You are interested in determination of a steady state temperature profiles in a cylinder conducting the electrical current in axial direction, the flow of which results in a uniform volumetric heat generation within its volume. Write the appropriate form of a heat diffusion equation in the following cases:
i) The cylinder is very long and its entire surface is exposed to a uniform thermal environment
ii) The cylinder is short and all surfaces are exposed to the same thermal environment
iii) The cylinder is very long and one half of its circumference is insulated while the other half exposed to a uniform thermal environment.