question_answer

A wall has two layers A and B made of different materials. The thickness of both the layers is the same. The thermal conductivity of A and B are \[{{K}_{A}}\] and \[{{K}_{B}}\] such that \[{{K}_{A}}=3{{K}_{B}}\]. The temperature across the wall is \[20{}^\circ C\]. In thermal equilibrium

A) The temperature difference across \[A=15{}^\circ C\]

B) The temperature difference across \[A=5{}^\circ C\]

C) The temperature difference across A is \[10{}^\circ C\]

D) The rate of transfer of heat through A is more than that through B.

Correct Answer: B

Solution :

In series rate of flow of heat is same

\[\Rightarrow \] \[\frac{{{K}_{A}}A({{\theta }_{1}}-\theta )}{l}=\frac{{{K}_{B}}A(\theta -{{\theta }_{2}})}{l}\]

\[\Rightarrow \] \[3{{K}_{B}}({{\theta }_{1}}-\theta )={{K}_{B}}(\theta -{{\theta }_{2}})\]

\[\Rightarrow \] \[3({{\theta }_{1}}-\theta )=(\theta -{{\theta }_{2}})\]

\[\Rightarrow \] \[3{{\theta }_{1}}-3\theta =\theta -{{\theta }_{2}}\]

\[\Rightarrow \] \[4{{\theta }_{1}}-4\theta ={{\theta }_{1}}-{{\theta }_{2}}\]

\[\Rightarrow \] \[4({{\theta }_{1}}-\theta )=({{\theta }_{1}}-{{\theta }_{2}})\]

\[\Rightarrow \] \[4({{\theta }_{1}}-\theta )=20\]

\[\Rightarrow \] \[({{\theta }_{1}}-\theta )=5{}^\circ C\]