A cylindrical rod of length l, thermal conductivity k and area of cross section A has one end in a furnace maintained at constant temperature. The other end of the rod is exposed to surrounding. The curved surface of the rod is well insulated from the surrounding. The surrounding temperature is `T_(0)` and the furnace is maintained at `T_(1) = T_(0) + DeltaT_(1)`. The exposed end of the rod is found to be slightly warmer then the surrounding with its temperature maintained `T_(2) = T_(0) + DeltaT_(2) [DeltaT_(2) ltlt T_(0)]`. The exposed surface of the rod has emissivity e. Prove that `DeltaT_(1)` is proportional to `DeltaT_(2)` and find the proportionality constant.

Answer

`(4esigmaLT_(0)^(3))/(k)+1`