## The Geometry of Scalar Fields

This sequence introduces various representations of scalar fields in the context of electrostatic potentials. While middle-division students have lots of experience with representations of functions of a single independent variable, many still need help with visualizing functions of two (or especially three) independent variables. This sequence introduces: equipotential curves and surfaces (contour plots), tangible dry-erasable plastic surfaces, computer-generated cross-sectional plots of various types, and using color to represent the value of the potential.

We prefer to start upper-division E & M with electrostatic potential $V$ before electric field $\vec{E}$. This choice allows students to struggle with the simpler idea of a scalar field (a number at every point in space) before moving on to the more complicated idea of a vector field (a vector at every point in space). To use this sequence in this way, it is necessary to direct students to avoid (temporarily) using reasoning about electric fields and electric field lines, in order to build up intuition about the relationship between charged sources and electrostatic potentials. A little later, during a review of electric fields, the relationship between potential and electric field can also be reinforced.

### Activities

• Electrostatic Potential due to a Point Charge (Estimated time: 5-20 minutes): This small whiteboard question typically results in an algebraic expression of one variable: the distance from the origin to the point charge. Discussions which will likely arise include notation of the distance from the origin to the point charge, the constants in the equation, and the dimensions of the equation. The representation used by students is predictably algebraic in form, however, the discussion can include other representations of a one-dimensional electrostatic potential.
• Drawing Equipotential Surfaces (Estimated time: 20 minutes): This small group activity encourages students to work in the plane of four point charges arranged in a square to find level curves of equipotential. Students construct a contour plot of the electrostatic potential in the plane of the four charges and explore the constructed scalar field close to the charges, far from the charges, and at important points in the field. 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 (Estimated time: 20 minutes): Students begin by brainstorming ways in which to represent three-dimensional scalar fields in two-dimensions and then use a Mathematica notebook to explore various representations for a distribution of point charges. This activity allows students to check their solutions to Drawing Equipotential Surfaces as well as explore other representations. Students recognize that the electrostatic potential is a function of three spatial variables which requires an alternative way to represent the potential such as the use of color, cross-sections, or plotting equipotential surfaces.
• Add the surfaces activity here.
• Electrostatic Potential due to a Pair of Charges (Estimated time: 30 minutes, 50 minutes with optional power series extension): Students are asked to find (algebraically) the electrostatic potential on the axis and in the plane of two point charges using the superposition principle. To accomplish this, students need to figure out to use the equation for the potential due to a point charge that is not at the origin. If appropriate for the course, this activity can conclude with student using power series approximations to calculate formulas for an approximation to the potential near the origin or far from the charges.

### Implementation

The first two activities, Electrostatic Potential due to a Point Charge and Drawing Equipotential Surfaces, can be paired together with little introduction. The third and fourth activities, Visualizing Electrostatic Potentials and (Surfaces), introduce new representations of scalar fields using the same charge distributions as Drawing Equipotential Surfaces and can be used as effective follow-ups to that activity. These four activities can be used effectively in immediate succession. (Does the surfaces project have a paper recommending a particular sequence?) The last activity Electrostatic Potential Due to a Pair of charges is the capstone of the sequence. It benefits from some extra set-up on the distance formula which can be accomplished by the Star Trek activity.

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