- Homework Statement:
- A very thin disc of radius R, made of a material with extremely high thermal conductivity is placed inside a large, homogeneous medium with thermal conductivity κ. In the plane of the disk, concentrically with it, there is a thin circular loop of radius R√2 made of conductive wire and connected to a battery. The heating power generated by this loop per unit length is uniform, equal P/(2πR√2) and does not change with time. The effects of the wires connecting the element to the battery can be neglected. Suppose the steady state is reached, which means that the temperature of each point in space does not change with time. Determine the difference ΔT between the temperature of the disk and the temperature at a point of the medium very far from the disk. Neglect thermal radiation.
- Relevant Equations:
- Fourier's heat equation, Laplace's equation
I've tried to explicitly solve the Fourier's equation in cylindrical coordinates but I'm getting some messy integrals which cannot be solved analytically. Additionally my instructor said that there's a neat trick for this problem and it's possible to obtain the answer in a rather elementary way.Could someone point out to me this neat trick?