BOSTES HSC Physics 2015 Question 29

Question:

In the Large Hadron Collider, protons move at a speed about 0.99999 c.
(a) What are the advantages of using superconductors to produce magnetic fields in the Large Hadron Collider? 

(b) Discuss concepts in special relativity that are related to the high speed of protons in the Large Hadron Collider.

Marking Guidelines:

29(a) Criteria	Marks
• Identify two advantages	2
• Identify an advantage	1

Sample answer:

Strong magnetic fields are required to guide the path of protons due to their high speeds and the mass dilation of protons. To produce strong magnetic fields, very high electric currents are required and this is feasible by using superconductors.

29(b) Criteria	Marks
• Discusses concepts in special relativity that are related to high speed of protons in the Large Hadron Collider.	3

Sample answer:

The protons are moving at almost the speed of light and this is related to the concept of mass dilation. Mass dilation implies that an increase amount of electrical energy is needed to accelerate the protons. (Physics teachers may accept answers that include length contraction/time dilation or non-inertial frame of reference.)
(Source: https://www.boardofstudies.nsw.edu.au/hsc_exams/2015/guides/2015-hsc-mg-physics.pdf)

Comments:

There are at least three advantages of using superconductors to produce the magnetic fields and guide protons around the Large Hadron Collider may include strong magnetic fields, minimal generation of thermal energy, and lower electrical power consumption.

1. Strong magnetic fields: The magnetic fields of Large Hadron Collider can be as high as about 8 Teslas and it is about 10 000 times stronger than the earth’s magnetic fields. Strong magnetic fields are required to deflect the protons that are moving at high speeds. To produce strong magnetic fields, high electric currents are required and superconductors allow very high electric currents.

2. Minimal generation of thermal energy: As mentioned earlier, high electric currents are required to achieve strong magnetic fields. If there is electrical resistance in the magnets, there will be ohmic heating, and thus, an increase in temperature of the magnets or even overheating. However, the electrical resistance of a superconductor is effectively zero, and the generation of thermal energy is minimal.

3. Lower electrical power consumption: In the words of Lederman (1993), “[o]ne could ramp the model magnets from zero current to 5,000 amperes in 10 seconds, and the superconductivity persisted. In 1978-79 a production line began producing twenty-one-foot magnets with excellent properties, and in 1983 the Tevatron began operating as a superconducting ‘afterburner’ at the Fermilab complex. The energy went from 400 GeV to 900 GeV, and the power consumption was reduced from 60 megawatts to 20 megawatts, with most of that used to produce liquid helium (p. 234).” In other words, the reduction in electrical power consumption also results in cost savings.

In part (b) of the question, students are expected to discuss the application of special relativity to the protons in the Large Hadron Collider. The sample answer involves the concept of mass dilation that means an increasing amount of energy is needed to accelerate the proton. However, in an article titled The concept of mass (mass, energy, relativity), Okun (1989) writes that “there is in the theory of relativity essentially just one term, mass, and all the others come ‘from the devil’ (p. 118).” In other words, he disagrees with the use of terms such as relativistic mass, transverse mass, and longitudinal mass. 

Okun (1989) has five arguments against the concept of relativistic mass and the definition of mass in terms of E = mc². The five arguments are summarized as shown below:

1. Aesthetic: The equations E² – p² = m² and p = Ev are elegant rather than E = mc².

2. Ethical: It creates the illusion that E/c² is a universal measure of inertia or universal gravitational mass.

3. Philosophical: There is not a complete equivalence of mass and energy as suggested by E = mc².

4. Terminological: There are confusions in the notation and terminology. Okun prefers the equation E₀ = mc² instead of E = mc².

5. Pedagogical: Students who learn the concept of relativistic mass cannot truly understand the essence of the theory of relativity.

On the contrary, in an article titled In defense of relativistic mass, Sandin (1991) writes that “The one equation nearly every student brings to an introductory treatment of relativity is E = mc². From the pro-relativistic mass point of view (the pro-view), this famous equation states with elegant simplicity that energy and mass are equivalent (p. 1032).” Sandin’s arguments to support relativistic mass are shown below:

1. Aesthetic: The equation E = mc² is elegant and simple.

2. Ethical: It is consistent to accept the equivalence of inertial mass and gravitational mass (both equal to E/c²).

3. Philosophical: Energy and mass are equivalent as suggested by E = mc². Thus, there is a direct proportionality between energy and mass.

4. Terminological: To be precise, one may use equations such as Δm = ΔK_i/c².or Δm = E_ex/c².

5. Pedagogical: The concept of relativistic mass is consistent and simple within relativity, whereas four-vectors can be unnecessarily complicated for students.

There is no agreement on the use of terms such as relativistic mass among physicists. Some physicists are also against teaching the concept of length contraction and time dilation in the theory of special relativity.

Feynman’s insights or goofs?:

Feynman states that “[t]he mass varies with velocity according to the law m = m₀/√(1−v²/c²) where m₀ is the mass of the body at rest and c is the speed of light (Feynman et al., 1963, section 10-5 Relativistic momentum).” In other words, Feynman is open to the concept of relativistic mass or velocity-dependent mass. Furthermore, Feynman 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’ (Feynman et al., 1963, 7–8 Gravity and relativity).” That is, it is possible that a beam of light has ‘mass’ because it has energy. This is based on the principle of equivalence of energy and mass. However, this approach is objected by particle physicists such as Okun.

On the contrary, Feynman also provides the alternative view: “the following relations are easily proved, and turn out to be very useful: E² − P²c² = m₀²c⁴ and Pc = Ev/c (Feynman et al., 1963, section 16–5 Relativistic energy).” In other words, the two equations do not need the concept of mass dilation or relativistic mass. Simply put, Feynman does not care so much about terminologies and he has a pragmatic attitude toward physics. (In short, it is more important for him to check whether the equations work.) Similarly, in his Nobel speech, Feynman (1965) opines that “[m]any different physical ideas can describe the same physical reality (p. 30)” and that “equation guessing might be the best method to proceed to obtain the laws for the part of physics which is presently unknown (p. 31).”

Note: 

1. Okun (1989) quotes Einstein’s letter to Barnett in 1948, “It is not good to introduce the concept of the mass, m_r = m₀ /√(1 – v²/c²) of a moving body for which no clear definition can be given.” However, in Autobiographical notes, Einstein is supportive of the idea of how kinetic energy may contribute to mass: “…the theory had to combine the following things: 1. From general considerations of special relativity theory it was clear that the inert mass of a physical system increases with the total energy (therefore, e.g., with the kinetic energy). 2. From the very accurate experiments… it was empirically known with very high accuracy that the gravitational mass of a body is exactly equal to its inert mass (Einstein, 1949, p. 61).” 

2. In Lederman’s (1993) words, “[t]he system is self-correcting. If the particle gains too much energy (mass), its radius will increase and it will arrive later at the gap and see a decelerating voltage, which will correct the error (p. 219).” Lederman was the director of Fermilab and he was awarded Nobel (Physics) Prize 1988 for the neutrino beam method and the demonstration of the doublet structure of the leptons through the discovery of the muon neutrino. 

References:

1. Einstein, A. (1949/1979). Autographical notes (Translated by Schilpp). La Salle, Illinois: Open court.

2. Feynman, R. P. (1965). The development of the space-time view of quantum electrodynamics. In Brown, L. M. (ed.), Selected papers of Richard Feynman. 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. Lederman, L. & Teresi, D. (1993). The God Particle: If the universe is the answer, what is the question?. New York: Delta Books.

5. Okun, L. B. (1989). The concept of mass (mass, energy, relativity). Soviet Physics Uspekhi, 32(7), 629.

6. Sandin, T. R. (1991). In defense of relativistic mass. American Journal of Physics, 59(11), 1032-1036.

BOSTES HSC Physics 2015 Question 28

Question: 
A light trolley (attached to a copper plate) moves at an initial velocity (v) towards a magnet fixed to a support as shown below. Sketch a graph of the trolley’s velocity traveling from A to D. Provide detailed explanations on your graph.

Marking Guidelines:

Criteria	Marks
• Sketch a graph of trolley’s velocity showing where it has constant velocities and where its velocity is decreasing. • Provide correct explanations on the graph: the effects of changing magnetic flux through the copper plate.	5

(Source: https://www.boardofstudies.nsw.edu.au/hsc_exams/2015/guides/2015-hsc-mg-physics.pdf)

Sample answer:

The trolley’s velocity is decreased within the region under the magnet (from B to C) because its kinetic energy is converted into heat energy within the copper plate. The change in magnetic flux is caused by the movement of the copper plate through the magnetic field. This generates induced currents in the copper plate (Faraday’s law) that result in a magnetic field that opposes the changing magnetic flux (Lenz’s law), and thus, it produces a force that slows down the trolley.

Comments:
According to the sample answer, there is a transformation of kinetic energy into “heat energy in the copper plate.” On the contrary, Zemansky (1970) argues that we should avoid the phrase “heat in a body,” and disagrees with the use of the term “thermal energy” which may mean heat energy or internal energy. More importantly, it is widely reported that students have difficulty in distinguishing heat and internal energy (e.g. Meltzer, 2004). Furthermore, the first law of thermodynamics is based on three key concepts: internal energy, heat, and mechanical work. However, the sample answer could be rephrased as “a transformation of kinetic energy into internal energy of the copper plate.”

On the other hand, the sample answer is vague when it states that “this induces currents in the Cu plate that produce a magnetic field that opposes the changing flux and hence produces a force that decelerates the trolley.” Importantly, there are two different forces that reduce the speed of the trolley. As a suggestion, we may explain the effects of changing flux through the plate as a two-step process as follows: 
(1) Repulsive force: firstly, as the trolley moving toward the magnet, the changing flux in the copper plate causes a repulsive force to slow down the trolley (Fig 1). 

Fig 1: When the trolley is moving toward the magnet

(2) Attractive force: secondly, as the trolley moving away from the magnet, the changing flux in the copper plate causes an attractive force to further slow down the trolley. The directions of the two forces (repulsive and attractive) are toward the left (Fig 2). 

Fig 2: When the trolley is moving away from the magnet

Based on the sample answer, it seems to suggest induced currents in the copper plate is due to Faraday’s law of electromagnetic induction, whereas the production of a magnetic field that opposes the changing flux is due to Lenz’s law of electromagnetic induction. However, we could explain that Lenz’s law is embedded in the Faraday’s law. Simply put, the direction of induced currents can be explained by both Faraday’s law and Lenz’s law.

Feynman’s insights or goofs?:

In Feynman’s words, “[w]hen we gave “the flux rule” that the emf is equal to the rate of change of the flux linkage, we didn’t specify the direction of the emf. There is a simple rule, called Lenz’s rule, for figuring out which way the emf goes: the emf tries to oppose any flux change. That is, the direction of an induced emf is always such that if a current were to flow in the direction of the emf, it would produce a flux of B that opposes the change in B that produces the emf (Feynman et al., 1963, section 16–2 Transformers and inductances).” In short, Lenz’s law is expressed by the “minus sign” in the Faraday’s law, E_s = -N_s dΦ/dt.

However, Feynman explains that “[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).” In addition, Feynman mentions that “for a monatomic gas, the kinetic energy is the total energy. In general, we are going to call U the total energy (it is sometimes called the total internal energy (Feynman et al., 1963, section 39–2 The pressure of a gas).” Feynman’s definitions of heat energy and internal energy are mainly about the kinetic energy of an object. In a sense, Feynman is sloppy in defining heat energy and it is potentially confusing or misleading for introductory students.

Note:

1. Feynman also explains that “Faraday’s observations led to the discovery that electric and magnetic fields are related by a new law: in a region where the magnetic field is changing with time, electric fields are generated. It is this electric field which drives the electrons around the wire — and so is responsible for the emf in a stationary circuit when there is a changing magnetic flux. The general law for the electric field associated with a changing magnetic field is ∇ × E = −∂B/∂t. We will call this Faraday’s law. It was discovered by Faraday but was first written in differential form by Maxwell, as one of his equations (Feynman et al., 1964, section 17–1 The physics of induction).”

2. In the words of Feynman, “[a]ccording to Lenz’s law, these currents are in such a direction as to oppose the increasing field. So the induced magnetic moments of the atoms are directed opposite to the magnetic field (Feynman et al., 1964, section 34–1 Diamagnetism and paramagnetism).” 

3. “There are three main infelicities of expression that are indulged in by writers who are trying to come down to the level of introductory physics or chemistry. They are (1) Referring to the 'heat in a body.' (2) Using 'heat' as a verb. (3) Combining heat and internal energy into one undefined concept 'thermal energy,' which on one page means heat and on the next page means internal energy (Zemansky, 1970, p. 298).”

References:

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

2. Meltzer, D. E. (2004). Investigation of students’ reasoning regarding heat, work, and the first law of thermodynamics in an introductory calculus-based general physics course. American Journal of Physics, 72(11), 1432-46.

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

BOSTES HSC Physics 2015 Question 27

Question: Maxwell’s theory of electromagnetism explained the nature of light and predicted the existence of other electromagnetic waves. Explain how Hertz performed experiments that validate Maxwell’s theory.

Marking Guidelines

Criteria	Marks
• Describes how Hertz performed experiments to test Maxwell’s theory and his predictions. • Describe how Hertz’s experiments validated Maxwell’s theory.	6

(Source: https://www.boardofstudies.nsw.edu.au/hsc_exams/2015/guides/2015-hsc-mg-physics.pdf)

Comments: 

This question is about an understanding of Hertz’s experiments that validate Maxwell’s theory. Students are expected to describe how Hertz performed experiments to test Maxwell’s predictions. However, we could provide the following points: 

1. Oscillator: To produce electromagnetic waves, Hertz made an oscillator - an induction coil connected by two brass knobs - that generates sparks in an air gap between the two knobs under high voltages.

2. Detector: To detect electromagnetic waves, Hertz simply took a piece of copper wire, bent it into a circular shape, and created a short air gap between its two ends. 

3. A “surprising” phenomenon: When a spark is generated at the terminals of the induction coil, we could observe another spark in the air gap of the wire when the oscillator is reasonably close to the detector.

4. Theoretical explanation: According to Maxwell’s theory, electromagnetic waves are produced while the sparks are being generated. In addition, the electromagnetic waves propagate to the detector and set up oscillating electric and magnetic fields in the wire of the detector. (In other words, there is a resonance between the oscillator and the detector.)

5. Properties of electromagnetic waves: Hertz’s experiments show that electromagnetic waves have the following properties of light: 1. reflection; 2. refraction; 3. interference; 4. diffraction; 5. polarization.

6. Mathematical validation: Hertz was able to set up a stationary wave pattern by using a metal reflector, and determine the wavelength (λ) of the electromagnetic waves by measuring the distance between two nodes. Furthermore, by determining the oscillating frequency (f) of the electric current, Hertz calculated the speed of electromagnetic waves in air (v = fλ) which equals to the speed of light.

Essentially, Hertz developed the first primitive radio that can transmit electromagnetic waves. In a sense, Hertz’s experiments validate Maxwell’s equivalence of light and electromagnetic wave. Interestingly, Duhem, for example, proposed that Helmholtz’s electrodynamics could be another alternative to explain Hertz’s experiments (O’Rahilly, 1965). Thus, Hertz’s experiments do not validate Maxwell’s theory completely and conclusively. It is worth mentioning that Hertz intended to demonstrate that Maxwell’s theory is incorrect. In Hertz’s (1893) words, “I reflected that it would be quite as important to find out that electric force was propagated with an infinite velocity, and that Maxwell’s theory was false, as it would be, on the other hand, to prove that this theory was correct, provided only that the result arrived at should be definite and certain (p. 8).”

Feynman’s insights or goofs?:

Feynman has objections on the correctness of Maxwell’s theory. For instance, Feynman proposes that point charges interact only with other charges, but the interaction is related to both advanced and retarded waves. In addition, he explains that “[l]ight behaves like photons. It isn’t 100 percent like the Maxwell theory. So the electrodynamics theory has to be changed. We have already mentioned that it might be a waste of time to work so hard to straighten out the classical theory, because it could turn out that in quantum electrodynamics the difficulties will disappear or may be resolved in some other fashion. But the difficulties do not disappear in quantum electrodynamics (Feynman et al., 1964, section 28–5 Attempts to modify the Maxwell theory).” Essentially, light has both particle-like and wave-like characteristics. However, there are still difficulties in Maxwell’s theory even after modifications are made with quantum mechanics.

Feynman’s another objection is Maxwell’s ether. In an invited talk presented at the symposium The Past Decade in Particle Theory, Feynman (1970) elaborates that “in the case of ‘the ether never being found,’ it was ultimately realized that there wasn’t any ether at all, the ether was one of these scaffoldings to create a theory. It was later realized that the ether was an irrelevant complication and it may be that the partons are also nonexistent (p. 812).” In other words, Feynman believes that it is difficult to define the ether which cannot be detected conclusively. However, Einstein (1920) writes that, ‘[m]ore careful reflection teaches us however, that the special theory of relativity does not compel us to deny ether (p. 13).” In a sense, the concept of the ether does not fade away, but it is currently expressed by the term, fields (Wilczek, 1999).

Note:

1. In an article titled The Persistence of Ether, Wilczek (1999) writes that “[h]ow did I provoke Feynman? I asked him, ‘Doesn’t it bother you that gravity seems to ignore all we have learned about the complications of the vacuum?’ To which he immediately responded, ‘I once thought I had solved that one. I had a slogan: "The vacuum is empty. It weighs nothing because there’s nothing there."’ It was then he got wistful. I was deeply impressed to realize that Feynman had been wrestling with the problem of the cosmological term already in the 1940s, long before it became a widespread obsession and frustration. You have to admit that his slogan is catchy. So just maybe, despite everything I’ve said up to this point, eventually we really may have to do without ether (p. 13).”

2. Wilczek (1999) writes that “they are all surface manifestations of a single more basic entity, the electron field, an ether that pervades all space and time uniformly (p. 13)”.

References:

1. Einstein, A. (1920). Ether and the Theory of Relativity. In A. Einstein (1922). Sidelights on Relativity. London: Methuen. 

2. Feynman, R. P. (1970). Partons. In R. P., Feynman & L. M., Brown (Eds.), Selected Papers of Richard Feynman: With Commentary (pp. 773-813). Singapore: World Scientific.

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. 

4. Hertz, H. (1893). Electric Waves: Being Researches on the Propagation of Electric Action with Finite Velocity Through Space. London: Macmillan.

5. O’Rahilly, A. (1965). Electromagnetic Theory: A Critical Examination of Fundamentals, vol. 1. New York: Dover. 

6. Wilczek, F. (1999). The Persistence of Ether. Physics Today, 52(1), 11-13.

BOSTES HSC Physics 2015 Question 26

Question: 
We can use two mathematical models to calculate the work done when a 300 kg satellite is moved from Earth’s surface to an altitude of 200 km. Model A uses the mathematical equation, W = mg∆h, whereas Model B uses the mathematical equation, ΔE = −GMm/(R + h) – (–GMm/R).
(a) State the assumptions made on Earth’s gravitational field in models X and Y.
(b) Explain why models X and Y produce results that are approximately the same.

Marking Guidelines

Question 26 (a) Criteria	Marks
• Identifies correct assumptions.	2
• Identifies an assumption with correct information.	1

Possible answer: 
Assumption for Model X: Earth’s gravitational field is uniform.

Assumption for Model Y: Earth’s gravitational field varies with altitude.

Question 26 (b) Criteria	Mark
• Correct reason.	1

Possible answer: Variations in Earth’s gravitational field from the surface to an altitude of 200 km are sufficiently small.

(Source: https://www.boardofstudies.nsw.edu.au/hsc_exams/2015/guides/2015-hsc-mg-physics.pdf)

Comments: 
In part (a), possible answer includes “the Earth’s gravitational field is uniform.” Alternatively, students might answer that the gravitational field is approximately constant on the Earth’s surface and up to an altitude of 200 km (as specified in the question). That is, we may assume the gravitational field strength to be constant (about 9.8 N/kg) if the satellite is not going beyond a certain height. Therefore, the work done in lifting the satellite can be mathematically expressed by the equation, W = mg∆h, because the gravitational force (mg) is assumed to be constant from Earth’s surface to the altitude (∆h) of 200 km. However, this assumption does not always hold for a low Earth orbit satellite which may have an altitude between 160 km and 2,000 km.

On the other hand, the sample answer states that “Model Y assumes the gravitational field changes with altitude.” To be precise, students could answer that the gravitational field follows the inverse square law (g = GM/r²). However, some students might include additional assumptions such as the density of the Earth is constant, the Earth is not rotating, and the Earth is spherical. Strictly speaking, the gravitational field is not even constant or the same at different locations on the Earth’s surface.

In part (b), the sample answer states that “variations in gravitational attraction from the Earth’s surface to an altitude of 200 km are sufficiently small.” However, 200 km is still quite a long distance. Thus, we should explain that the altitude of 200 km is negligible if it is compared to the radius of the Earth which is about 6,370 km. We can clarify this fact mathematically as shown below: 
Change in gravitational potential energy, ΔE = −GMm/(R + h) – (–GMm/R)
= (−GMm/R) (1 + h/R)^–1 – (–GMm/R) ≈ (−GMm/R) (1 – h/R – 1) 
= (GMmh/R²) = mgh
Note: By using (1 + x)^–1 = 1 – x for small values of x.

The approximation is appropriate because the altitude of 200 km is relatively short as compared to the radius of the Earth (i.e. h/R << 1). However, we can also explain that the error in work done increases as the altitude increases. The error is due to the neglected terms such as (−GMm/R) (h/R)².
Note: (1 + x)^–1 = 1 – x + x² – x³ + …

Feynman’s insights or goofs?:
Interestingly, Feynman explains that “[i]f we have a gravitational field that is uniform, if we are not going to heights comparable with the radius of the earth, then the force is a constant vertical force and the work done is simply the force times the vertical distance (Feynman et al., 1963, section 14-3 Conservative forces).” That is, the work done lifting an object is equal to the gravitational force times the vertical distance of the object raised. However, this simplification is possible only if the object is not moving to heights that are comparable to the radius of the earth. In other words, if the height increases, the error in calculating the work done increases.

More importantly, in Feynman’s words, “[t]he gravitational field of the earth is not precisely uniform, so a freely falling ball has a slightly different acceleration at different places — the direction changes and the magnitude changes (Feynman et al., 1964, section 42-5 Gravity and the principle of equivalence).” Simply put, the gravitational field of the Earth is not strictly constant and it can vary depending on the location of the Moon and the Sun. Essentially, the gravitational field of the Moon can affect the shape of the Earth and this further change the gravitational field of the 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. L. (1964). The Feynman Lectures on Physics, Vol II: Reading, MA: Addison-Wesley.