Two bodies having masses m₁ and m₂ along with specific heat capacities s₁ and s₂ are joined by a rod of length l , cross-sectional area A , thermal conductivity K , and negligible heat capacity. The entire system is thermally insulated. At time t = 0 , the temperatures of the first and second bodies are T₁ and T₂ respectively ( T₂ > T₁ ). Determine the temperature difference between the two bodies at time t .

Step-by-step solution

Let the temperatures of the two bodies at time t be ₁ and ₂ respectively. The rate of heat flow from the second body to the first body is given by: dQ dt = KA( ₂ - ₁) l The heat gained by the first body and lost by the second body changes their temperatures. The rates of change of temperatures are: d ₁ dt = 1 m₁ s₁ dQ dt = KA( ₂ - ₁) l m₁ s₁ d ₂ dt = - 1 m₂ s₂ dQ dt = - KA( ₂ - ₁) l m₂ s₂ The rate of change of the temperature difference = ₂ - ₁ is: d( ) dt = d ₂ dt - d ₁ dt = - KA l ( ₂ - ₁) ( 1 m₁ s₁ + 1 m₂ s₂ ) d