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.