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## Homework Statement

Suppose we have a cylinder of radius R, in an environment with temperature T

_{env}, and heat is generated in the wire at a rate P per unit volume (for example due to a current - the exact nature is irrelevant). The heat flux from the surface of the wire is A(T(R)-T

_{env}) for constant A (Newton's law of cooling). Find the temperature T(r).

## Homework Equations

Thermal diffusion equation,

∂T/∂t=(κ/C)∇

^{2}T+H/C

where κ is the thermal conductivity of the medium, C the heat capacity per unit volume of the medium, and H the rate of production of heat per unit volume in the medium.

In the steady state

κ∇

^{2}T+H=0

## The Attempt at a Solution

I'm assuming we want the steady state solution - however, if we had say a cup of tea and we had Newton cooling, the temperature would decay exponentially to T

_{env}, so the steady state would just be T

_{env}- is it the fact that we have some heat production P that allows a non trivial steady state?[/B]

Anyway, for some reason I believe I am supposed to convert the heat flux into a heat gained per unit volume, which is done by

-A(T(R)-T

_{env})*(2πRl/πR

^{2}l)=-2A(T(R)-T

_{env})/R

**If correct, why am I allowed to do this - how do I know it is lost uniformly throughout the cylinder?**

Then

H=P-2A(T(R)-T

_{env})/R

Using the laplacian in cylindrical polars (no z or angular dependence)

(κ/r)d/dr(rdT/dr)=-H

T=-(Hr

^{2}/4κ)+Alnr+B

We need A=0 for T to be finite at r=0, and we need T(R) at r=R so B=T(R)+(HR

^{2}/4κ).

Then

T(r)=T(R)+(H/4κ)(R

^{2}-r

^{2})

with

H=P-2A(T(R)-T

_{env})/R.

**However I don't really like leaving T(R) in the answer - how am I supposed to find out what this is?**

Thanks for any help with the bold bits :)

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