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One of our first computational exercises is the culmination of a sequence of activities that asks students to represent the electrostatic potential due to a ring of charge in several different ways. Working in small groups, students first find an expression for the potential, valid everywhere in space. Their job is to take a general, abstract equation for the potential due to a continuous distribution of charge and, wrestling through the geometry of the problem, come up with an equation that represents the specific problem at hand, which has been stated, simply, in words. This is the step that always needs to be done first, for the professional to use computational resources effectively. Students are surprised to find that the resulting expression, an elliptic integral, cannot be computed in closed form. Next, each separate group is assigned the task of computing a series expansion for their expression, valid in some region of symmetry: on the axis or in the plane of the ring, near the origin or far from the ring. A class discussion centers around the physical interpretation or these limiting cases. Then, the groups are asked to sketch (by hand, on a 2×3 sq. ft. whiteboard) the electrostatic field due to some simple charge distributions! A dipole, a quadrupole, and the ring. They confront, for themselves, the difficulty of representing a scalar function of three variables on a flat graph. 0 nly after students have had all of these initial experience s, do we introduce a Maple or Mathematica worksheet that shows equipotential surfaces and possible cross-sections of the potential and compares them to the various limiting cases that the students have already calculated. Students are impressed that the computer algebra packages can “do” the elliptical integral numerically. We discuss, in particular, how the symmetry of the problem allows us to represent all of the desired information in a three-dimensional graph of the potential as a function of just the variables r and z. Even with all of this preparation, the resulting graph (see Figure x) is tricky for even the faculty to make sense of. It is an excellent exercise in visualization.

A similar emphasis on helping students master multiple representations for a single concept plays out when we discuss electric fields. In the past, pictures of electric fields in textbooks almost always represented 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 are studying electromagnetic waves. What, exactly, is waving? The former emphasis on field lines arose in large part because a static graph of arrows in three dimensional space appears as an overlapping indistinguishable mess. (See Figure y). Now, with the aid of modern computer graphics in a program such as Maple or Mathematica, students can rotate such a graph with a simple sweep of their mouse. This engages the part of the brain that processes the visualization of objects in spaced 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 (acknowledge Shannon Mayer) uses Maple to verify Gauss’s law for a point charge somewhere inside a unit cube. In code that students can easily examine and alter, the electric field vectors due to a point charge are plotted. Then the value of the integrand of the flux through a side of the box is plotted. This intermediate steps 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 area are relevant. And finally, the integral of the flux through each side of the box and the total flux through all the sides of the box are calculated and shown to be proportional to the value of the charge enclosed by the box. The fun of this activity comes when the students interactively move the charge around with respect to the box. If the charge is outside the box, then the total flux is zero, even though the fluxes through the individual side are nonzero. If the charge is at the center of the cube or in a few other special places, then the 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. If the charge is on a side, edge, or vertex of the cube, then the total flux is proportional to only that fraction of the charge that is “inside” the box! Students can then sketch field lines for this case and see which ones actually pass through the sides of the box.

In the middle if the junior year, a final sequence of activities with potentials arises when the Central Forces paradigm discusses classical orbits. After a fairly traditional lecture introducing the effective potential, small groups of students have the chance to experiment with how the shape of the diagram depends on the various parameters such as reduced mass or the z-component of angular momentum. This is a case where it is valuable to us a (prewritten) worksheet in a computer algebra package such as Maple or Mathematica, rather than a prettier, but “black box” simulation such as a Java applet, so that students can actually see that mathematical expressions that are being plotted. Next, we would like to show the students another worksheet that integrates the equations of motion and plots both the effective potential and the corresponding orbits for various values of the parameters. Here, a black box simulation is fine. The details of how to do the numerical solution of the differential equation are a significant enough departure from the main flow of our course, that we choose not to address them. Unfortunately, without intervention, many students do not immediately see how these two different graphical representations are related to each other. We address this issue by inserting the following kinesthetic activity. A single student, carefully chosen to be someone who will be comfortable with being put on the spot, is asked to come to the front of the room to act out the part of the orbiting planet while the teacher plays the part of the sun. Most students, on their first attempt, will walk around the teacher—after all, this is what planets do. When directed to refer to the effective potential diagram, with its apparent classical turning points, most students on their second attempt will move towards and away from the teacher in a straight line, with an embarrassed laugh for the obvious absurdity of their motion. It takes a significant class discussion to bring out the role of the angular momentum in resolving the paradox. After this classroom experience, most students explore the computational simulation of the orbits in more depth. Where do the minimum and maximum radii occur?