In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.
1. << Cube Charge | Integration Sequence | Total Charge >>
Calculating Total ChargeEach group will be given one of the charge distributions given below: (\(\alpha\) and \(k\) are constants with dimensions appropriate for the specific example.)
- Spherical Symmetery
- A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density \(\rho (\vec{r}) = \alpha\, r^{3}\)
- A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density \(\rho (\vec{r}) =\alpha\, e^{(kr)^{3}}\)
- A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density \(\rho (\vec{r}) = \alpha\, \frac{1}{r^{2}}\, e^{(kr)}\)
- Cylindrical Symmetry
- A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \(\rho (\vec{r}) = \alpha\, s^{3}\)
- A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \(\rho (\vec{r}) =\alpha\, e^{(ks)^{2}}\)
- A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \(\rho (\vec{r}) = \alpha\, \frac{1}{s}\, e^{(ks)}\)
For your group's case, answer the following questions:
- Find the total charge. (If the total charge is infinite, decide what you should calculate instead to provide a meaningful answer.)
- Find the dimensions of the constants \(\alpha\) and \(k\).
We usually start with a mini-lecture reminder that total charge is calculated by integrating over the charge density by chopping up the charge density, multiplying by the appropriate geometric differential (length, area, or volume element), and adding up the contribution from each of the pieces. Chop, Multiply, Add is a mantra that we want students to use whenever they are doing integration in a physical context.
The students should already know formulas for the volume elements in cylindrical and spherical coordinates. We recommend Scalar Surface and Volume Elements as a prerequisite.
We start the activity with the formulas \(Q=\int\rho(\vec{r}')d\tau'\), \(Q=\int\sigma(\vec{r}')dA'\), and \(Q=\int\lambda(\vec{r}')ds'\) written on the board. We emphasize that choosing the appropriate formula by looking at the geometry of the problem they are doing, is part of the task.
Each student group is assigned a particular charge density that varies in space and asked to calculate the total charge. This activity is an example of https://paradigms.oregonstate.edu/whitepaper/compare-and-contrast-activity.
This activity helps students practice the mechanics of making total charge calculations.
You might ask two groups to present their solutions, one spherical and one cylindrical so that everyone can see an example of both. Examples (b) and (f) are nice illustrative examples.