AP Physics 1 2015 Free Response Question 2

Question: 

Students are expected to describe an experimental procedure that could be used to answer the following question: “do fewer electrons leave a light bulb than enter the bulb in one second?” Next, they have to explain how experimental data can be used to answer the abovementioned question. (Assume that students are provided with voltmeters that are marked in 0.1 V increments and ammeters that are marked in 0.01 A increments.)

Scoring Guidelines:

Describe an analytical method of using experimental data and explain how the data can be used to answer the abovementioned question. Example: If the electric current measured is the same on both sides of the light bulb, then the number of electrons entering the bulb is the same as those leaving the bulb in one second.

1 point

(Source: http://apcentral.collegeboard.com/home)

Comments: 
In this question, students are expected to have a good concept of series circuits. The main purpose of the question is to assess students’ knowledge of experimental design and their ability in data analysis by using ammeters and voltmeters. Students were also asked to account the uncertainty of electrical measurement and to discuss how it affects the experimental results.

In the first part of this question, two ammeters are connected in series on both sides of the bulb to measure the electric current entering and leaving the bulb (based on the scoring guidelines). Importantly, we should not conclude that “the number of electrons per second entering and leaving the bulb is the same” if the electric current is measured to be the same on both sides of the bulb. There are both theoretical and experimental issues in this question. 

1. Theoretical issues: If the electric current through the ammeters on both sides of the bulb are the same, then the student expects the number of electrons per second entering and leaving the bulb are the same. This is based on a “simplified” definition of electric current: the number of electrons per second. However, to quote Feynman, “[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).” Firstly, the electric current could be contributed by a flow of positive charge carriers such as copper atoms in the wire, but they are “almost essentially” stationary. Secondly, the magnitude of electric current is also dependent on the drift velocity of electrons. Thus, the electric current is not simply the number of electrons per second moving through the bulb.

2. Experimental issues: In this question, the ammeters are marked in 0.01 A increments. However, one may prefer to use clamp meters which have 0.001 A resolution and they do not introduce additional electrical resistance. More importantly, if the uncertainty in measuring an electric current is 0.01 A, this may imply a possible error of 0.01 C in one second. Therefore, the uncertainty in measurement corresponding to the number of electrons per second could be of the order 10¹⁷ because the electric charge of an electron is 6.02 × 10^-19 C. In a sense, this experiment is analogous to the use of a meter ruler to conclude that the atoms and electrons have the same length. From an empiricist’s perspective, one should not comment about the number of electrons per second entering and leaving the bulb are the same when there are limitations (or significant uncertainty) in measuring the electric current accurately.

Currently, an ampere is defined as the electric current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these two conductors a magnetic force that is equal to 2 × 10^–7 N/m. In other words, the ampere is not defined as one electron at a particular time based on current technology. Furthermore, the definition of an ampere in terms of two infinitely long conductors cannot be precisely implemented in practice. Recently, Dr. Giblin (2016) and researchers from National Physical Laboratory’s Quantum Detection Group and the University of Cambridge have reported the counting of electrons trapped in an electron pump (a small region of a specially designed semiconductor device) with a fidelity of one part per million. Thus, the electron pump could be the new primary reference standard of electric current. However, there are still significant errors in counting the number of electrons.

Feynman’s insights or goofs?:

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).” In other words, an electric current is a net drift or flow of charge carriers such as electrons. However, Feynman’s definition of the electric current can be improved. For example, physics teachers may prefer to define the electric current as the time rate of flow of charge carriers. That is, we can distinguish the rate of flow with respect to time from the rate of flow with respect to displacement. Furthermore, the electric current can be more comprehensively defined as the rate of flow of ‘free’ electrons (or other charge carriers) per unit time due to a potential difference across the ends of an electrical conductor, under constant circuit conditions.

On the other hand, some physicists advocate the use of operational definitions. However, Feynman did not provide an operational definition of electric current. For example, Karplus (2003) writes that: “Operational definition: Electric current is measured by the dial reading of a standard ammeter (p. 315).” Furthermore, an ampere is defined as the electric current that can produce an attractive force of 2 × 10⁻⁷ N per meter of length  between two straight, parallel conductors of infinite length placed one meter apart in a vacuum. Importantly, a problem of the operational definition of electric current is that an uncertainty of 0.01 ampere in measuring an electric current corresponds to an uncertainty of 10¹⁷ electrons per second. Note that this operational definition is a measure of an effect of electric current that is a magnetic force.

Interestingly, Feynman has an insightful explanation on the effects of electric current: “[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. Intuition would seem to tell us that the electrons get their energy from being pushed along the wire, so the energy should be flowing down (or up) along the wire. But 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).”

Note:
1. You may want to visit this website:

http://feynman-answer.blogspot.sg/2016/08/electric-current-flow-of-electrons-or.html

2. For another discussion of this question, you can visit the following website:
https://www.youtube.com/watch?v=OpbYvr8yhqQ

References:

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

2. Giblin, S. P., See, P., Petrie, A., Janssen, T. J. B. M., Farrer, I., Griffiths, J. P., ... & Kataoka, M. (2016). High-resolution error detection in the capture process of a single-electron pump. Applied Physics Letters, 108(2), 023502.
3. Karplus, R. (2003). Introductory physics: A Model approach (2nd ed.). Buzzards Bay: Captain’s Engineering Services.

AP Physics 1 2015 Free Response Question 1

Question: 
This question is based on a simple Atwood’s machine: two blocks are connected by a massless string passing over massless pulleys which are frictionless (See figure 1 below). Block 2’s mass (m₂) is greater than block 1’s mass (m₁). Students have to derive the magnitude of the acceleration of block 2 when the two blocks are released from rest. The answer should be expressed in terms of m₁, m₂, and g.

Fig. 1

It could be of interest to mention that a similar question can be found in Feynman’s Tips on Physics. The question involves the calculation of frictional force: “Two masses, m₁ = 4 kg and m₃ = 2 kg, are connected with cords of negligible weight over essentially frictionless pulleys to a third mass, m₂ = 2 kg. The mass m₂ moves on a long table with a coefficient of friction…(Feynman et al., 2006, p. 144).”

Scoring Guidelines:

Using Newton’s second law for block 1 m1a = T – m1g	1 point
Using Newton’s second law for block 2 m2a = m2g – T	1 point
Eliminate T and derive the acceleration: T = m1a + m1g m2a = m2g – m1a – m1g (m2 + m1)a = (m2 – m1)g a = (m2 – m1)g / (m1 + m2)	1 point

(Source: http://apcentral.collegeboard.com/home)

Comments:

The main purpose of the question is to assess students’ ability to apply Newton’s second law of motion to a modified Atwood’s machine. Students are expected to have a basic knowledge of free body diagrams and an understanding of factors that determine the acceleration of the system. However, they may have difficulty with the use of conventions for force and acceleration because one block is moving upward and the other block is moving downward.

We may expect this question to be simply answered by using Newton’s Second law of motion as shown below:

To find the acceleration of block 2, we can visualize an “equivalent question” in which the forces acting on block 1 and block 2 are shown in figure 2 below: Fig. 2 The larger force, m2g, acting on block 2 is the driving force and the smaller force, m1g, acting on block 1 is the opposing force. For block 2: the driving force (m2g) is in the direction to the right, whereas the tension (T) is in the direction to the left. By using Newton’s Second Law of motion, ΣF = ma we have m2g – T = m2a ------ (1) (Convention: The forces that are in the direction to the right are assigned positive.) For Block 1: The tension (T) is in the direction to the right and the opposing force (m1g) is in the direction to the left. By using Newton’s Second Law again, we have T – m1g = m1a ------(2) By solving the two equations (1) and (2), a = (m2 – m1)g / (m1 + m2) However, we can visualize block 1 and block 2 as a system. In this case, the total mass is m1 + m2 and the external forces on the system are m2g – m1g. Thus, a = F/m = (m2 – m1)g / (m1 + m2) Would students be penalized by using only one equation?

Feynman's insights?:
Alternatively, Feynman explains that “the laws of Newton could be stated not in the form F = ma but in the form: the average kinetic energy less the average potential energy is as little as possible for the path of an object going from one point to another (Feynman et al., 1964, section 19–1 A special lecture—almost verbatim).” That is, it is possible to use Euler-Lagrange’s equation to answer this question. Interestingly, Mr. Bader introduced the principle of least action to his student, Feynman. In addition, Dias, Araújo, Silva, Santos, Barros, & Carvalho-Santos (2012), for example, propose to introduce Euler-Lagrange’s equation in introductory physics. However, it is unclear whether students who use a more advanced method could be awarded extra credit or penalized instead. (I happen to know of a high school student who was able to apply Euler-Lagrange’s equation to solve mechanics problems that are even more difficult.)

It is not difficult to use Euler-Lagrange’s equation to solve this question on Atwood’s machine. It can be accomplished by using only a few steps as shown below: 
The kinetic energy of the system (T) is given by T = ½ m₁ẋ² + ½ m₂ẋ² 

The potential energy of the system (U) is given by U = – m₂gx – m₁g(l – x) 

Thus, the Lagrangian can be written as 

T – U = ½m₁ẋ² + ½m₂ẋ² + m₂gx + m₁g(l – x)

By using Euler-Lagrange’s equation of motion, d/dt(∂L/∂ẋ) – ∂L/∂x = 0

We get (m₁ + m₂)ẍ – (m₂ – m₁)g = 0

Therefore, ẍ = (m₂ – m₁)g/(m₁ + m₂)

(Would some physics teachers penalize students for using Euler-Lagrange’s equation to get the correct answer?)

Note:
1. In Feynman’s words, “when I was in high school, my physics teacher—whose name was Mr. Bader—called me down one day after physics class and said, ‘You look bored; I want to tell you something interesting.’ Then he told me something which I found absolutely fascinating, and have, since then, always found fascinating. Every time the subject comes up, I work on it. In fact, when I began to prepare this lecture I found myself making more analyses on the thing. Instead of worrying about the lecture, I got involved in a new problem. The subject is this—the principle of least action (Feynman et al., 1964, section 19–1 A special lecture—almost verbatim).”

2. For another discussion of this question, you can visit the following websites:
https://www.youtube.com/watch?v=NrNG6OPijeY
https://www.youtube.com/watch?v=ff6SNrgUFt0

References:
1. Dias, C. F., Araújo, M. A., Silva, G. M., Santos, C. A., Barros Jr, P., & Carvalho-Santos, V. L. (2012). Adaptation of the Euler-Lagrange equation for studying one-dimensional motions in a constant force. arXiv preprint arXiv:1209.2197.

2. Feynman, R. P., Gottlieb, M. A., Leighton, R. (2006). Feynman's tips on physics: reflections, advice, insights, practice: a problem-solving supplement to the Feynman lectures on physics. San Francisco: Pearson Addison-Wesley.

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

IB Physics 2015 Higher level Paper 2 Question 7

Question: Diffracted electrons through a thin layer of graphite are incident on a fluorescent layer. Students are expected to explain how an observable pattern demonstrates that electrons have wave properties.

Mark Scheme: 
(1) bright and dark rings / circles / circular fringes. 
(2) maximum and minimum / constructive and destructive. 
(3) mention of interference / mention of superposition. 
(4) link to interference being characteristic of waves.

Comments:

Based on the mark scheme, students are expected to state circular fringes (or similar descriptions) and link the observable pattern to a characteristic of waves. That is, they have to specify the observable pattern when electrons are incident on a fluorescent layer in the tube. As a suggestion, students could mention that “bright and dark rings or circular fringes could be observed at the end of the tube.” The bright rings (or circular fringes) are due to constructive interference, whereas dark rings are due to destructive interference. Importantly, the circular fringes can be linked to the wave-like properties of electrons instead of particle-like properties.

Furthermore, students could explain that “the circular fringes are a result of interference (or superposition) which is a wave-like property of electrons.” That is, the formation of fringes follows the principle of superposition because electrons behave like waves. However, interference is not the only characteristic of a wave. For example, Knight (2004) suggests that wave properties could be described as non-localized, continuous, and collective (Knight, 2004). On the other hand, in Hertz’s experiment, “[t]he waves were found to exhibit the properties of: 1. reflection; 2. refraction; 3. interference; 4. diffraction; 5. polarization; and 6. they travelled at c (the speed of light) (Warren, 2003, p. 108).”

However, in Physics for the IB diploma, it is stated that: “when an electron moves inside a crystal whose interatomic spacing has similar dimensions as the de Broglie wavelength will diffraction take place (Tsokos’s 2008, p. 395).” That is, wave-like properties of electrons may include “diffraction” and “de Broglie wavelength.” Thus, students could specify that “the de Broglie wavelength of the diffracted electrons is dependent on their speeds and it should have similar dimensions as the interatomic spacing.” It is worth mentioning that the circular fringes occur where the path difference of the electron waves from the sources is zero or they differ by an integral multiple of the de Broglie wavelength.

Feynman insights?:

Interestingly, Feynman mentions 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).” In essence, diffraction is an important wave property of neutrons when they move inside a block of graphite. However, it is the slowest neutrons that pass through the long block of graphite. These neutrons have longer wavelengths and behave more like waves.

Importantly, Feynman explains that “[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. So, we shall not worry about whether it is interference or diffraction (Feynman et al., 1963, section 30–1 The resultant amplitude due to n equal oscillators).” In short, there is diffraction in the phenomenon “interference,” and there is interference in the phenomenon “diffraction.” In other words, interference involves diffraction or spreadings of waves, whereas diffraction involves interference or summings of waves.

In addition, Feynman clarifies that “[h]istorically, the electron, for example, was thought to behave like a particle, and then it was found that in many respects it behaved like a wave. So it really behaves like neither (Feynman et al. 1963, section 37–1 Atomic mechanics).” Simply phrased, the electrons are neither particles nor waves. More importantly, Feynman elaborates that “[t]he electrons arrive in lumps, like particles, and the probability of arrival of these lumps is distributed like the distribution of intensity of a wave. It is in this sense that an electron behaves sometimes like a particle and sometimes like a wave (Feynman et al., 1963, section 37–5 The interference of electron waves).” In short, it is the distribution of electrons that is guided by a wave function or a probability wave. Moreover, electrons could be observed to have particle-like properties or wave-like properties depending on the experimental set-up.

Note:

In a sense, the phrase “wave properties” in the question should be changed to “wave-like properties.” For example, in Feynman’s own words, “the particle has wavelike properties (Feynman et al., 1966, section 3-1 The laws for combining amplitudes).”

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. L. (1966). The Feynman Lectures on Physics, Vol III: Quantum mechanics. Reading, MA: Addison-Wesley.

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

4. Tsokos, K. A. (2008). Physics for the IB diploma (5th ed.). Cambridge: Cambridge University Press.

5. Warren, N. (2003). Excel HSC Physics. Glebe, NSW: Pascal.

IB Physics 2015 Higher level Paper 2 Question 6

Question: Students are expected to provide a definition of the enhanced greenhouse effect. 

Mark Scheme: addition of greenhouse gases / named greenhouse gas to the atmosphere; increasing the temperature of Earth’s surface / global warming. 

Comments: 

In Physics for the IB diploma, the enhanced greenhouse effect is defined as “the additional warming of the caused by increased quantities of greenhouses gases. The increase in the greenhouse gas concentrations is mainly due to human activity (Tsokos, 2008, p. 824).” In a sense, the mark scheme is better than the textbook definition because it specifies the effect more clearly. That is, the mark scheme includes “increasing the temperature of the Earth’s surface,” whereas the textbook definition only mentions “additional warming.” On the other hand, the textbook definition states that “the increase in the greenhouse gas concentrations is mainly due to human activity.” However, this fact is not found in the mark scheme.

Importantly, the mark scheme and textbook definition of enhanced greenhouse effect can still be improved. For example, they could specify greenhouse gases such as carbon dioxide and methane. Furthermore, the mark scheme and textbook definition could clarify that the increase in the greenhouse gas concentrations can be due to human activities such as burning fossil fuels and agricultural farming. Conversely, Freeman Dyson, for example, opines that good scientists should be skeptical of the global warming. However, the majority of climate scientists believe that the global warming is “very likely” caused by human greenhouse gas emissions. Currently, there are different physical models on the enhanced greenhouse effect.

Feynman insights?:

One simple model to understand the distribution of various gases on the Earth is to assume we have a column of gas extending to a great height at thermal equilibrium, as well as without winds and other kinds of disturbance. Based on this model, Feynman explains that “if we have different kinds of molecules with different masses, they go down with different exponentials. The ones which were heavier would decrease with altitude faster than the light ones. Therefore, we would expect that because oxygen is heavier than nitrogen, as we go higher and higher in an atmosphere with nitrogen and oxygen the proportion of nitrogen would increase. This does not really happen in our own atmosphere, at least at reasonable heights, because there is so much agitation which mixes the gases back together again. It is not an isothermal atmosphere. Nevertheless, there is a tendency for lighter materials, like hydrogen, to dominate at very great heights in the atmosphere, because the lowest masses continue to exist, while the other exponentials have all died out (Feynman et al., 1963, section 40–1 The exponential atmosphere).” Essentially, the densities of various gases in earth’s atmosphere decrease exponentially with height based on the assumption of constant temperature and constant gravitational field.

Furthermore, Feynman elaborates how water vapors radiate heat to the sky. In his own word, “when you go up in altitude the air is colder. The ground is heated by the sun, and the re-radiation of heat to the sky comes from water vapor high in the atmosphere; so at high altitudes, the air is cold — very cold — whereas lower down it is warm. You may say, “Then it’s very simple. Warm air is lighter than cold; therefore the combination is mechanically unstable and the warm air rises.” Of course, if the temperature is different at different heights, the air is unstable thermodynamically. Left to itself infinitely long, the air would all come to the same temperature. But it is not left to itself; the sun is always shining (during the day). So the problem is indeed not one of thermodynamic equilibrium, but of mechanical equilibrium (Feynman et al., 1964, section 9–4 Thunderstorms).” That is, a realistic model of earth’s atmosphere for greenhouse effect cannot be based on thermodynamic equilibrium.

Note:

Physicists prefer to define heat as a process rather than a noun (Romer, 2001; Wong, Chu, & Yap, 2014). Thus, it is good to avoid phrases such as “heat-trapping gases” or “too much heat is trapped on Earth.”

References:

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

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

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

4. Tsokos, K. A. (2008). Physics for the IB diploma (5th ed.). Cambridge: Cambridge University Press. 

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

IB Physics 2015 Higher level Paper 2 Question 5

Question: This question is related to a charge-coupled device (CCD). Students are expected to provide a definition of quantum efficiency of a pixel.

Mark Scheme: ratio of the no. of electrons produced to no. of photons (of a particular energy) incident on the pixel.

Comments: 
The question can be improved by at least two ways. Firstly, the question could be rephrased as “define the quantum efficiency of a charge-coupled device’s pixel.” This is to distinguish from the quantum efficiency of a solar cell. For example, the quantum efficiency of a solar cell can be defined as the ratio of the number of charge carriers collected by the solar cell to the number of photons of a specific range of wavelength absorbed. Secondly, there could be an interesting context such as a selfie or how a photograph can be taken by using a smartphone. That is, the question can be related to the photo-sensors of the smartphone.

On the other hand, the definition specified in the marking scheme is slightly different from the textbook’s definition. In Physics for the IB diploma, the quantum efficiency of a pixel is defined as “the ratio of the number of emitted electrons to the number of incident photons (Tsokos, 2008, p. 466).” The mark scheme uses the phrase “electrons produced” instead of “emitted electrons.” It should be noted that electron-holes pairs are produced instead of only an electron. Next, the marking scheme uses the phrase “number of photons” instead of “incident photons.” However, the mark scheme could include physical requirements such as “normal lighting conditions.” Physics teachers should explain that intense photons emitted from a laser may damage the photo-sensors.

Feynman’s insights?:
In general, quantum efficiency is a measure of the photoelectric sensitivity of a device. Interestingly, Feynman opines that the photoelectric effect is not simply about how photons eject electrons from a classical view. In Feynman’s (1942) words, “one attempts to list those phenomena which seem to indicate that light is quantized, the first type of phenomenon which comes to mind are like the photoelectric effect or the Compton effect. One is however, struck by the fact that since these phenomena deal with the interaction of light and matter their explanation may lie in the quantum aspects of matter, rather than requiring photons of light. This supposition is aided by the fact that if one solves the problem of an atom being perturbed by a potential varying sinusoidally with the time, which would be the situation if matter were quantum mechanical and light classical, one finds indeed that it will in all probability eject an electron whose energy shows an increase of hν, where ν is the frequency of variation of the potential (pp. 3-4).” Importantly, the probability of ejection of the electron should be related to the work function of the matter from a quantum mechanical view.

In The Feynman Lectures on Physics, Feynman does not explicitly discuss the photoelectric effect. From a perspective of classical physics, Feynman explains that “it turns out that in a piece of metal, electrons are attracted to the ions, or to atoms, of the metal. They are attracted, if we may say it crudely, to the metal. In order to get an electron out of a piece of metal, it takes a certain amount of energy or work to pull it out. This work varies with the different kinds of metal (Feynman et al., 1963, section 42–2 Thermionic emission).” In addition, Feynman defines the work function as follows: “W is equal to q_eϕ, where ϕ is the so-called work function, or the voltage needed to pull an electron off the surface (Feynman et al., 1963, section 42–2 Thermionic emission).” In other words, the work function is related to an electric field or potential difference that can remove the electron from the metal’s surface.

More importantly, Feynman elaborates that “it turns out that the behavior of electrons in a metal is not correctly described by the classical theory, but by quantum mechanics, but this only changes the factor in front a little. Actually, no one has ever been able to get the thing straightened out very well, even though many people have used the high-class quantum-mechanical theory for their calculations. The big problem is, does W change slightly with temperature? (Feynman et al., 1963, section 42–2 Thermionic emission).” Essentially, it is possible that the work function of the metal varies with its temperature. From a perspective of quantum mechanics, the probability of emission of electrons from the metal’s surface is dependent on the temperature of the metal.

Note:
1. Based on the classical Maxwell’s theory of light, if the intensity of incident light rays is increased, the kinetic energy of photoelectrons ejected from the metal can be increased. However, this does not happen in the single-photon photoelectric effect. This photoelectric effect experiment illustrates the particulate nature of light rays and the increase in intensity of light rays does not increase the kinetic energy of photoelectrons. Strictly speaking, an increase of light intensity can increase the maximum of kinetic energy of photoelectrons emitted. For example, a more intense light source such as a laser of a particular wavelength can produce the multiple-photon photoelectric effect in which the maximum kinetic energy of photoelectrons can be increased (Georges, 1995). That is, an electron at the surface of a metal could absorb energy from more than one photon such that it can leave the metal’s surface with more kinetic energy.

2. In the famous paper titled On a Heuristic Point of View Concerning the Production and Transformation of Light, Einstein (1905) writes that “it seems to me that the observations of ‘blackbody radiation,’ photoluminescence, production of cathode rays by ultraviolet light, and other related phenomena associated with the emission or transformation of light appear more readily understood if one assumes that the energy of light is discontinuously distributed in space. According to the assumption considered here, in the propagation of a light ray emitted from a point source, the energy is not distributed continuously over ever-increasing volumes of space, but consists of a finite number of energy quanta localized at points of space that move without dividing, and can be absorbed or generated only as complete units (p. 178).” 

References:
1. Einstein, A. (1905). On a Heuristic Point of View Concerning the Production and Transformation of Light. In J. Stachel (ed.), Einstein’s Miraculous year: Five papers that changed the face of physics. Princeton: Princeton University Press. 
2. Feynman, R. P. (1942). Feynman thesis: A New approach to Quantum Theory. Singapore: World Scientific. 
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. Georges, A. T. (1995). Theory of the multiphoton photoelectric effect: A stepwise excitation process. Physical Review B, 51(19), 13735. 
5. Tsokos, K. A. (2008). Physics for the IB Diploma (5th ed.). Cambridge: Cambridge University Press.

IB Physics 2015 Higher level Paper 2 Question 4

Question: This question is about the entropy of an ideal gas in a cyclic process. Students are expected to explain whether there is any change in the entropy of the gas immediately after it has completed the cyclic process. 

Mark Scheme: (1) The entropy is unchanged. (2) The gas returned to its original state. 

Comments: 
Based on the mark scheme, one may expect a reasonably good answer is “the entropy is unchanged because the gas has returned to its original state.” However, physics teachers should also accept explanation such as “entropy is a state function which is path independent.” Alternatively, Feynman mentions that “we have found another quantity which is a function of the condition, i.e., the entropy of the substance. Let us try to explain how we compute it, and what we mean when we call it a function of the condition (Feynman et al., 1963, section 44-6 Entropy).” Simply phrased, entropy is a function of the condition and the change of entropy is dependent on the initial and final condition of the gas.

On the other hand, the cyclic process ABCDA is not a Carnot cycle. Furthermore, the definition of entropy as dQ/T is based on the condition that the process is reversible. In general, a reversible process is a quasi-static process whose direction can be “reversed” by means of infinitesimal change. In Cheng’s (2006) words, “[a]ny arbitrary reversible cyclic process can be approximated by a very large number of infinitesimal Carnot cycles (p. 48).” In other words, the combination of these large number of Carnot cycles is a “good approximation” to the cyclic process when the temperature difference ΔT between the isothermal processes of any two neighboring Carnot cycles approaches zero. Importantly, the adiabatic processes of the neighboring processes cancel each other because their directions are opposite to each other.

Feynman insights?:
There are at least three insights that we can learn from Feynman’s lectures.

1. Entropy is a state function of temperature and volume: In the words of Feynman, “[w]e can, therefore, say that there is a certain function, which we call the entropy of the substance, that depends only on the condition, i.e., only on the volume and temperature (Feynman et al., 1963, section 44–6 Entropy). More importantly, Feynman elaborates that “[c]onsider the system in two different conditions, much as we had in the experiment where we did the adiabatic and isothermal expansions. (Incidentally, there is no need that a heat engine has only two reservoirs, it could have three or four different temperatures at which it takes in and delivers heats, and so on.) We can move around on a pV diagram all over the place, and go from one condition to another. In other words, we could say the gas is in a certain condition a, and then it goes over to some other condition, b, and we will require that this transition, made from a to b, be reversible. Now suppose that all along the path from a to b we have little reservoirs at different temperatures, so that the heat dQ removed from the substance at each little step is delivered to each reservoir at the temperature corresponding to that point on the path. Then let us connect all these reservoirs, by reversible heat engines, to a single reservoir at the unit temperature. When we are finished carrying the substance from a to b, we shall bring all the reservoirs back to their original condition. Any heat dQ that has been absorbed from the substance at temperature T has now been converted by a reversible machine, and a certain amount of entropy dS has been delivered at the unit temperature as follows: dS = dQ/T (Feynman et al., 1963, section 44–6 Entropy).” Essentially, the entropy of a system is a state function and this is based on the assumption of reversible heat engines.

2. A reversible process is a theoretical idealization: According to Feynman, “we will lose something if the engines contain devices in which there is friction. The best engine will be a frictionless engine. We assume, then, the same idealization that we did when we studied the conservation of energy; that is, a perfectly frictionless engine (Feynman et al., 1963, section 44-3 Reversible engines).” That is, the reversible heat engines should be ideally frictionless. Similarly, in the words of Landau and Lifshitz (1978), “[r]eversible processes are those in which the entropy of the closed system remains constant, and which can, therefore, take place in the reverse direction. A strictly reversible process is, of course, an ideal limiting case; processes actually occurring in Nature can be reversible only to within a certain degree of approximation (p. 33).” In short, a reversible process is an idealized process that is unlikely possible in the real world. Thus, one may explain that the entropy of the ideal gas is unchanged because the process is reversible, but it involves a theoretical idealization and approximation.

3. The entropy of an isolated system or universe increases: Feynman clarifies that “in any process that is irreversible, the entropy of the whole world is increased. Only in reversible processes does the entropy remain constant. Since no process is absolutely reversible, there is always at least a small gain in the entropy; a reversible process is an idealization in which we have made the gain of entropy minimal (Feynman et al., 1963, section 44-6 Entropy).” An important point here is that it is always likely to have an increase in the entropy of the universe. Furthermore, in Feynman Lectures on Computation, Feynman (1996) elaborates that “[f]or an irreversible process, the equality is replaced by an inequality, ensuring that the entropy of an isolated system can only remain constant or increase (p. 141).” More important, the entropy of an isolated system can remain constant or increase depending whether the process is reversible or irreversible.

Note:
1. In the words of Clausius (1862), “I prefer going to the ancient languages for the names of important scientific quantities, so that they mean the same thing in all living tongues. I propose, accordingly, to call S the entropy of a body, after the Greek word [τρoπη], ‘transformation’. I have designedly coined the word entropy to be similar to energy, for these two quantities are so analogous in their physical significance, that an analogy of denominations seems to me helpful.”

2. You may want to take a look at this website:
http://feynman-answer.blogspot.sg/2016/06/entropy-remains-unchanged-or-increases.html

References:

1. Cheng, Y.-C. (2006). Macroscopic and Statistical Thermodynamics. Singapore: World Scientific. 

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

3. Landau, L. D., & E.M. Lifshitz, E. M. (1980). Statistical Physics. Vol 1. Oxford: Pergamon Press.

4. Clausius, R. (1862). XXIX. On the application of the theorem of the equivalence of transformations to the internal work of a mass of matter. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 24(160), 201-213.