The figures below show the scalar potential and the electric fields associated with a radiating dipole (e.g., a radio antenna). Using MATLAB, students can develop such animations both to explore what the equations mean, and also to check if their own approaches are yielding physically sensible answers. Double click on an image to see the action, or go back to top.

Visualization can also be helpful in understanding how mathematical approaches work. For example, in electrostatics, one often has to solve Laplace's equation. Visualization allows us to understand how the different techniques for solving Laplace work. The figures below show two different approaches: the "relaxation method", which exploits the fact that solutions to Laplace are local averages, and separation of variables, which uses an expansion of sines, cosines, and exponentials to solve to problem. Visualization allows us easily to see that both approaches give similar results, but also that the separation of variables method gives a good answer much more quickly than the relaxation method . Double click on an image to see the action, or go back to top.