Consider a system consisting of a colloidal particle of radius and charge Q-+20e (e is the charge of an electron) stationary in the center of a spherical cavity of radius R=5. Its counterions have negligible size and can be at any distance between and R. Estimate the counterion distribution function, g(r), as a function of distance using a numerical solution of the Poisson-Boltzmann equations as described in the notes. In particular, perform the calculations at conditions so that the Bjerrum length = (4okB7) is (a) = 0/5 (b) the Bjerrum length is 50/50 and (c) =/2. Also obtain the total energy of the system for these three cases. Note that is easier to work in reduced units, with length measured in units of and energy in units of e/400.