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## Representations of Scalar Fields

We prefer to start upper-division E & M with electrostatic potential before electric field since 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). The electrostatic potential is defined by spatial variables which students should have some familiarity through previous mathematics and physics courses. The potential has a value for every point in space which can cause confusion when the potential is described by more than one spatial variable. One spatial variable describing the potential at every point in space allows students to use the representation typically used in mathematics course: a plot of a function, $V(r)$, with respect to its variable, $r$.

When the potential becomes a function of two spatial variables, the corresponding representation is not as clear. Potentials of two variables can be represented using a variety of means including, but not limited to, a surface, a contour plot, and a color varying plot.

When extending to a potential dependent on three spatial variables, the ways in which the potential can be represented become more limited than the two-dimensional case. This is due to the constraints of working in three-dimensional space. Therefore, it is most obvious to represent the potential with equipotential surfaces. The recognition of symmetries within scalar fields allows other representations to be used such as cross sectional contour plots.

This sequence addresses various representations of scalar fields in the context of electrostatic potentials. The first two activities, Electrostatic Potential due to a Point Charge and Drawing Equipotential Surfaces, can be easily paired together with little introduction. The third activity, Visualizing Electrostatic Potentials, can extend the representations of scalar fields which students work with using the same charge distribution as Drawing Equipotential Surfaces and can be used as a strong follow-up to that activity. These three activities can be used in immediate succession, followed by the small group activity Electrostatic Potential Due to a Pair of charges. Typically, this last activity requires more introduction than the preceding activities because more mathematics is involved including series expansions, cylindrical coordinates, and some vector calculus.

### 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: 45 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 and plotting equipotential surfaces.
• Electrostatic Potential due to a Pair of Charges (Estimated time: 30 minutes): 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.

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