Physics problem help??

Question

A spherical shell has inner and outer radii a and b, respectively, and the temperatures at the inner and outer surfaces are T2 and T1. The thermal conductivity of the material of which the shell is made is K . Derive an equation for the total heat current through the shell.

Express your answer in terms of the given quantities and appropriate constants.

-> I see that i must use dQ/dt = -k*A*dT/dx, but i cant apply this to the questions! how would you?

Answer 1

If U is internal thermal energy,

Q = dU/dt

Q = - kAΔT/x

δQ = - kdAdT/dx

For a sphere, x becomes r, and dA = r^2sinθdθdφ

δQ = - kr^2sinθdθdφdT/dr

δQ = - 4πkr^2dT/dr

- δQdr/r^2 = 4πkdT

Q(1/b - 1/a) = 4πk(T1 - T2)

Q = 4πk(T2 - T1)/(1/a - 1/b)

Q = 4πkab(T2 - T1)/(b - a)