Each group will be given one of the charge distributions given below: (\(\alpha\) and \(k\) are constants with dimensions appropriate for the specific example.)

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 \(k\).
   • Spherical Symmetry - A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density:

     1. \(\rho (\vec{r}) = \alpha\, r^{3}\)

     2. \(\rho (\vec{r}) =\alpha\, e^{(kr)^{3}}\)

     3. \(\rho (\vec{r}) = \alpha\, \frac{1}{r^{2}}\, e^{(kr)}\)

   • Cylindrical Symmetry - A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density:

     1. \(\rho (\vec{r}) =\alpha\, e^{(ks)^{2}}\)

     2. \(\rho (\vec{r}) = \alpha\, \frac{1}{s}\, e^{(ks)}\)

     3. \(\rho (\vec{r}) = \alpha\, s^{3}\)