Temperature distribution in an embedded sphere, a sphere of radius R and thermal conductivity k₁ is embedded in an infinite solid of thermal conductivity k₀. The center of the sphere is located at the origin of coordinates, and there is a constant temperature gradient A in the positive z direction far from the sphere. The temperature at the center of the sphere is T°. The steady-state temperature distributions in the sphere T₁ and in the surrounding medium T₀ have been shown to be: 

(a) What are the partial differential equations that must be satisfied by Eqs 11B.8-1 and 2? 

(b) Write down the boundary conditions that apply at r = R.

(c) Show that T₁ and To satisfy their respective partial differential equations in (a). 

(d) Show that Eqs. 1 l B.8-1 and 2 satisfy the boundary conditions in (b).