Two chunks of metal with heat capacities `C_(1)` and `C_(2)`, are interconnected by a rod length `l` and cross-sectional area `S` and fairly low heat conductivity `K`. The whole system is thermally insulated from the environment. At a moment `t = 0` the temperature difference betwene the two chunks of metal equals `(DeltaT)_(0)`. Assuming the heat capacity of the rod to be negligible, find the temperature difference between the chucks as a function of time.

Explanation

Suppose the chunks have temperatures `T_1, T_2` at time `t` and `T_1 - dT_1, T_2 + dT_2` at time `dt + t`.
Then `C_1 dT_1 = C_2 dT_2 = (kS)/(l)(T_1 - T_2)dt`
Thus `d Delta T = -(kS)/(l) ((1)/(C_1) + (1)/(C_2)) Delta T dt ` where `Delta T = T_1 - T_2`
Hence `Delta T = (Delta T)_0 e^(-t//tau)` where `(1)/(tau) = (ks)/(l) ((1)/(C_1) + (1)/(C_2))`.