## Direct Integration to Determine Electrostatic Potential

### Activities

• The Distance Between Two Points--Star Trek (Estimated time: 20-30 minutes): This kinesthetic activity the class work together to generalize the $\frac{1}{r}$ factor, found in many introductory physics courses, to $\frac{1}{|\vec{r}-\vec{r}'|}$. Many students in the middle-division have prior experience with vectors, however, few have used vectors to describe the $\frac{1}{r}$ factor found in the electrostatic potential when expressed as in many introductory level courses as $V=\frac{1}{4\pi\epsilon_0}\frac{q}{r}$. Many problems in electricity and magnetism require evaluating and approximating $\frac{1}{|\vec{r}-\vec{r}'|}$ in various geometries therefore this concept is important for students to thoroughly understand.
• 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.
• Electrostatic Potential due to a Ring (Estimated time: 40 minutes, Wrap Up: 10 minutes): In this small group activity, students aim to generalize the expression for the potential by applying what they have learned about charge densities, power series approximations, and various geometries. In order to accomplish this, students practice breaking the problem up into “manageable” pieces: finding a relationship for the charge density, understanding the geometry of the problem, setting up the correct integral, and then using power series approximations to evaluate the potential in some given region. Solving this problem is a big step for many students and also prepares them for the more difficult problem of finding the electric field due to a ring of charge.

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