Visualization of electric fields

In the past, representations of electric fields in textbooks were almost always depictions of field lines, a representation that can be extremely powerful when applied to Gauss’s law and the concept of divergence. But students run into problems if this is the only representation in their mental toolbox when they attempt to understand electric fields as the gradients of potentials or when they study electromagnetic waves. What, exactly, is waving? The former emphasis on field lines arose in large part because static graphs of arrows (add a picture here) in three-dimensional space may appear as an overlapping indistinguishable mess. Now, with the aid of modern computer graphics in a program such as Maple or Mathematica, students can rotate a graph such as Fig. 2 (add a link to David's simulation here) with a simple sweep of the mouse. This engages the part of the brain that processes the visualization of objects in space and the image immediately takes on a three-dimensional effect.

Electric field vectors are also a useful representation when students are learning about flux. One of our favorite visualization activities uses Maple to verify Gauss’s law for a point charge somewhere inside a unit cube. Using code that students can easily examine and alter, this worksheet first plots the electric field vectors due to a point charge as discussed above. Then the value of the integrand of the flux through the top of the cube is plotted as in Fig. 3 (Add a link to David's figure). This intermediate step allows for a class discussion about where one might expect the integrand to be large or small - both the distance of the charge from the area in question and the angle of the electric field vector with respect to the directed area are relevant. And finally, the integral of the flux through each face of the cube and the total flux through all the faces of the cube are calculated and shown to be proportional to the value of the charge enclosed by the cube. The fun of this activity comes when the students interactively move the charge with respect to the cube. If the charge is outside the cube, then the total flux is zero, even though the fluxes through the individual faces are nonzero. If the charge is at the center of the cube or in a few other special places, then Maple calculates the flux analytically, in closed form. At all other points, the calculation is automatically numerical and students see this effect - the answer is quoted to a certain number of digits of accuracy. Students rapidly seek the special cases that they think might test the bounds of the code by placing the charge on a face, edge, or vertex of the cube. Here they discover, for example, that a charge on the face contributes only ½ the flux of an interior charge. They can then sketch field lines for this case and see which ones actually pass through the faces of the cube.