question_answer

Two rods A and B are of equal length. Each rod has its ends at temperatures \[{{T}_{1}}\] and \[{{T}_{2}}\] . What is the condition that will ensure equal rates of flow of heat through the rods A and B?                                                                   

Answer:

                Let \[x\] be the length of each rod. The rates of flow of heat through the rods A and B will be equal if                 \[\frac{{{K}_{1}}{{A}_{1}}({{T}_{1}}-{{T}_{2}})}{x}=\frac{{{K}_{2}}{{A}_{2}}({{T}_{1}}-{{T}_{2}})}{x}\]                 or \[{{K}_{1}}{{A}_{1}}={{K}_{2}}{{A}_{2}}\] or \[\frac{{{A}_{1}}}{{{A}_{2}}}=\frac{{{K}_{2}}}{{{K}_{1}}}\]                 Hence for equal rates of flow of heat, the areas of cross-section of the two rods should be inversely proportional to their coefficients of thermal conductivity.