JEE MAIN - Physics (2021 - 26th February Morning Shift - No. 12)
The temperature $$\theta$$ at the junction of two insulating sheets, having thermal resistances R1 and R2 as well as top and bottom temperatures $$\theta$$1 and $$\theta$$2 (as shown in figure) is given by :
_26th_February_Morning_Shift_en_12_1.png)
$${{{\theta _1}{R_2} + {\theta _2}{R_1}} \over {{R_1} + {R_2}}}$$
$${{{\theta _1}{R_1} + {\theta _2}{R_2}} \over {{R_1} + {R_2}}}$$
$${{{\theta _1}{R_2} - {\theta _2}{R_1}} \over {{R_2} - {R_1}}}$$
$${{{\theta _2}{R_2} - {\theta _1}{R_1}} \over {{R_2} - {R_1}}}$$
Explanation
Temperature at the junction is $$\theta$$.
so using the formula
$${{{T_2} - T} \over {{R_1}}} = {{T - {T_1}} \over {{R_2}}}$$
$${{{\theta _2} - \theta } \over {{R_2}}} = {{\theta - {\theta _1}} \over {{R_1}}}$$$${R_1}({\theta _2} - \theta ) = {R_2}(\theta - {\theta _1})$$
$${R_1}{\theta _2} - {R_1}\theta = {R_2}\theta - {R_2}{\theta _1}$$
$${R_1}\theta + {R_2}\theta = {R_1}{\theta _2} + {R_2}{\theta _1}$$
$$\theta = {{{R_1}{\theta _2} + {R_2}{\theta _1}} \over {{R_1} + {R_2}}}$$
so using the formula
$${{{T_2} - T} \over {{R_1}}} = {{T - {T_1}} \over {{R_2}}}$$
$${{{\theta _2} - \theta } \over {{R_2}}} = {{\theta - {\theta _1}} \over {{R_1}}}$$$${R_1}({\theta _2} - \theta ) = {R_2}(\theta - {\theta _1})$$
$${R_1}{\theta _2} - {R_1}\theta = {R_2}\theta - {R_2}{\theta _1}$$
$${R_1}\theta + {R_2}\theta = {R_1}{\theta _2} + {R_2}{\theta _1}$$
$$\theta = {{{R_1}{\theta _2} + {R_2}{\theta _1}} \over {{R_1} + {R_2}}}$$
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