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Geometry of Scalar Fields

Electromagnetism is one of the first areas of physics in which students come into contact with scalar and vector fields. Moreover, students often learn about the electric field first, and then describe the electric potential in terms of the electric field. But from the geometry, the potential is easier for students to wrestle with, as they need not worry about direction. This approach, to study the geometry of the electrostatic potential and scalar fields first, is the approach Paradigms takes.

Activities

  • Drawing Equipotential Surfaces: Most students are familiar with the elementary equation of the electrostatic potential, but few reconcile the equation with the geometry of a scalar field. This small group activity forces students to explicitly work out the geometry of the potential of a quadrupole, allowing them to realize what's “scalar” about the electrostatic potential.
  • Visualizing Electrostatic Potentials: After the students have had a chance at working out the electrostatic potential of an elementary charge distribution by hand, they move on to more complicated charge distributions and use a computer to help visualize the potential. As students saw in the previous activity, the electrostatic is immediately of a function of space, a function of three variables. In order to determine how to plot the value of the potential, students think of alternative ways for representing the potential, including the use of color.

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