• # question_answer 55) Two rods of length ${{d}_{1}}$and ${{d}_{2}}$ and coefficients of thermal conductivities ${{K}_{1}}$and ${{K}_{2}}$ are kept touching each other. Both have the same area of cross-section the equivalent of thermal conductivity is A)  ${{K}_{1}}+{{K}_{2}}$B)  ${{K}_{1}}{{d}_{1}}+{{K}_{2}}{{d}_{2}}$C)  $\frac{{{d}_{1}}{{K}_{1}}+{{d}_{2}}{{K}_{2}}}{{{d}_{1}}+{{d}_{2}}}$D)  $\frac{{{d}_{1}}+{{d}_{2}}}{({{d}_{1}}/{{K}_{1}}+{{d}_{2}}/{{K}_{2}})}$

When two rods are connected in series $Q=\frac{A({{T}_{1}}-{{T}_{2}})t}{\frac{{{d}_{1}}}{{{K}_{1}}}+\frac{{{d}_{2}}}{{{K}_{1}}}}=\frac{A({{T}_{1}}-{{T}_{2}})t}{({{d}_{1}}+{{d}_{2}})/K}$ $\therefore$ $\frac{{{d}_{1}}+{{d}_{2}}}{K}=\frac{{{d}_{1}}}{{{K}_{1}}}+\frac{{{d}_{2}}}{{{K}_{2}}};$ $\therefore$ $K=\frac{({{d}_{1}}+{{d}_{2}})}{\frac{{{d}_{1}}}{{{K}_{1}}}+\frac{{{d}_{2}}}{{{K}_{2}}}}$