Two bodies of masses `m_(1)` and `m_(2)` and specific heat capacities `S_(1)` and `S_(2)` are connected by a rod of length `l`, cross-section area `A`, thermal conductivity `K` and negligible heat capacity. The whole system is thermally insulated. At time `t=0` , the temperature of the first body is `T_(1)` and the temperature of the second body is `T_(2)(T_(2)gtT_(1))` . Find the temperature difference between the two bodies at time `t`.

Answer

A, B, C, D

Explanation

Let `T_1` be the temperature of `C_1` and `T_2` the temperature of `C_2` at some intant of time. Further let T be the temperature difference at that instant.

Then,
`C_1(-(dT_1)/(dt)) = H = T/(l//KA) = (KA)/l (T)`
and `C_2 (+(dT_2)/(dt)) = H = T/(l//KA) = (KA)/l (T)`
`:. -(dT)/(dt) = -(dT_1)/(dt)+ (dT_2)/(dt)`
`:. -(dT_1)/(dt) = (KA)/(lC_1) (T)`
and `+(dT_2)/(dt) = (KA)/(lC_2)`
Further, `-(dT)/(dt) = -(dT_1)/(dt) + (dT_2)/(dt)`
`=(KA(C_1+C_2))/(lC_1C_1) T`
or `int_(DeltaT_0)^T (dT)/(T) = -(KA(C_1+C_2))/(lC_1C_2) int_(0)^(t) dt`
Solving, we get `T = Delta T_0 e^(-alphat)`
where, `alpha = (KA(C_1+C_2))/(lC_1C_2)`.