by Len Cerny

At first it may seem as if a concept as simple as current, measured as charge per unit time, would be entirely unproblematic for upper-division physics students. However, nearly all of our incoming juniors have an incompletely developed understanding of current. Here are some of the things students often do not thoroughly understand as they enter upper-division courses:

• Current as a vector
• Linear, surface, and volume current densities
• The type of “gate” one would use to measure linear, surface, and volume current densities
• Recognizing that for a steady current $I = dq/dt = Q/T = \lambda v$
• Recognizing that a linear current density is simply the current
• Understanding current distributions that produce non-zero magnetostatic fields vs. zero magnetic fields

• Steady current
• Current as a flux

While students are very capable of acquiring a more complete understanding of current, if lessons are taught assuming that students already have these understandings, students may spend an unnecessarily long time struggling with conceptualizing current. We therefore advocate that some class time be spent bolstering students' understanding of current.

When asked to find the magnetic vector potential in all space due to spinning ring of charge, many of our students assumed that the current would be the same everywhere on the ring. Consequently, many pulled current out of the integral. Students often did not recognize that although the magnitude of the current was the same everywhere on the ring, the direction of that current was different at each location.

With circuit problems in introductory physics courses, the direction of the current is always along the wires and the curving of the wires in the circuit is usually ignored when solving simple LRC circuit problems. Students may have faced several problems where they needed to establish the direction of the current as either positive or negative along the direction of a straight wire and then apply the right-hand rule to find forces or magnetic fields. However, even in solenoid problems, most of the “hard thinking” has been done by the instructor or the text, and students become familiar with equations like $B = \mu_{o}IN/L$, where current is again addressed as a scalar. When first addressing problems where current is in an integrand, and considering the vector nature is essential, students may need their attention specifically drawn to the role that the direction of the current plays in the problem.

Prior to upper-division courses, many students have never dealt with surface or volume current densities. In fact, prior to upper-division courses, students may have spent very little time dealing with densities other than mass density. In middle school and high school students have drilled into their brains that density is mass over volume. Incoming upper-division students often need a few minutes of discussion, demonstration, or activities to wrap their minds around linear and surface charge densities. The idea of a current density introduces an additonal factor beyond charge density because it now includes motion and flow along with the idea of density.

In our case, we spend about 10 minutes of class time doing a kinesthetic activity in which students are asked various questions as they act as charges and walk around in a circle. A day later, when addressing the ring problem, we only give students the general formula for magnetic vector potential in terms of $J$, the volume current density. Students are expected to realize that in the case of a spinning ring that there is only a linear current density, which reduces $J$ to $I$ and reduces a tripple integral to a single integral. Different groups of students require different amounts of time to deal with this. Some groups transition to the linear version of the equation in only a few seconds, while others spend a minute or longer, and occasionally may even need an instructor to help them. In our view, this is time well spent that allows students to see more clearly the relationship between general formulas and specific cases. Students who can do this do not need to be thinking that there are three separate formulas for find magnetic vector potential for currents - one for linear currents, one for surface currents, and one for volume currents. Instead, our students know one general formula and know that they can reproduce any of the others if they are needed.

When students think of measuring a current density, it is not immediately obvious to them that the “gate” the current would flow through has one fewer dimensions than the current density flowing through it. Thus, a point gate can be used to measure a linear current density, a line segment is used to measure a surface density, and so on. When measuring a surface charge density, students need to measure charge per unit area, whereas when measuring a surface currentdensity students need to measure current per unit length. We have found that students are initially confused by this and that physically acting it out, with students representing charges in a surface density walking side by side past a gate, is helpful in developing their understanding.

When we first assigned students to find the magnetic vector potential due to a ring of radius $R$ and charge $Q$ spinning with period $T$, we assumed that finding the magnitude of the current would be trivial and students would immediately recognize that the magnitude of the current was $Q/T$. However, since we had given students only the general formula in terms of the volume current density, $J$, and we had previously had students thinking about the linear charge density, $\lambda$, most groups of students took a more circuitous route involving charge density before they eventually reached a valid conclusion. The median group took about seven minutes to determine that the current density was $Q/T$. Students frequently had interesting discussions about the relationship between current density and charge density. In addition, students often spent time discussing the angular velocity $\omega$ and what role, if any, the radius $R$ would play.

Most students, at some point during their discussion of current, would make a statement or write an equation that was incorrect. However, to our surprise, there was an enormous diversity in the types of incorrect statements made. The group problem-solving process allowed for group members to help fellow students correct mistakes and develop stronger connections to correct understandings. Instructor input to individual groups also helped students develop their understanding. In the end, most students successfully arrived at $I = Q/T$, but in the process, they strengthened the resources that they brought to bear on the remainder of the problem.

It has been noted in PER literature that when students are asked to give a situation that represents a changing acceleration they will frequently give an example of constant acceleration and when asked to give an example of a constant acceleration they will often give an example of an object moving with constant velocity. There has been similar research showing problems with students understanding the type of current needed to produce a constant magnetic field. We have found that these problems persist past the sophomore year. For example, when asked how a series of concentric spinning charged rings could be used to produce a non-zero non-changing magnetic field, some of our juniors answered that different rings would have to be spinning different directions to cancel out the field of other rings. We also found other examples in which students confused a non-zero magnetostatic field with a zero magnetic field.

With our student population it is obvious to us that most students need some refresher of concepts “covered” in introductory physics courses. The aforementioned kinesthetic activity, in which students act as moving charges, is one of the ways we address this.

Throughout the junior year we find ourselves often repeating the idea that students “know less but are a lot smarter than we give them credit for.” The ideas behind this are twofold. The first idea is that students have often forgotten concepts that they have only seen a few times, and that students are often expected to know things that they have never been taught at any point in their studies. The second idea is that students are often surprisingly capable of learning quickly once they have the sufficient prerequisite understandings and resources to address a problem. Once students are given the opportunity to fill in gaps in their knowledge, they are often able to solve complex problems with surprising sophistication.

By recognizing and addressing aspects of electric current that students have forgotten or have never been asked to consider, one can lay the groundwork for deeper understanding and addressing higher level concepts.