## Building a Model of a Spherically Symmetric Electric Field

One of the major hurdles students face when learning about electricity and magnetism is the invisibility of point particles, vector fields, equipotential surfaces, and so on. Drawings used to illustrate various charge and current distributions and their electric and magnetic fields are an invaluable tool for teaching, but they are not without shortcomings.

Two dimensional drawings of field lines can fall flat when they are used to motivate symmetry arguments employed in calculations of electric fields using Gauss’s Law. The symmetry arguments are powerful, but often misunderstood. I found that many students have difficulty interpreting a perspective drawing and inferring the three-dimensional nature of some object from its orthographic projection drawings.

Using 3-D field models and building them as a part of in-class exercises helps students develop mental models and gain a better understanding of how two-dimensional drawings of a field relate to its three-dimensional representation.

Here, I describe preparation of materials used in such a model-building exercise for a spherically symmetric charge distribution and its electric field. If you are as pedantic as I am when it comes to symmetry, you will appreciate the method.

Using 3-D field models and building them as a part of in-class exercises helps students develop mental models and gain a better understanding of how two-dimensional drawings of a field relate to its three-dimensional representation.

Here, I describe preparation of materials used in such a model-building exercise for a spherically symmetric charge distribution and its electric field. If you are as pedantic as I am when it comes to symmetry, you will appreciate the method.

### Making the Model

Step 1: I got some styrofoam spheres, which can be found in most craft stores. These are 2 inches (5.1 cm) in diameter and you can dye them with styrofoam-safe acrylic paint (beware that some paints may melt them!). They look best after about two or three coats of paint.

Step 2: To figure out where to place the toothpicks so that their distribution is uniform and symmetric, I made paper models of Platonic solids so that the styrofoam spheres can inscribe them. The number of vertices of a solid determines how many equally spaced field lines I can place on the sphere; to demonstrate the difference between fields of spheres with different amounts of electric charge, one can use the solids with a different number of vertices.

Step 3: I chose to work with a dodecahedron, which has twenty vertices, because twenty field lines nicely illustrate the spherical symmetry, while the work is manageable. I made and printed out a dodecahedron net, ensuring that the length of its side gives me the right inscribing radius of 2 in, so that my styrofoam ball can snuggly fit inside. The equation relating the inscribing radius r and the side-length of a dodecahedron a gives r ≈ 1.1 a. For an inscribing sphere with a 2 in diameter, the side of the dodecahedron should be about 0.9 in (2.3 cm). I eyeballed mine, but a well fitting net keeps the sphere snug inside, so that the toothpicks can be placed more accurately.

Step 4: I pasted the net on a sheet of sturdier drawing paper and cut it out, keeping some extra tabs on select edges so that I can glue them together.

Step 5: (a) I glued all but three faces so that I can reuse the dodecahedron for a number of spheres. (b) I used masking tape to close up the remaining three faces (seven edges).

Step 6: I pushed a toothpick through each vertex and into the styrofoam sphere by about 2/8 in, making sure that it is sticking out at the same angle relative to all three edges coming out of the vertex (like the central toothpick in the bottom right image). The angle is important because the field lines should come out perpendicularly to the surface of the sphere, especially if you are using it to model a conducting ball.

Step 7: I removed all the toothpicks, took the sphere out, and marked the puncture holes with a color pencil so that students know where to place the “field lines.”

To see the final product, I stuck the toothpicks back into the ball and voila! The “field lines” should look the same regardless from which side the object is viewed. As you may imagine, the symmetry is not perfect, but it does the job.

I repeated steps 5-7 fourteen times, leaving Step 8 for the in-class activity. By the time I was done, the dodecahedron was in a pretty bad shape. If you make it out of a more durable material, it can probably be used for more than 14 spheres.

I also made a couple of fancier models to use as class demos.

You can use painted spheres to distinguish between positive and negative charges — the direction of the field arrows can be labeled with a marker on each toothpick.

You can use skewers instead of toothpicks to get a field that spreads out further. It is probably a bit much, but it makes the weakening of the field more obvious.

I attempted to build a model the electric field of an electric dipole with some semi-flexible wire. The tricky part with using a dodecahedron for this is that each field line is in its own plane that passes through the axis connecting the spheres. As a result, making a connection to the standard 2-D drawings of the dipole field is not as straight forward.

This was a relaxing, but a time consuming exercise. I suggest it for a cold night when there is nothing else to do.