computer Mathematica Activity

30 min.

Using Technology to Visualize Potentials

Static Fields 2023 (6 years)

electrostatic potential visualization

Begin by prompting the students to brainstorm different ways to represent a three dimensional scalar field on a 2-D surface (like their paper or a whiteboard). The students use a pre-made Sage code or a Mathematica worksheet to visualize the electrostatic potential of several distributions of charges. The computer algebra systems demonstrate several different ways of plotting the potential.

group Small Group Activity

30 min.

Number of Paths

E&M Conservative Fields Surfaces

Student discuss how many paths can be found on a map of the vector fields \(\vec{F}\) for which the integral \(\int \vec{F}\cdot d\vec{r}\) is positive, negative, or zero. \(\vec{F}\) is conservative. They do a similar activity for the vector field \(\vec{G}\) which is not conservative.

group Small Group Activity

30 min.

The Hill

Vector Calculus II 23 (7 years)

Gradient

Gradient Sequence

In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.

group Small Group Activity

30 min.

DELETE Navigating a Hill

Static Fields 2023 (5 years) In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.

accessibility_new Kinesthetic

10 min.

Acting Out the Gradient

Static Fields 2023 (7 years)

gradient vector fields electrostatics

Gradient Sequence

Students are shown a topographic map of an oval hill and imagine that the classroom is on the hill. They are asked to point in the direction of the gradient vector appropriate to the point on the hill where they are "standing".

group Small Group Activity

30 min.

Electric Field Due to a Ring of Charge

Static Fields 2023 (8 years)

coulomb's law electric field charge ring symmetry integral power series superposition

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in small groups to use Coulomb's Law \[\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the electric field, \(\vec{E}(\vec{r})\), everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for \(\vec{E}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

group Small Group Activity

30 min.

Directional Derivatives

Vector Calculus I 2022

Directional derivatives

Gradient Sequence

This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.

group Small Group Activity

30 min.

The Hillside

Vector Calculus I 2022 (2 years)

Gradient Sequence

Students work in groups to measure the steepest slope and direction on a plastic surface, and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).

format_list_numbered Sequence

Gradient Sequence

This sequence starts with an introduction to partial derivatives and continues through gradient. While some of the activities/problems are pure math, a number of other activities/problems are situated in the context of electrostatics. This sequence is intended to be used intermittently across multiple days or even weeks of a course or even multiple courses.