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## Gauss's Law

Gauss's law equates the total charge to the flux of the electric field by $\int{ \vec{E} \cdot d\vec{A}}=\frac{Q_{enclosed}}{\epsilon_0}$. This sequences develops the essential components of using Gauss's law: making symmetry arguments, calculating total charge, and using a flux integral to find the electric field.

Gauss's law can be used with charge densities which have high symmetry, however, some students have not developed the skills to make coherent symmetry arguments in introductory courses. In Paradigms, students begin to make explicit symmetry arguments to simplify their calculations in Gauss's law by using *Proof by Contradiction*. In *Proof by Contradiction*, the opposite of what you are trying to prove is assumed and then arguments are made using this assumption until a statement is made that is clearly false. For example, an infinite sheet of uniform charge density has high symmetry and explicit symmetry arguments and application to Gauss's law are made here as an example.

Students may struggle with understanding what $Q_{enclosed}$ means in Gauss's law. Charge distributions can have high symmetry in rectangular, cylindrical, and spherical coordinates, so strong integration skills in each of those coordinate systems, in various dimensions, are essential. Additionally, charge distributions are described by various terms such as uniform and constant which many students may not be familiar.

Gauss's law directly relates the flux of an electric field with the enclosed charge. Most students are introduced to Gauss's law in introductory physics, however, many do not understand the mathematics of the flux integral. Therefore, developing understanding of flux as done in the Geometry of Flux can assist students in properly using and understanding Gauss's law.

### Activities

**Acting Out Charge Densities***(Estimated time: 10 minutes)*: A kinesthetic activity in which students act as individual charges and move about the classroom to demonstrate linear, surface, and volume charge distributions. Students learn to distinguish between uniform and non-uniform charge distributions in various dimensions.

**Total Charge***(Estimated time: 30 minutes)*: In this small group activity, students calculate the total charge in spherical or cylindrical dielectric shells from charge densities which vary in space. Students practice finding the total charge of highly symmetric charge densities which may be similar to densities encountered while using Gauss's law. Through this activity, students build their integration skills in cylindrical and spherical coordinates.

**The Geometry of Flux Sequence**: This sequence of activities addresses the geometry of flux as well as allowing students ample practice in the mathematics which is used to calculate flux through various surfaces. This sequence culminates in applying Gauss's law to find the electric field due to particular charge densities. Because Gauss's law equates the flux of the electric field to the total charge, the flux is important for students to understand conceptually and be able to calculate fluently. In middle-division electrostatics, Gauss's law is used for a static electric field and therefore it is important to avoid explanations which imply time dependence such as amount of stuff which “flows through” or “gets through”. Instead the emphasis should be on describing flux as the sum of the normal components of a vector field through a surface. Given a strong geometric understanding of flux can provide students with some physical intuition with Gauss's law.

**Gauss's Law--the integral version***(Estimated time: 30 minutes)*: This lecture introduces Gauss's law in integral form and serves as an introduction to the Gauss's Law activity.

**Gauss's Law***(Estimated time: 60 minutes)*: This small group activity is the final activity in the Geometry of Flux sequence where students calculate the electric field due to various charge densities in spherical and cylindrical shells. Students are asked to make explicit symmetry arguments using*Proof by Contradiction*.