“Necklace” Model for Current in Simple (Series) Electric Circuits

I’m teaching first-year electricity and magnetism this semester, and we are using the textbook College Physics by Knight, Jones, and Field. Students find it very clear, and it’s worked out well (we used the same textbook last year). The workbooks are very effective, as they walk students through concept development very nicely.

As teacher resources, I use Teaching Introductory Physics by Arnold Arons, and Tutorials in Introductory Physics by Lillian McDermott et al (also see the Physics Education Group at the University of Washington).

Misconceptions abound among students of elementary physics. An example relevant to my course is the following question:

“A battery is connected to two identical light bulbs in series. When the battery is switched on, does the light bulb closest to the battery shine first?”

If your image of an electric circuit is something like an empty pipe, and the “juice” from the battery is like water being poured into the pipe, then you are liable to say that the light bulb closest to the battery does indeed shine first. However, this is not correct.

The real situation in a conducting wire is quite complicated. Even when no current flows, the free electrons in the wire move randomly. When current flows, there is a somewhat organized flow in the wire superposed on the random motion.

In the process of answering a question in class yesterday, I came up with an analogy that seemed to help students see their way through understanding the question posed above. (I got quite a few smiles of satisfaction when I brought up the analogy, so I figure that it struck a chord with at least some of the students.) I told them to imagine that the electrons flowing in a series circut are all tied together, like the beads on a necklace. This is obviously false, but it is a helpful fiction.

The analogy makes it clear that current flow in a series circuit occurs simultaneously at all points of the circuit. That is, when an electron leaves the negative terminal of the battery, an electron simultaneously enters the positive terminal from the other end of the circuit. When an electron enters one side of the lightbulb, another electron simultaneously leaves the other end. This can be visualized as the necklace of electrons simply “rotating” through the circuit.

The physical reason behind this behaviour is that the battery sets up an electric field in the circuit, which occurs virtually instantly after the switch is thrown to connect the battery to the circuit. The fact that the electric field is present everwhere in the circuit explains why the electrons move everywhere in the circuit simultaneously.

The necklace analogy also makes the conservation of current clear. The same understanding could be obtained using the water-pipe analogy, as long as we imagine that the pipe is always full of water. Then the battery is a pump, pumping water through the pipe; but of course, every bit of water pumped into one end of the pipe is matched by water flowing into the pump from the other side of the pipe … conservation of current.

Another situation that caused all kinds of problems is this one: Imagine that you connect two non-ideal wires, with two different resistivities, in series with a 10 V battery. Question: Does more current flow in the first wire or the second wire? Answer: The same amount of current flows in both wires, by the conservation of current. The necklace model still applies.

What makes this hard for students to understand is the following: If, as a second situation, you take one of the wires and replace it with another that has yet a different resistivity, wouldn’t the current in just the new wire change? If so, then the current in the two wires would be different in this situation, violating the conservation of current.

The reason for this misconception is that students expect the potentials at each point in the circuit to be fixed, independent of changes in the second situation. This is not true. When the wires are changed in the second situation, the potentials change in such a way that the current is the same in both wires.

To make this concrete, suppose that your 10 V battery is connected in series in the first situation to non-ideal wires of resistance 1 Ω and 9 Ω. The potential difference (“voltage drop”) across the first wire is 1 V, and the potential difference across the second wire is 9 V. By Ohm’s law, the current in each wire is 1 A.

In the second situation, say we take replace the 9 Ω wire by a wire with resistance 99 Ω. Then the potential difference across the first wire is reduced to 0.1 V, and the potential difference across the new second wire is 9.9 V. The current in each wire is now 0.1 A, by Ohm’s law. The fact that the current is the same in each wire in the second situation is consistent with conservation of current.

What confused some students before I showed them the explicit examples is this: They expected that in the second situation the current would change only in the second wire, and therefore the current would be different in the two wires. The explicit examples made clear where the misconception lay: they did not realize that changing one of the resistances would change the potential differences for both of the resistances.

One final point: In the necklace analogy, the current represents the rate at which the necklace is moving through the circuit. The battery voltage represents the additional potential energy that each electron (more precisely, each coulomb of electrons) obtains by going through the battery. The image is that each bead of the necklace (each electron) is filled with some fluid (i.e., some energy) by the battery; the fluid then leaks out as the electrons make there way around the circuit. The magic of the situation is that the electrons always return to the battery “empty,” to be refilled for the next trip around the circuit.

A gravitational analogy takes the magic out of the previous paragraph. Imagine a child climbing the ladder to the top of a slide at a playground. The child then slides down a series of slides, connected by level “landings,” until the child reaches the ground, then runs back to the ladder for another round trip. Climbing the ladder represents an electron going through the battery, sliding down represents an electron going through a resistor, and the level landings represent ideal wires (no resistance). Changing the number and/or the value of resistances corresponds to changing the number and/or slopes of the slides; but the last slide always ends up at ground level, just as the electrons always return to the battery with zero potential energy after going through the circuit.

(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 25 January 2021.)