in this way: the time-rate-of-change of a quantity called momentum is proportional to the force (Feynman et al., 1963, section 9–1 Momentum and force).” To be precise, Feynman uses the term momentum instead of motion that is adopted by Newton. Furthermore, he specifies force as the time-rate-of-change of momentum. This is more precise because the rate of change of momentum could be with respect to displacement instead of time. However, Feynman’s statement can be further improved. First, we can be more precise by using the term linear momentum that distinguishes from angular momentum. Better still, the word proportional can be replaced by directly proportional.

Feynman also mentions that “Newton’s Second Law may be written mathematically this way: F = d(mv)/dt. Now there are several points to be considered. In writing down any law such as this, we use many intuitive ideas, implications, and assumptions which are at first combined approximately into our ‘law.’ … First, that the mass of an object is constant; it isn’t really, but we shall start out with the Newtonian approximation that mass is constant, the same all the time, and that, further, when we put two objects together, their masses add. These ideas were of course implied by Newton when he wrote his equation, for otherwise it is meaningless. For example, suppose the mass varied inversely as the velocity; then the momentum would never change in any circumstance, so the law means nothing unless you know how the mass changes with velocity (Feynman et al., 1963, section 9–1 Momentum and force).” However, particle physicists prefer Newton’s second law to be written as F = d(γmv)/dt in which the Lorentz factor, γ, equals to 1/(1 – v²/c²)^1/2 and c is the speed of light. This is related to the concept of invariant mass that is velocity-independent.

Moreover, Feynman explains that “there is another interesting consequence of Newton’s Second Law, to be proved later, but merely stated now. This principle is that the laws of physics will look the same whether we are standing still or moving with a uniform speed in a straight line. For example, a child bouncing a ball in an airplane finds that the ball bounces the same as though he were bouncing it on the ground. Even though the airplane is moving with a very high velocity, unless it changes its velocity, the laws look the same to the child as they do when the airplane is standing still. This is the so-called relativity principle. As we use it here we shall call it ‘Galilean relativity’  to distinguish it from the more careful analysis made by Einstein, which we shall study later (Feynman et al., 1963, section 10–2 Conservation of momentum).” In other words, Newton’s second law is valid in an inertial frame of reference in which every free particle moves with a constant velocity.

3. Problems of defining force

Newton’s second law of motion is commonly known as a law of force or a definition of force. Feynman would discuss problems of defining force such as context, precision, and circularity as shown below: 

Context: Feynman mentions that “[m]omentum is not the same as velocity. A lot of words are used in physics, and they all have precise meanings in physics, although they may not have such precise meanings in everyday language (Feynman et al., 1963, section 9–1 Momentum and force).” Similarly, the term force has alternative definitions in the everyday context and technical context. For example, a definition of force in a dictionary or everyday language is “energy.” Moreover, Feynman clarifies that “[t]he first term is the mass times acceleration, and the second is the derivative of the potential energy, which is the force (Feynman et al., 1964, section 19–1 A special lecture—almost verbatim).” Depending on the context, force may be defined as “mass times acceleration,” “time rate of change of linear momentum,” or “derivative of the potential energy”, and thus, the term force could be confusing to students.

Precision: Feynman explains that “[t]he student may object, ‘I do not like this imprecision, I should like to have everything defined exactly; in fact, it says in some books that any science is an exact subject, in which everything is defined.’ If you insist upon a precise definition of force, you will never get it! First, because Newton's Second Law is not exact, and second, because in order to understand physical laws you must understand that they are all some kind of approximation (Feynman et al., 1963, section 12–1 What is a force?).” To illustrate this fact, Feynman gives the example in which the mass of a chair can be defined only approximately. He argues that it is difficult to distinguish the atoms that are chair, air, dirt, or paint.

Circularity: Feynman provides the following insights: “[w]e could also define force to mean that a moving object with no force acting on it continues to move with constant velocity in a straight line. If we then observe an object not moving in a straight line with a constant velocity, we might say that there is a force on it. Now such things certainly cannot be the content of physics, because they are definitions going in a circle (Feynman et al., 1963, section 12–1 What is a force?).” In a sense, it suggests that Newton’s first law and second law are circular: the first law states that zero force does not result in a change in velocity, whereas second law states that a force results in a change in velocity. Thus, both statements are essentially similar and the first law may be considered as a special case of second law. However, this is different from another circularity in which force and mass are defined based on Newton’s second law. That is, one should not define force by using the equation F = ma, and then define mass by using the equation m = F/a. (Some prefer to define mass using E/c².)

To conclude, Newton’s second law of motion can be stated based on the equation F = ma or F = dp/dt. To be more accurate, the concept of force should be defined as the time rate of change of linear momentum instead of simply the product of mass and acceleration. Importantly, Feynman disagrees that Newton’s second law is simply a definition because it is not exactly correct and it can be falsified by experiment. Furthermore, there are idealization and approximations in this physical law of force as well as it is valid in an inertial frame of reference. However, we should be cognizant of problems in defining force. 

References:

1. Breithaupt, J. (2000). Understanding Physics for Advanced Level (4th ed). Cheltenham: Stanley Thorne. 

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. Knight, R. D. (2004). Physics for Scientists and Engineers with Modern Physics. California: Addison-Wesley. 

5. Landsberg, G. S. (1971/2000). Textbook of Elementary Physics, Volume I. (A. Troitsky, Transl.) Honolulu, Hawaii: University Press of the Pacific.