Question

The temperature $ \mathrm{\theta }$ at the junction of two insulating sheets, having thermal resistances $ {\mathrm{R}}_{1} \mathrm{and} {\mathrm{R}}_{2}$ as well as the bottom and top temperatures $ {\mathrm{\theta }}_{1} \mathrm{and} {\mathrm{\theta }}_{2}$ (as shown in the figure) is given by:

• A. $\frac{({\mathrm{\theta }}_1{\mathrm{R}}_2+{\mathrm{\theta }}_2{\mathrm{R}}_1)}{({\mathrm{R}}_1+{\mathrm{R}}_2)}$
• B. $\frac{({\mathrm{\theta }}_1{\mathrm{R}}_2–{\mathrm{\theta }}_2{\mathrm{R}}_1)}{({\mathrm{R}}_2–{\mathrm{R}}_1)}$
• C. $\frac{({\mathrm{\theta }}_2{\mathrm{R}}_2–{\mathrm{\theta }}_1{\mathrm{R}}_1)}{({\mathrm{R}}_2–{\mathrm{R}}_1)}$
• D. $\frac{({\mathrm{\theta }}_1{\mathrm{R}}_1+{\mathrm{\theta }}_2{\mathrm{R}}_2)}{({\mathrm{R}}_1+{\mathrm{R}}_2)}$

Answer 1

The correct option is A

$\frac{({\mathrm{\theta }}_1{\mathrm{R}}_2+{\mathrm{\theta }}_2{\mathrm{R}}_1)}{({\mathrm{R}}_1+{\mathrm{R}}_2)}$

The explanation for the correct option:

A: $\frac{({\mathrm{\theta }}_1{\mathrm{R}}_2+{\mathrm{\theta }}_2{\mathrm{R}}_1)}{({\mathrm{R}}_1+{\mathrm{R}}_2)}$

The expressed used for the calculation of temperature at the junction $\mathrm{\theta }$:

$\frac{({\mathrm{T}}_2–\mathrm{T})}{{\mathrm{R}}_2}=\frac{(\mathrm{T}–{\mathrm{T}}_1)}{{\mathrm{R}}_1}$

putting required values that are given in above diagram:

$\begin{array}{rcl}({\mathrm{\theta }}_2–\mathrm{\theta })/{\mathrm{R}}_2& =& (\mathrm{\theta }–{\mathrm{\theta }}_1)/{\mathrm{R}}_1\\ {\mathrm{R}}_1({\mathrm{\theta }}_2–\mathrm{\theta })& =& {\mathrm{R}}_2(\mathrm{\theta }–{\mathrm{\theta }}_1)\\ {\mathrm{R}}_1{\mathrm{\theta }}_2–{\mathrm{R}}_1\mathrm{\theta }& =& {\mathrm{R}}_2\mathrm{\theta }–{\mathrm{R}}_2{\mathrm{\theta }}_1\\ {\mathrm{R}}_1\mathrm{\theta }+{\mathrm{R}}_2\mathrm{\theta }& =& {\mathrm{R}}_1{\mathrm{\theta }}_2+{\mathrm{R}}_2{\mathrm{\theta }}_1\\ \mathrm{\theta }& =& ({\mathrm{R}}_1{\mathrm{\theta }}_2+{\mathrm{R}}_2{\mathrm{\theta }}_1)/({\mathrm{R}}_1+{\mathrm{R}}_2)\end{array}$

Final Answer: Therefore, Option A is the correct option.