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

\frac{θ_{1} R_{2} + θ_{2} R_{1}}{R_{1} + R_{2}}

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\frac{θ_{1} R_{2} - θ_{2} R_{1}}{R_{2} + R_{1}}

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\frac{θ_{1} R_{2} - θ_{1} R_{1}}{R_{2} - R_{1}}

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\frac{θ_{1} R_{1} + θ_{2} R_{2}}{R_{1} + R_{2}}

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Solution

The correct option is A $\frac{θ_{1} R_{2} + θ_{2} R_{1}}{R_{1} + R_{2}}$
Temperature at the junction is $θ$ .

Since the two sheets are in series combination , so the thermal current flowing through them is same .

$\frac{T_{2} - T}{R_{2}} = \frac{T - T_{1}}{R_{1}}$

$\frac{θ_{2} - θ}{R_{2}} = \frac{θ - θ_{1}}{R_{1}}$

$R_{1} (θ_{2} - θ) = R_{2} (θ - θ_{1})$

$R_{1} θ_{2} - R_{1} θ = R_{2} θ - R_{2} θ_{1}$

$R_{1} θ + R_{2} θ = R_{1} θ_{2} + R_{2} θ_{1}$

$θ = \frac{R_{1} θ_{2} + R_{2} θ_{1}}{R_{1} + R_{2}}$

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The temperature θ at the junction of two insulating sheets, having thermal resistances R1 and R2 as well as bottom and top temperatures θ1 and θ2 (as shown in figure) is given by :

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