Question

A rectangular iron bar (coefficient of thermal conductivity = K1) and a
rectangular brass bar (coefficient of thermal conductivity = K2) are soldered
end to end. The length of the iron bar is double that of the brass bar while
its area of cross section is half that of the brass bar.
If the free ends of the iron bar and the brass bar are maintained at
temperatures of T1 and T2, respectively, obtain an expression for the
(equilibrium) temperature of the junction of the two bars.
​

Answer 1

The length of the combined rod is 2L.

The thermal resistance of iron rod is Riron = (1/K1)(L/A)

The thermal resistance of brass rod is  Rbrass= (1/K2)(L/A)

The equivalent thermal resistance is R = Riron + Rbrass

The thermal conductivity of the combined rod will be -

R = R1+R2

R = 1/K1(L/A)+1/K2(L/A)

  = (L/A)(1/K1+1/K2)

  = 1/K(2L/A)

Comparing we will get -

2/K = 1/K1+1/K2

Therefore, expression =  K= 2K1K2/K1+K2