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.

Feynman's answers?

Mechanics

Explain two problems of this statement of Newton’s First Law of Motion.

Why doesn't the Moon fall into the Earth?

Thermodynamics

Is heat a noun or a verb?

Does the entropy of an ideal gas remain unchanged or increase?

Optics

What are the characteristics of an image formed by a plane mirror?

Waves

Why is the sky blue?

Electromagnetism

What do you understand by the term electric current?

Are the input potential difference and the output electromotive force of an ideal transformer in phase or out of phase?

Modern physics

Based on Heisenberg’s uncertainty principle, which of the diagram below shows the correct position where Δx and Δp are defined?

Is the nature of energy discrete or continuous?

The nature of heat (noun or verb?)

Question: Explain the nature of heat.

There is no consensual definition of heat. For example, Baierlein (1994) opines that heat is an adjective, Zemansky (1970) disagrees that heat is a verb, and Romer (2001) proposes that heat is not a noun. Essentially, heat can be distinguished as a “transfer” of energy (verb) and “energy” transferred (noun). However, it can be confusing for students when heat may mean “a form of energy” or “process of energy transfer” within a textbook.

In the nineteenth century, there were two competing concepts of heat: “caloric (or material) theory” and “kinetic (or mechanical) theory” (Chang 2004). In the early twentieth century, the term heat was even more confusing because there were at least three different definitions of heat: (1) energy in transition from a hot to a cold body, with the usual symbol Q; (2) internal energy or energy stored in a body, with the symbols E or U; and (3) enthalpy and it can be represented by the function U + PV, with the symbol H (Stuart, 1938). Currently, you may find different definitions of heat in biology, chemistry, and physics. Importantly, there are differences in opinion whether it is appropriate to use words such as flow or transfer in definitions of heat because they have the connotations that heat is a form of fluid or substance.

How would Feynman answer?

       It is possible that Feynman would explain heat as a noun and a verb (or a process) and discuss problems of defining heat as shown below.

1. Heat is a noun:

During a British Broadcasting Corporation (BBC) television interview, Feynman (1994) explains that “you can either have the idea that heat is some kind of a fluid which flows from a hot thing, and leaks into the cold thing; or you can have a deeper understanding, which is closer to the way it is – that the atoms are jiggling, and their jiggling passes their motion on to the others (p. 127).” However, the notion of heat as a form of fluid can be attributed to the caloric theory of heat in the nineteen century or earlier. In addition, de Berg (2008) clarifies that “[t]he terms, heat flow, or energy flow, are remnants of the old caloric theory of heat in which heat was considered as a material fluid that had the capacity to flow. Identifying heat as motion is also a remnant of the early kinetic ideas of the 19th century (p. 80).” Thus, some scientists and educational researchers might consider Feynman to be misleading the public.

Similarly, in The Feynman Lectures on Physics, Feynman mentions that “the jiggling motion is what we represent as heat: when we increase the temperature, we increase the motion (Feynman et al., 1963, section 1–2 Matter is made of atoms).” Furthermore, he elaborates that “we can change the amount of heat. What is the heat in the case of ice? The atoms are not standing still. They are jiggling and vibrating (Feynman et al., 1963, section 1–2 Matter is made of atoms).” Feynman considers heat to be due to the kinetic energy of atoms or atomic vibrations. Essentially, the amount of heat is dependent on the temperature, and thus, heat is a noun.

Feynman elaborates that “[t]he heat is ordinarily in the form of the molecular motion of the hot gas (Feynman et al., 1963, section 1–4 Chemical reactions).” This would suggest that heat is the internal energy of a system. This description of heat is commonly found in biology textbooks (Doige & Day, 2012). Therefore, students may find it confusing because heat may mean “internal energy” and “energy in transit.” More importantly, when we define the First Law of Thermodynamics in terms of ΔU = Q + W, it becomes necessary to distinguish the internal energy U and heat Q as the energy in transit due to a temperature difference. In other words, we should not define heat as internal energy (U) and energy in transit (Q) that can be found in the same equation.

2. Heat is a verb:

In Feynman’s own words, “[i]f we heat the water, the jiggling increases and the volume between the atoms increases, and if the heating continues there comes a time when the pull between the molecules is not enough to hold them together and they do fly apart and become separated from one another (Feynman et al., 1963, section 1–2 Matter is made of atoms).” Feynman also uses the term heat as a verb. However, this is different from some physicists and physics educator who only use heat as a verb or define heat as a process of energy transfer. The use of heat as a process also clearly means that heat is not a form of substance or fluid.

In formulating the first law of thermodynamics, Feynman mentions that “[l]et us begin by stating the first law, the conservation of energy: if one has a system and puts heat into it, and does work on it, then its energy is increased by the heat put in and the work done. We can write this as follows: The heat Q put into the system, plus the W done on the system, is the increase in the energy U of the system; the latter energy is sometimes called the internal energy: Change in U = Q + W (Feynman et al., 1963, section 44–1 Heat engines; the first law).” In short, the term heat may be used when there is a non-mechanical transfer of energy into a system. Nevertheless, we can be more precise by defining heat as a method of energy transfer due to a temperature difference.

Interestingly, Feynman explains that “when we stretch a rubber band it heats, and when we release the tension of the band it cools. Now our instincts might suggest that if we heated a band, it might pull: that the fact that pulling a band heats it might imply that heating a band should cause it to contract (Feynman et al., 1963, section 44–1 Heat engines; the first law).” Conversely, Feynman states that “[w]hen we stretch a rubber band, we find that its temperature falls (Feynman et al., 1963, section 45–2 Applications).” That is, Feynman contradicts himself in the previous chapter by saying that the temperature falls. However, this is likely a careless mistake because the temperature should increase when the rubber band is stretched. It can be simply explained by the first law of thermodynamics, ΔU = ΔQ + FΔL. Feynman’s mistake could be related to the use of a mathematical expression for work done by the rubber band, −FΔL.

3. Problems of defining heat: 

Some physicists prefer to define heat in terms of the first law of thermodynamics. Nevertheless, Feynman explains that “[i]f we have a hot thing and a cold thing, the heat goes from hot to cold. So the law of entropy is one such law. But we expect to understand the law of entropy from the point of view of mechanics. In fact, we have just been successful in obtaining all the consequences of the argument that heat cannot flow backwards by itself from just mechanical arguments, and we thereby obtained an understanding of the Second Law. Apparently, we can get irreversibility from reversible equations… Since our question has to do with the entropy, our problem is to try to find a microscopic description of entropy (Feynman et al., 1963, section 46–4 Irreversibility).” Thus, it is possible that Feynman would relate a problem of defining heat to the law of entropy.

On the other hand, Canagaratna (1969) argues that another problem of defining heat is a problem of defining a measure of thermal interactions. He explains that “the ice-calorimetric method and the heat capacity method are unable to define q for irreversible processes taking place between any two bodies. Since the concept of heat has no necessary connection with reversible processes, it must be concluded that q can be defined in all its generality only through the use of the first law (Canagaranta, 1969, p. 683).” In essence, he proposes operational definitions of heat by using an ice-calorimetric method and heat capacity method. Moreover, Canagaranta (1969) opines that heat should be defined only by using the first law of thermodynamics, but it may involve irreversible mechanical work experimentally.

Lastly, and ideally, a scientific term should have only a definition such that there is no confusion when the term is used. Currently, there are daily definitions of heat that are not related to science. In addition, the term heat has a variety of definitions that can be used differently in biology, chemistry, and physics. For instance, definitions of heat may mean internal energy in biology and include terms such as “in contact” in chemistry. There are also different opinions how heat should be defined in physics.

       To conclude, Feynman would use the word heat as a noun or a verb. Importantly, some scientists and educational researchers disagree with him in explaining heat as a form of fluid, and defining heat as the internal energy. However, Feynman might discuss problems of defining heat or how heat could be used differently depending on the context.

Note: 

1. During the Messenger Lectures, Feynman (1965) mentions that “[h]eat is supposed to be jiggling, and the word for a hot thing is just the word for a mass of atoms which are jiggling. But for a while, if we are talking about heat, we sometimes forget about the atoms jiggling (p. 124).” 

2. In the words of Feynman, “[w]e call this form of energy heat energy, but we know that it is not really a new form, it is just kinetic energy — internal motion (Feynman et al., 1963, section 4–4 Other forms of energy).”

References:

1. Baierlein, R. (1994). Entropy and the second law: A pedagogical alternative. American Journal of Physics, 62(1), 15–26.

2. Canagaratna, S. G. (1969). Critique of the definitions of heat. American Journal of Physics, 37(7), 679–683. 

3. Chang, H. (2004). Inventing temperature. Oxford, United Kingdom: Oxford University Press. 

4. De Berg, K. C. (2008). The Concepts of Heat and Temperature: The Problem of Determining the Content for the Construction of an Historical Case Study which is Sensitive to Nature of Science Issues and Teaching–Learning Issues. Science & Education, 17(1), 75–114. 

5. Doige, C. A. & Day, T. (2012). A typology of undergraduate textbook definitions of ‘heat’ across science disciplines. International Journal of Science Education, 34(5), 677–700.

6. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press. 
7. 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. 

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

9. Romer, R. H. (2001). Heat is not a noun. American Journal of Physics, 69(2), 107–109. 

10. Stuart, M. C. (1938). Use and Meaning of the Term Heat. American Journal of Physics, 6(1), 40.

11. Zemansky, M. W. (1970). The use and misuse of the word “heat” in physics teaching. The Physics Teacher, 8(6), 295–300.

Electric current (A flow of electrons or positive charges?)

Question: What do you understand by the term electric current? OR Explain what is meant by an electric current.

In a study conducted by Garnett and Treagust (1992), students studying both physics and chemistry were more confused about the concept of electric currents in metallic conductors as compared to students who were only studying chemistry. They propose that the physics syllabus should adopt the electron flow model of electric current in metallic conductors because some students had conceptual problems with the different conventions used in chemistry and physics. On the contrary, Arons (1990) argues that the positive current convention should be maintained in physics for the following four reasons: “(1) the definitions of electric field strength and potential difference; (2) the treatment of capacitive and inductive circuit elements; (3) all the standard mnemonics of electromagnetism and the Maxwell’s equations; and (4) the standard notations in diagrams of electronic circuits (p. 179).” However, electric current can be defined as a flow of charge carriers such as electrons and holes depending on the context.

How would Feynman answer?

Feynman might discuss charge carriers, nature of the electric current, and problems of defining the electric current as shown below.

1. Charge carriers: An important feature of a definition of electric current is the charge carriers. For instance, Feynman states that “[e]lectric currents are electrons or other charges in motion with a net drift or flow (Feynman et al., 1964, section 13–2 Electric current; the conservation of charge).” That is, an electric current is a net drift or flow of charge carriers such as electrons. In addition, this is dependent on the context (or a conducting medium) such as a copper wire or metallic conductor. Thus, Feynman mentions that “[i]n a normal conductor, like copper, the electric currents come from the motion of some of the negative electrons — called the conduction electrons — while the positive nuclear charges and the remainder of the electrons stay fixed in the body of the material (Feynman et al., 1964, section 13–6 The relativity of magnetic and electric fields).”

In general, electric charge can be positive and negative, and charge carriers may include electrons and holes. Therefore, Feynman elaborates that “[o]ne can also have both holes and electrons together. If there are not too many, they will all go their way independently. With an electric field, they will all contribute to the current. For obvious reasons, electrons are called the negative carriers and the holes are called the positive carriers. We have so far considered that electrons are put into the crystal from the outside, or are removed to make a hole. It is also possible to ‘create’ an electron-hole pair by taking a bound electron away from one neutral atom and putting it some distance away in the same crystal. We then have a free electron and a free hole, and the two can move about as we have described (Feynman et al., 1966, section 14–1 Electrons and holes in semiconductors).” In short, the charge carriers include electrons and holes in the context of semiconductors.

On the other hand, there are also electric currents in the earth’s atmosphere. In fact, Feynman mentions that “[d]ue to the air currents, ions, and water drops on ice particles in a thunderstorm, positive and negative charges are separated. The positive charges are carried upward to the top of the cloud, and the negative charges are dumped into the ground in lightning strokes. The positive charges leave the top of the cloud, enter the high-altitude layers of more highly conducting air, and spread throughout the earth. In regions of clear weather, the positive charges in this layer are slowly conducted to the earth by the ions in the air — ions formed by cosmic rays, by the sea, and by man’s activities. The atmosphere is a busy electrical machine! (Feynman et al., 1964, section 9–5 The mechanism of charge separation).” In other words, we should not only visualize electric currents in terms of electric circuits and metallic conductors.

2. Nature of electric current: In textbooks, a simple definition of electric current is a flow of electrons. During a BBC interview, Feynman (1994) explains that “[y]ou can imagine the city electricity as being like a water system if you like. You can visualize the electrons going in pipes all over the place, with a big pump at one end. In fact, look at any book and they’ll tell you that the voltage is the pressure and the amperage is the amount of flow (p. 135).” Essentially, electric current is about the amount of flow. Importantly, Feynman also provides a good analogy in which the electric current is like the flow of water and the potential difference is similar to the pressure difference. However, electric current could be more precisely defined as the rate of flow of charge carriers instead of simply the flow of charge carriers.

Mathematically, Feynman states that “the current I can be written as ρ₋vA (Feynman et al., 1964, section 13–6 The relativity of magnetic and electric fields).” Note that ρ₋ refers to charge density of the conduction electrons, A refers to the area of a cross-section of the wire, and v is the velocity of the conduction electrons. Simply phrased, the rate of flow is dependent on the velocity of charge carriers. In addition, Feynman mentions that “[t]he total charge passing per unit time through any surface S is called the electric current, I. It is equal to the integral of the normal component of the flow through all of the elements of the surface: I = ʃS j · n dS (Feynman et al. 1964, section 13–2 Electric current; the conservation of charge).” This provides another mathematical definition of electric current as compared to I = dq/dt which means the instantaneous rate of flow of electric charge, q.

Better still, electric current can be defined as a rate of flow of ‘free’ electrons due to a potential difference across the ends of the metallic conductor. Importantly, it is the electric field that causes the flow of electrons in a copper wire. In Feynman’s words, “in an atom with three protons in the nucleus exchanging photons with three electrons – a condition called a lithium atom - the third electron is further away from the nucleus than the other two (which have used up the available space), and exchanges fewer photons. This causes the electron to easily break away from its own nucleus under the influence of photons from other atoms. A large number of such atoms close together easily lose their individual third electrons to form a sea of electrons swimming around from atom to atom. This sea of electrons reacts to any small electrical force (photons), generating a current of electrons - I am describing lithium metal conducting electricity (Feynman, 1985, p. 113).” In essence, the flow of electrons is due to the presence of the electric field and it is mediated by photons.

3. Problems of Defining electric current: The definition of electric current has several definitional problems which may not be easily resolved. Firstly, electric current is sometimes defined as “the rate of flow of electric charge, I = q/t” and electric charge is defined by the equation, “electric charge (q) = electric current (I) × the duration of time for which it flows (t).” This suggests a problem of circularity in which both electric current and electric charge are defined by using the same equation. However, this circularity problem should not be resolved by considering electric current to be a fundamental concept and thus it is indefinable or it cannot be defined in terms of simpler, more basic concepts.

Another problem is that the Poynting theory pertaining to the flow of energy in a wire seems “crazy” to Feynman. In his own words, “[w]e ask what happens in a piece of resistance wire when it is carrying a current. Since the wire has resistance, there is an electric field along it, driving the current. Because there is a potential drop along the wire, there is also an electric field just outside the wire, parallel to the surface. There is, in addition, a magnetic field which goes around the wire because of the current. The E and B are at right angles; therefore there is a Poynting vector directed radially inward, as shown in the figure. There is a flow of energy into the wire all around. It is, of course, equal to the energy being lost in the wire in the form of heat. So our “crazy” theory says that the electrons are getting their energy to generate heat because of the energy flowing into the wire from the field outside… the theory says that the electrons are really being pushed by an electric field, which has come from some charges very far away, and that the electrons get their energy for generating heat from these fields (Feynman et al. 1964 section 27-5 Examples of energy flow).” Intuitively speaking, Feynman feels that the theory is incomplete because it does not seem to make sense that the field energy could flow from the battery to infinity, and then back into the wire as suggested by Poynting vector.

Furthermore, Feynman mentions that “if currents are made to go through a piece of material obeying Ohm’s law, the currents distribute themselves inside the piece so that the rate at which heat is generated is as little as possible. Also, we can say (if things are kept isothermal) that the rate at which energy is generated is a minimum. Now, this principle also holds, according to classical theory, in determining even the distribution of velocities of the electrons inside a metal which is carrying a current. The distribution of velocities is not exactly the equilibrium distribution because they are drifting sideways. The new distribution can be found from the principle that it is the distribution for a given current for which the entropy developed per second by collisions is as small as possible. The true description of the electrons’ behavior ought to be by quantum mechanics, however. The question is: Does the same principle of minimum entropy generation also hold when the situation is described quantum-mechanically? I haven’t found out yet (Feynman et al., section 19–2 A note added after the lecture).” Simply phrased, the theory of electric current is also incomplete from a perspective of quantum mechanics. 

       In summary, a comprehensive definition of electric current is a rate of flow of free electrons due to a potential difference (or electric field) across the ends of the conductor, under constant circuit conditions. More importantly, Feynman would discuss charge carriers, the nature of electric current in different contexts and equations, as well as definitional problems of electric current.

Note:

In Feynman Lectures on Computation, Feynman (1996) explains that “[g]ood conductors have a plentiful supply of free electrons under normal conditions, the band gap energy being tiny or non-existent (filled and conduction bands can even overlap) (p. 214).”

References:
1. Arons, A. B. (1990). A guide to introductory physics teaching. New York: John Wiley & Sons.
2. Feynman, R. P. (1985). QED: The Strange Theory of Light and Matter. Princeton: Princeton Scientific Library.
3. Feynman, R. P. (1994). No Ordinary Genius: The Illustrated Richard Feynman. New York: W. W. Norton & Company.
4. Feynman, R. P. (1996). Feynman lectures on computation. Reading, Massachusetts: Addison-Wesley. 
5. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.
6. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol III: Quantum Mechanics. Reading, MA: Addison-Wesley.
7. Garnett, P. J., & Treagust, D. F. (1992). Conceptual difficulties experienced by senior high school students of electrochemistry: Electric circuits and oxidation-reduction equations. Journal of Research in Science Teaching, 29(2), 121-142.