Relaxation Method for a real parallel-plate capacitor

Sean Bartz
Macalester College,
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Summary

Students learn to solve Laplace's equation for the potential in free space under the influence of a variety of boundary conditions that would be difficult to solve analytically, using the method of relaxation.

Students will use this solution to determine the capacitance of a parallel-plate capacitor and compare this result to measurements done on a real system.

Learning Goals

Students will learn the value of Laplace's Equation for finding the electric potential, especially when boundary conditions are given instead of a charge distribution. They will also learn the utility of numerical methods for solving differential equations. They will learn about solving boundary value problems, which are generally emphasized less than initial value problems.

Students will learn model development, determining what parameters (grid size, number of iterations, etc) will give the best results. They will also learn to work with dimensionless units in computation, but relate those to SI units when they compare to lab measurements.

Because this is included as part of a laboratory activity, operation of lab equipment will also be emphasized. Finally, technical writing will be required in the writeup for assessment.

Context for Use

This computational exercise is part of a lab activity for an upper-level majors course in electrodynamics.

We plan one hour for introduction to the numerical tools, one hour for the students to learn the setup of the laboratory and how to take their measurements, and then students complete their measurements and analysis in their lab groups on their own schedule.

If a minimal working program is provided to the students, they need some familiarity with loops, MATLAB's built-in vector derivative functions, and 3D plotting functions.

For efficiency of calculation, it is best if the students are familiar with MATLAB vectorization techniques -- looping over the matrix elements greatly increases runtime. However, for pedagogical purposes, it may be worthwhile to use less-efficient methods that are more intelligible to students.

Description and Teaching Materials

This technique can be implemented in any program that allows for manipulation of arrays and 3-D plotting. I choose MATLAB because vectorization allows for great efficiency in the calculation. I want to emphasize the simplicity of the relaxation technique without going into details about improvements to the method, such as successive over-relaxation. MATLAB also has built-in gradient and Laplacian operators that allow us to get the electric field and charge density simply.
Relaxation Method for Parallel Plates (Matlab File 2kB Sep30 16)
Relaxation Method Handout (Acrobat (PDF) 70kB May9 19)
Measuring Capacitance Lab Handout (Acrobat (PDF) 2.7MB Sep30 16) Measuring Capacitance Lab Handout (Acrobat (PDF) 2.7MB Sep30 16)

Teaching Notes and Tips

This activity is best suited for after students have some experience solving Laplace's equation using the technique of separation of variables.

A more computationally-focused class could emphasize improvements to this method, such as the Gauss-Seidel method, successive over-relaxation, or multigrid techniques.

Assessment

This activity will be assessed as a laboratory activity by the writing of a lab report.

References and Resources

"Form and Capacitance of Parallel-Plate Capacitors" by Hitoshi Nishiyama and Mitsunobu Nakamura, IEEE TRANSACTIONS ON COMPONENTS, PACKAGING, AND MANUFACTURING TECHNOLOGY-PART A, VOL. 17, NO. 3. SEPTEMBER 1994