Here’s a fascinating question taken from Matter and Interactions to push your students toward a deeper understanding of circuits.

Let’s start with a battery, connected by a bent wire, like so:

Now, we know that the job of a battery is to separate positive and negative charge, making the positive terminal positively charged, and the negative terminal negatively charged. This charge separation on the battery creates an electric field in the space around the battery, as shown in the illustration below (I have chosen to show the field vectors only along the axis of the battery).

There’s a problem in the region indicated by the dotted line. Since this is a simple wire connected to a battery, we will have a current flowing from the positive terminal to the negative. However, in the dotted line region, the electric field due to the battery points opposite the direction of positive charge flow in the wire. This clearly can’t be right.

The solution to this paradox is to look at what happens to excess charges. If you notice, that electric field drives excess positive charge to the right hand side of the circuit. This will leave behind some negative charge on where the wire turns, which will set up an additional stronger, electric field, opposite the electric field of the battery, which will produce a net electric field to the left in the dotted region. Thus, positive charges in the wire in this region will feel a force to the left, as they should to create a current.

It turns out that a similar phenomenon of surface charges takes place even in a long straight wire carrying a steady current. A great question for students is—how can this be? If we have a steady current, we must have a uniform electric field inside the wire. But what sort of charge arrangement can produce this field?

Matt Greenwolfe has produced this wonderful vpython program to help students explore the answer to this question on their own. He uses this program after students discover that the voltage drop for every unit of length is constant along a wire connected to a battery, which implies that the electric field inside the wire is constant.

Here’s a video demonstrating the program:

Here’s a link to the code that produced the simulation.