Mathematical physics problem - heat conduction

[asynja]

Homework Statement

There's a radioactive isotope placed inside an iron sphere (R=2cm), which acts as a source of constant heat (P=1W). The isotope is uniformly distributed over a very thin spherical layer (r=1cm). How much higher is temperature in the center of the sphere compared to temperature on sphere's surface, which is constant all the time?

Homework Equations

spherical bessel functions?

The Attempt at a Solution

u(r,t)=$$\Sigma$$$$A_{n}$$($$j_{0}$$($$k_{n}$$r)-$$n_{0}$$($$k_{n}$$r)e$$^{-k^{2}_{n}tD}$$
I tried to divide problem in two parts: First we have spherical waves going inwards - that's to get the temp. in the centre; Then, we have spherical waves going outwards, where we know what temp. is at r=1cm and search for temp. at r=2cm. But I haven't been able to form the right equations. Also, I am lost at how to perform normalisation to get A$$_{n}$$ . Can anyone help?

 

[Mapes]

Spherical bessel functions aren't necessary here; the problem is much simpler. What is the governing equation for temperature in the sphere for $r<1\,\mathrm{cm}$ and $r>1\,\mathrm{cm}$, and what are the boundary conditions?