Laplace's Equation and the Trapezoidal Resistor

In undergraduate courses, the Laplace equation is usually applied only to highly symmetric (and highly unrealistic) configurations in which closed form solutions can be found by methods of applied mathematics. Although closed form solutions are useful for developing some intuition, real world problems require solving the Laplace equation for asymmetric shapes.

In PHYCS 3910 Laplace's equation is taught in a module in which the resistance of a thin uniform trapezoidal resistor is to be computed. The theory is developed with a text section specifically written for this course. Since the cross section of this resistor is not constant, there is no easy method for computing the resistance.

Students will use a finite element numerical method, on a computer, to find the resistance. This method is based on breaking the resistor into a mesh of triangles, and solving Laplace's equation in the triangles. The construction of the mesh and solution is automated in a finite element package. This is the tool a scientist or engineer would use to solve such a problem.

From the solution of Laplace's equation, the finite element package can find current flows when a potential difference is specified at the edge of the model. The student can therefore use the computer model to calculate the expected resistance of a trapezoidal model of any dimensions and any resistivity.

The student is given a thin film trapezoidal resistor, and computes the theoretical resistance, then measures the actual resistance. An understanding of the inaccuracies involved in both the modeling and the measurement are an important part of this learning experience.

All work (lectures, computation, measurements) are done in a single room, so the time needed can be given to the various aspects of understanding the material. There is no lab vs. lecture division imposed by the logistics of different rooms.