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 \(k\).