Representations of Fields

One of the difficulties many students have is connecting an algebraic expression with the associated geometry. This is particularly noticeable when students study potential and force fields. This sequence of activities aims to help students understand the geometry of scalar and vector fields, and how to connect them to algebraic expressions.

Activities: 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.

  • 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.

Activities: Geometry of Vector Fields

After students have spent time working with both the algebra and geometry of scalar fields, they move to the more complex electric field. Building off of the previous activities on the electrostatic potential, students begin to wrestle with the more complicated geometry of vector fields.

  • Drawing Electric Field Vectors: In direct analogy to Drawing Equipotential Surfaces, this small group activity has students work out by hand the electric field of a quadrupole. Comparing the two activities allows students to connect the geometries of the electric field, a vector field, to the electrostatic potential, a scalar field. Most of all, students observe how the geometry leads to a physical relationship between the electric field and the potential.
  • Visualizing Gradient: Most students coming into upper-division electromagnetism are somewhat familiar with the expression $\vec{E}=-\vec{\nabla}V$, however many do not understand the geometry behind this relationship. Building on the previous activity, this computer visualization activity allows students to observe the geometric connection between $\vec{E}$ and $V$.
  • Visualizing Flux: Electric field vectors are also a useful representation when students are learning about flux. Most students are familiar with Gauss's law and have preconceived notions of flux integrals, but they often confuse flux of an electric field with the “flow” of an electric field. This computer visualization activity allows students to visualize the electric flux of a point charge in a cube.
  • Visualizing Divergence: Many incoming students are familiar with the algebraic expression of the divergence of a vector field, however many have difficulty in giving a geometrical interpretation of the quantity. This computer visualization activity allows students to predict the divergence, a scalar quantity, at any point by considering the vector field near that point. In this way students can use geometry to interpret divergence.
  • Visualizing Curl: Similar to the previous activity, this computer visualization activity aims to help students gain a geometric perspective of the curl of a vector field. Upon completion of this activity, many students are capable of predicting the value of the curl at point by looking at the vector field near that point.

FIXME Add quantum activities, check link formats, add a verbal description of this sequence including problems visualizing scalar field in 3 dimensions–use of color.

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