One end of a metal rod having a cross-sectional area of 1.0 cm ^2 is being heated. At a certain instant, the temperature gradient is 5.0^ C cm ⁻¹ at cross section A and 2.5^ C cm ⁻¹ at cross section B . Determine the rate at which the temperature is increasing in the section AB of the rod. The heat capacity of the section AB is 0.40 J ^ C ⁻¹ , and the thermal conductivity of the rod's material is 200 W m ⁻¹ ^ C ⁻¹ .

Step-by-step solution

The rate of heat flow at cross-section A is given by Fourier's law of heat conduction: H_A = K A | dT dx |_A Substituting the given values: H_A = 200 (1.0 10⁻⁴) (5.0 10^2) = 10 W Similarly, the rate of heat flow at cross-section B is: H_B = K A | dT dx |_B H_B = 200 (1.0 10⁻⁴) (2.5 10^2) = 5 W The net rate of heat accumulation in the section AB is the difference between the heat entering at A and leaving at B : dQ dt = H_A - H_B = 10 - 5 = 5 W The rate at which the temperature is increasing in section AB is related