## Flux Integrals

Flux integrals are used in electricity and magnetism to find total current and in Gauss's law. Therefore, it is important for physics students to be mathematically and conceptually proficient with flux. Mathematically, flux has the differential area vector, $d\vec{A}$, which is likely new to students in cylindrical and spherical coordinates. Conceptually, the flux of a static vector field is the amount of vector fields that “points through” a given area in the plane of the area. Many students will have an existing notion of flux as a “flow through” an area. Therefore, avoiding terms which imply time dependence (“flows through” or “gets through”) is important because flux is introduced in the context of electrostatics. Finally, there are important physical applications for flux such as determining total current and using Gauss's law which require mathematical and conceptual understandings of flux.

### Activities

• Surface and Volume Elements in Cylindrical and Spherical Coordinates (Estimated time: 20 minutes): This small group activity is similar to Scalar Distance, Surface, and Volume Elements except relies on student understanding of $d\vec{r}$ to construct area elements in cylindrical and spherical coordinates using, $d\vec{r_1}\times d\vec{r_2}=d\vec{A}$. Students construct differential area elements, $d\vec{A}$, in coordinates commonly used in physics.
• The Concept of Flux (Estimated time: 5 minutes): This kinesthetic activity introduces students to the concept of flux by having students create a vector field with rulers while the instructor uses a hula hoop as a surface element. The whole class discussion then focuses on what it means geometrically to take the dot product of a vector with a surface element and then extends to flux by adding up all of these dot products over a surface which gives the flux through the surface.
• Acting Out Current Density (Estimated time: 10 minutes): This kinesthetic activity has students move around the classroom to act out linear, surface, and volume current densities to provide students with an intuitive understanding of current density and total current. The class discussion which occurs during or following the activity focuses on the meaning of several words commonly used to describe current in physics: steady, constant, uniform, and linear. Additionally, current is introduced as the flux of the current density.
• Total Current: (Homework?)
• Calculation of Flux (Estimated time: 30 minutes): In this small group activity, students calculate the flux of a simple vector field through a cone. Students work in either cylindrical or spherical coordinates to determine the flux which requires constructing an appropriate $d\vec{A}$ on the cone. The wrap-up discussion reinforces the concept that only the component of the vector field perpendicular to the surface contributes to the flux and how to find the differential area element on a surface that is not a “coordinate equals constant” surface. This is an exercise in the mathematical calculation of flux and can be used to check student understanding of flux both conceptually and computationally.

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