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.