Today I handed back the graded mid-term tests in my first-year electricity and magnetism course, and the results were OK. The 2-hour test consisted of five problems and then a final 7-part question that probed for conceptual understanding.
In the past I have asked students to write a paragraph about this or that, but this time around I chose a true/false + explanation format. Presented with a statement, you first have to indicate whether the statement is true or false, and then you have to explain in a sentence or two. Each such question is worth 2 marks, with the understanding that just indicating true or false without any explanation is likely to lead to a grade of 0 for the question.
I like such questions much more than multiple-choice questions, because in the latter one has no idea what students are thinking. Requesting an explanation gives some insight into what misconceptions students have, which you can then go about trying to remedy, both in the current course and (one hopes) in subsequent editions of the same course. As an instructor, one is informed about what students understand and what they find difficult.
I first encountered such questions when I was a first-year student at Queen’s University at Kingston. These questions were a feature of every test and exam of Selwyn Caradus, my linear algebra professor.
Here are the questions, followed by answers, some good responses from students, and some common errors and misconceptions I noted (113 students wrote the exam.)
(a) At a point in space where the electrical potential is zero, the electric field is also necessarily zero.
False. The electric field might or might not be zero. The value of the potential at a point is not related to the electric field at that point. The magnitude of the electric field is the (negative of the) rate of change of the potential; more accurately, the electric field is the negative gradient of the potential:
$\vec{{\bf E}} = -\nabla V$
So in other words, the electric field at a point does not depend on the value of the potential at that point, but rather on how fast the potential is changing at that point.
It’s hard for students to adopt a “functional” approach; that is, they are used to solving problems in high-school physics by determining the difference in the values of some quantity at two different points (initial and final velocity, for example). It’s a step to consider that the velocity (or whatever other quantity we’re interested in) is a function, and has a value at every point of some space.
This also engenders confusion between $V$ and $\Delta V$. For example, one incorrect answer argued that, “If $V = 0$, then $E = 0$, because $E$ is proportional to $V$ ($E = \dfrac{\Delta V}{d}$).”
Another student response that seems to indicate the same confusion between $V$ and $\Delta V$ is: “If there is no potential, then there is no electric field.” Another student said that there has to be an electric field in a place where there is a potential; if there is no potential, then there can be no electric field either. I believe that this also results from a misunderstanding of the formula $E = \dfrac{\Delta V}{d}$, and a confusion between potential and potential difference.
Another student commented that potential is “not useful,” an unfortunate interpretation of something I said repeatedly in class. I mentioned that only potential differences were physically meaningful, and that the potential values can be increased by a global constant value without changing the corresponding electric field. The electric field value at each point is physically meaningful, because a charged particle experiences a force based on the value of the electric field at its current position.
Some students, who correctly said that the statement in Part (a) is false, gave the example of a parallel-plate capacitor. The potential could be zero someplace within the capacitor, say at one of the plates, and yet the electric field is constant and non-zero there (i.e., where the potential is zero). A nice counterexample to the incorrect statement in Part (a).
(b) The magnitude of the electric field is greater where the equipotentials are closer together.
True. This is a standard fact about the connection between an electric field and its equipotential surfaces (also called equipotentials for short).
Some students made use of an analogy with topographical maps that we discussed in class; the contour lines on a topographical map are lines of constant height, and also lines of constant gravitational potential, so they are also equipotentials. Where the contour lines on the topographical map are closer together, the terrain is steeper.
(c) It’s not possible for two neutral objects to attract each other with an electric force; at least one of them must be charged.
False. Students were familiar with the example of a charged object attracting a neutral object from numerous examples discussed in class. They seemed to understand that a charged object brought near a neutral object tends to induce charge polarization in the neutral object.
However, many students did not realize that two neutral objects could both be polarized “naturally,” without the necessity for having a charged object nearby to induce the polarization. For example, water molecules are neutral yet polarized, and this causes some of water’s unusual properties (such as the fact that it expands as it is cooled through the last few degrees up to its freezing point).
(d) When a metal sphere is placed in a region of space where there is an electric field present, the electric field inside the metal sphere is slightly increased, due to polarization effects.
False. It is said that liars tend to embellish their stories with a little too much detail, and I believe I may have fallen into that error in this question.
This question probes for a standard fact about good conductors. When a good electrical conductor is placed in a region of space where there is an electric field present, charges flow to the surface of the conductor in such a configuration that the net electric field inside the conductor (and also within any hollows inside the conductor) is zero. This occurs once the conductor reaches equilibrium, which is typically within a small fraction of a second.
An applications of this concept: Electrical shielding. Coaxial cables have a metal sheath towards the outside of the cable, which shields the central, signal-carrying, wire from disturbances from external electric fields. Sensitive electrical equipment is often shipped within a foil pouch for the same reason. The external enclosure does not even have to be a solid metal; a mesh can sometimes do the same job. For example, metal structures such as bridges are often effective at shielding AM radio signals, as you might have experienced while driving on a bridge with your radio on. FM signals are typically not affected. So, for a mesh, the shielding effect is frequency-dependent (and depends also on the size of the mesh).
A few students made the mistake of thinking that the electric field inside any object (not just a conductor at equilibrium) must be zero. A few inevitably don’t realize that the conductor must be in equilibrium for the electric field inside the conductor to be zero. Thus, for a current-carrying wire, the electric field within the wire is certainly not zero.
(e) Charge cannot be transferred from one insulator to another, because current cannot flow in an insulator.
False. A major point of confusion is not distinguishing between charge and current. Just because it is difficult for a current to flow in an electrical insulator (an enormous electric field is needed), it is quite possible for a small charge to be transferred into or out of an electrical insulator.
Quite a number of students erroneously stated that because a current “cannot” flow in an insulator, charge cannot be transferred from one insulator to another.
Students realize that insulators have no free electrons, because all electrons in an insulator are tightly bound to atoms. But a few of these electrons can be ripped from an insulator (or added) by large forces. Friction is one such force; that is, electrons can be transferred from one insulator to another by rubbing. You can do this by combing your hair, rubbing a balloon on your head, or rubbing your shod feet on a carpet.
The same cause results in “static cling” in your dryer. The small sheets some people place in their dryer to eliminate static cling act by coating the clothes so that they are more slippery. The friction between them is reduced, resulting in little or no charge transfer between the clothes (and between the clothes and the dryer).
(f) If two light bulbs are connected in a row (“in series”) and then connected to a battery, the light bulb closer to the negative terminal of the battery will be brighter, because some of the current is used up in the first bulb.
False. This question was mostly answered quite well, perhaps due to the “necklace model” introduced in lectures. Maybe more students understand and retain things better if there are images attached to what they learn? This is the reason that proverbs and fables were used to pass on wisdom in oral societies, such as in our past: they are much more memorable that mere statements of fact. Moral for us teachers: Tell stories!
The question probed to see if students understood the “necklace model” of current flow in a series circuit. The point is that current is conserved, so the same current flows at all points of a series circuit simultaneously. Therefore, the bulbs light up simultaneously. As long as the bulbs are identical, they will also be equally bright.
No current is used up in the light bulbs; rather, some of the potential energy of each electron is converted to heat and light.
(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 25 January 2021.)