A metal rod of cross sectional area 1.0 cm^{2} is being heated at one end. At one time, the temperatures gradient is 5.0°C cm^{−1} at cross section A and is 2.5°C cm^{−1} at cross section B. Calculate the rate at which the temperature is increasing in the part AB of the rod. The heat capacity of the part AB = 0.40 J°C^{−1}, thermal conductivity of the material of the rod = 200 W m^{−1}°C^{−1}. Neglect any loss of heat to the atmosphere

#### Solution

Let the temperatur

Let the temperatures at the ends A and B be T_{A} and T_{B}, respectively.

Rate of flow of heat at end A of the rod is given by

`(dQ_A)/(dr)= KA. d/dt (T_A)`

Rate of flow of heat at end B of the rod is given by

`(dQ_B)/dt ( T_B)`

Heat absorbed by the rod = ms∆T

Here, s is the specific heat of the rod and ∆T is the temperature difference between ends A and B.

Rate of heat absorption by the rod is given by

`(dQ)/dt =ms (dT)/(dr)`

∴ ` ms (dT)/(dr)= (KA. dT_A

)/(dl)`

`⇒ (0.4). (dT)/ dt= 200 xx 1xx 10^-4xx (5-2.5) `

`dt/dr=12.5^circ` C/sec