Total Charge in Curvilinear Coordinates


  • Acting Out Charge Density (Estimated time: 10 minutes): This kinesthetic activity provides students with an embodied understanding of charge density and total charge by using their bodies to represent charges. Students move around the room to act out linear, surface, and volume charge densities which prompts a whole class discussion on the meaning of constant charge density, the geometric differences between linear, surface, and volume charge densities, and what is “linear” about linear charge density.
  • Total Charge in Rectangular Coordinates (Estimated time: ): This small group activity involves the instructor providing a specific volume charge density, $\rho$, in Cartesian coordinates. Students then integrate to find the total charge contained within a specified box which should be a review from multivariable calculus courses.
  • Curvilinear Coordinates (Estimated time: ): This lecture introduces students to curvilinear coordinates. The notation difference in physics and mathematics should be highlighted: that $\theta$ and $\phi$ are switched in each discipline. This is likely to be review for students who have been previously introduced to curvilinear coordinates in math courses.
  • Scalar Distance, Area, and Volume Elements (Estimated time: ): In this small group activity students derive expressions for infinitesimal distances in order to find area and volume elements in cylindrical and spherical coordinates. This activity can be done with Pineapples and Pumpkins to give students a three dimensional object to explore the geometry and construction of a volume element.
  • Pineapples and Pumpkins (Estimated time: ): This activity can be done in small groups or as an instructor led whole class activity. A pineapple (for cylindrical) and/or pumpkin (for spherical) can be cut to demonstrate the geometry of an infinitesimal volume element used in integration. If done as a small group activity, it can be combined with Scalar Distance, Area, and Volume Elements. If done as a whole class activity, the instructor cuts a pumpkin and/or pineapple prompted by students answering a series of small whiteboard questions. This emphasizes the construction of infinitesimal volume elements using a three dimensional representation.
  • Total Charge (Estimated time: 30 minutes): In this small group activity, students calculate the total charge within spherically or cylindrically symmetric volumes. Students use multivariable integration in various coordinate systems in order to find the total charge contained within the volume due to a specific charge density.

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