question_answer

Two rods having thermal conductivity in the ratio of 5 : 3 and having equal length and equal cross-sectional area. are joined face to face. If the temperature of free end of first rod is \[100{}^\circ C\] and temperature of free end of second rod is \[20{}^\circ C\], temperature of junction will be:

A) \[90{}^\circ C\]

B) \[85{}^\circ C\]

C) \[70{}^\circ C\]

D) \[50{}^\circ C\]

Correct Answer: C

Solution :

Key Idea: In the steady state, the rate of flow of heat in both the conductors will be the same. Taking the two conductors with same cross-section area A, joined in series, with same length d Let H be heat flow through this combination and \[\theta \] is temperature of common surface. In steady state rate of flow of heat is same hence, \[H=\frac{Q}{t}=\frac{{{K}_{1}}A({{\theta }_{1}}-\theta )}{d}=\frac{{{K}_{2}}A(\theta -{{\theta }_{2}})}{d}\] Given  \[{{\theta }_{1}}=100{{\,}^{o}}C,\,{{\theta }_{2}}={{20}^{o}}C\] and    \[{{K}_{1}}:{{K}_{2}}=\frac{5}{3}\] \[\therefore \]      \[{{K}_{1}}({{100}^{o}}-\theta )={{K}_{2}}(\theta -{{20}^{o}})\] \[\Rightarrow \]     \[\frac{{{K}_{1}}}{{{K}_{2}}}=\frac{\theta -{{20}^{o}}}{{{100}^{o}}-\theta }\] \[\Rightarrow \]      \[\frac{5}{3}=\frac{\theta -{{20}^{o}}}{{{100}^{o}}-\theta }\] \[\Rightarrow \]    \[8\theta ={{560}^{o}}\] \[\Rightarrow \]      \[\theta ={{70}^{o}}\]