B.2 we are interested in a three-dimensional potential u which obeys the laplace equation ▽pu-0. assuming spherical symmetry, the laplace equation in spher ical polar coordinates (r, θ, φ) becomes au (i) we can solve the laplace equation using a trial function u(r, θ) = r(r)0(θ). show that k, and where μ = cos θ and k is a constant. (ii) you may assume that the differential equation for θ has valid solutions given by the legendre polynomials pe41) only when k = 1(1+1 ), where f = 0, 1, 2, 3, are positive integers. show that r o r is a valid solution, and determine the two possible values of a in terms of the numbers t (iii) suppose that the angular dependence of the potential on the surface of the unit sphere is given by the surface of the sphere is not assumed to be source-free. using your re- sult from(i), find the potential both inside and outside the sphere, assuming that both regions outside the sphere are source-free hints: recall the second legendre polynom p()3-1), and 4 l m)