Student handout: Total Charge

Static Fields 2024 (9 years)
In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.

Calculating Total Charge

Each 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
    1. A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density \(\rho (\vec{r}) = \alpha\, r^{3}\)
    2. A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density \(\rho (\vec{r}) =\alpha\, e^{(kr)^{3}}\)
    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
    1. A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \(\rho (\vec{r}) = \alpha\, s^{3}\)
    2. A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \(\rho (\vec{r}) =\alpha\, e^{(ks)^{2}}\)
    3. A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \(\rho (\vec{r}) = \alpha\, \frac{1}{s}\, e^{(ks)}\)

For your group's case, answer the following questions:

  1. Find the total charge. (If the total charge is infinite, decide what you should calculate instead to provide a meaningful answer.)
  2. Find the dimensions of the constants \(\alpha\) and \(k\).


Author Information
Corinne Manogue, Tevian Dray
Keywords
charge charge density multiple integral scalar field coordinate systems differential elements curvilinear coordinates
Learning Outcomes