Diffraction of light waves

Question: Explain the meaning of diffraction of light.

Textbooks may not provide a comprehensive definition of diffraction. For example, Hewitt (2006) defines diffraction as “[t]he bending of light that passes around an obstacle or through a narrow slit, causing the light to spread (p. 578).” Essentially, this definition refers to a bending or spreading of light waves as a result of a narrow slit or an obstacle. However, we can explain that the diffraction of light is a spreading of light waves passing through a slit or obstacle whose size is comparable to the wavelength of the light waves and it results in bright and dark fringes (instead of light rays moving in a straight line).

How would Feynman answer?

Feynman would answer this question from the perspectives of diffraction through a single slit, diffraction grating (diffraction through multiple slits), and problems of defining diffraction.

1. Diffraction through a single slit:

Firstly, Feynman would explain that, “[a]ccording to the wave theory, there is a spreading out, or diffraction, of the waves after they go through the slit, just as for light. Therefore, there is a certain probability that particles coming out of the slit are not coming exactly straight. The pattern is spread out by the diffraction effect, and the angle of spread, which we can define as the angle of the first minimum, is a measure of the uncertainty in the final angle (Feynman et al., 1963, section 38–2 Measurement of position and momentum).” In short, the word diffraction means a spreading of waves.

In addition, Feynman would elaborate that “[t]o say it is spread means that there is some chance for the particle to be moving up or down, that is, to have a component of momentum up or down (Feynman et al., 1963, section 38–2 Measurement of position and momentum).” That is, Feynman views the spreading of waves from the perspective of particles. Simply phrased, this physical phenomenon can be explained by the uncertainty of particles in moving up or down. In a sense, it means that we can explain the wave theory of diffraction by using the uncertainty principle.

On the other hand, Feynman (1990) also says that “when you try to squeeze light too much to make sure it’s going in only a straight line, it refuses to cooperate and begins to spread out (p. 55).” Essentially, “it ‘smells’ the neighboring paths around it (p. 54)” while moving through a single slit before forming the diffraction pattern. This is based on his formulation of quantum electrodynamics that does not require an uncertainty principle. In other words, Feynman’s sum-over-paths recipe for a particle moving from a location A to another location B means that physicists need to consider all possibilities (or possible paths) between A to B. Thus, it is not simply about travelling in a straight line path, but one needs to include paths that include twists and turns.

2. Diffraction grating:

Feynman would discuss diffraction of waves through multiple slits or diffraction grating. According to Feynman, “a diffraction grating consists of nothing but a plane glass sheet, transparent and colorless, with scratches on it. There are often several hundred scratches to the millimeter, very carefully arranged so as to be equally spaced. The effect of such a grating can be seen by arranging a projector so as to throw a narrow, vertical line of light (the image of a slit) onto a screen. When we put the grating into the beam, with its scratches vertical, we see that the line is still there but, in addition, on each side we have another strong patch of light which is colored (Feynman et al., 1963, section 30–2 The diffraction grating).” This can be explained by the diffraction grating equation d sin θ = λ because the angle (θ) of spreading depends on lights of different colors or wavelengths.

Interestingly, a block of graphite may also function like a diffraction grating. More important, it is the slowest neutrons that pass through the long block of graphite. Thus, Feynman explains that “[i]f we take these neutrons and let them into a long block of graphite, the neutrons diffuse and work their way along. They diffuse because they are bounced by the atoms, but strictly, in the wave theory, they are bounced by the atoms because of diffraction from the crystal planes (Feynman et al., 1963, section 38–3 Crystal diffraction).” That is, these neutrons have longer wavelengths and behave more like waves. Note that neutrons having higher energy behave more like particles, whereas neutrons having lower energy behave more like waves.

3. Problems of defining diffraction:

It is possible that Feynman would discuss problems of defining diffraction. For example, in his own words, “[n]o one has ever been able to define the difference between interference and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them. The best we can do, roughly speaking, is to say that when there are only a few sources, say two, interfering, then the result is usually called interference, but if there is a large number of them, it seems that the word diffraction is more often used (Feynman et al., 1963, p. 30-1).” In essence, diffraction refers to a spreading of waves that includes the phenomenon interference, whereas interference refers to a superposition of waves that includes the phenomenon diffraction.

Feynman also clarifies that if there are only “two sources” of light, the phenomenon is commonly called interference; on the other hand, if there are a “large number of sources” of light, the phenomenon is known as diffraction. Importantly, we should include the nature of waves when we define the concept of diffraction. For example, we may speak of diffraction and interference of light waves and water waves. However, there is a spreading of elastic waves from a drum that has two-dimensional cylindrical symmetry instead of diffraction of one-dimensional string waves.

References:

1. Feynman, R. P. (1990). QED: The Strange Theory of Light and Matter. London: Penguin.

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. Hewitt, P. (2006). Conceptual Physics (10th ed.). San Francisco: Addison-Wesley.

Electric field

Question: Define electric field and explain why, when defining the electric field, the test particle must be stationary.

The electric field can be defined as the electric force per unit charge acting on a test charge that is stationary. An alternative definition of the electric field is the electric force per unit charge experienced by an infinitely small test charge placed in a vacuum. Mathematically, it can be expressed by the equation E = F/q, where F is the electric force experienced and q is the infinitely small test charge. This is because a large test charge can influence the charge carriers in the surrounding and distort the electric fields.

It is surprising that a physics teacher provides the following answer for the need of a stationary test charge: “a moving charge will generate electromagnetic waves that can interfere with the original electric field.” Strictly speaking, it is the accelerating charge carrier that produces electromagnetic waves. However, it is possible to have gravitational fields, electric fields, and magnetic fields in a region in space. Thus, physicists may specify the condition that the test particle is stationary because the net force could include a magnetic force for the moving test particle in addition to the electric (or electrostatic) force.

How would Feynman answer?

Feynman may answer this question from the perspectives of definition of the electric field, the condition of stationary test charge, and problems of defining the electric field.

1. Definition of the electric field:

Feynman would first explain that the definition of the electric field is based on Coulomb’s law and we need to assume that all other charges remain in the same positions. In his own words, “[w]hen applying Coulomb’s law, it is convenient to introduce the idea of an electric field. We say that the field E(1) is the force per unit charge on q₁ (due to all other charges). Dividing Eq. (4.9) by q₁, we have, for one other charge besides q₁, E(1) = (1/4πϵ₀)q²/r². Also, we consider that E(1) describes something about the point (1) even if q₁ were not there — assuming that all other charges keep their same positions (Feynman et al., 1964, section 4–2 Coulomb’s law; superposition).” Importantly, Feynman considers the electric field to be due charges that come in packages like electrons and protons, and think of them as being spread out in a continuous distribution.

To explain the concept of electric field, Feynman says that “[t]his potentiality for producing a force is called an electric field. When we put an electron in an electric field, we say it is “pulled.” We then have two rules: (a) charges make a field, and (b) charges in fields have forces on them and move (Feynman et al., 1963, section 2.2 Physics before 1920).” In other words, Feynman visualizes a charged particle creates a “condition” in space such that when we put another charged particle there, it feels an electric force. Furthermore, the nature of electric field can be illustrated with an analogy as follows: when two corks (charge particles) are floating in a pool of water, we can exert a force on a floating cork by giving another cork a gentle push. We can explain that there is a disturbance in the water (electric field) caused by a cork, and the water then disturbs the other cork “directly.”

2. Stationary test charge:

According to Feynman, the electrical force does not precisely follow Coulomb’s law that decreases inversely as the square of the distance between charges. His notion of electrical force includes the concept of magnetic force which a moving charged particle experiences. Thus, Feynman explains that “the electrical forces depend also on the motions of the charges in a complicated way. One part of the force between moving charges we call the magnetic force. It is really one aspect of an electrical effect. That is why we call the subject ‘electromagnetism.’ It is true that when charges are standing still the Coulomb force law is simple, but when charges are moving about the relations are complicated by delays in time and by the effects of acceleration, among others (Feynman et al., 1964, section 1–1 Electrical forces). In a sense, the magnetic force is a relativistic effect that complicates the electrical force on the moving charge.

On the other hand, Feynman elaborates that “[w]e find, from experiment, that the force that acts on a particular charge—no matter how many other charges there are or how they are moving — depends only on the position of that particular charge, on the velocity of the charge, and on the amount of charge. We can write the force F on a charge q moving with a velocity v as F = q(E + v×B) (Feynman et al., 1964, section 1–1 Electrical forces).” Essentially, the measured force is dependent on an observer’s frame of reference.

3. Problems in defining electric field:

Strictly speaking, the electric field is not simply equal to (1/4πϵ₀)q/r². Feynman would discuss problems of defining electric field at a point P as follows: “The electric field, E, is given by 

What do the various terms tell us? Take the first term, E= −qe_r′/4πϵ₀r′². That, of course, is Coulomb’s law, which we already know: q is the charge that is producing the field; e_r′ is the unit vector in the direction from the point P where E is measured, r is the distance from P to q. But, Coulomb’s law is wrong. The discoveries of the 19^th century showed that influences cannot travel faster than a certain fundamental speed c, which we now call the speed of light (Feynman et al., 1963, section 28–1 Electromagnetism).” That is, Coulomb’s law is not exactly correct because the influences (or signals) cannot travel faster than the speed of light and this can result in a delay in action. The time delay (or retarded time) is the time it takes for the influences moving at the speed of light to travel from the charge to the point P.

Moreover, Feynman might add that “[f]or those purists who know more (the professors who happen to be reading this), we should add that when we say that (28.3) is a complete expression of the knowledge of electrodynamics, we are not being entirely accurate. There was a problem that was not quite solved at the end of the 19^th century. When we try to calculate the field from all the charges including the charge itself that we want the field to act on, we get into trouble trying to find the distance, for example, of a charge from itself, and dividing something by that distance, which is zero. The problem of how to handle the part of this field which is generated by the very charge on which we want the field to act is not yet solved today (Feynman et al., 1963, section 28–1 Electromagnetism).” In other words, Feynman believes that there is no complete answer to the puzzle of self-force (or self-action).

          To conclude, Feynman would define electric field as the electric force per unit charge due to a distribution of charges (or protons and electrons). Experimentally, the measured force would include magnetic force if the test charge is not stationary. Importantly, he might discuss problems of defining electric field due to a time delay in the speed of influences and self-force of the electric charge.

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

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

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.