Question: As a result of radioactive decay, heat is generated uniformly throughout the interior of the earth at a rate of 30 watts per cubic kilometer. (A watt is a rate of heat production.) The heat then flows to the earth's surface where it is lost to space. Let F→ (x, y, z) denote the rate of flow of heat measured in watts per square kilometer. By definition, the flux of across F→ a surface is the quantity of heat flowing through the surface per unit of time.
(a) What is the value of div F→? Include units.
(b) Assume the heat flows outward symmetrically. Verify that F→ = αr→, where r→ = xi→ + yj→ + zk→ and α is a suitable constant, satisfies the given conditions. Find α.
(c) Let T (x, y, z) denote the temperature inside the earth. Heat flows according to the equation F→ = -k grad T, where k is a constant. Explain why this makes sense physically.
(d) If T is in °C, then k = 30,000 watts/km°C. Assuming the earth is a sphere with radius 6400 km and surface temperature 20°C, what is the temperature at the center?