A room at 20^oC is heated by a heater of resistance 20 ohm connected to 200 V mains. The temperature is uniform throughout the room and the heat is transmitted through a glass window of area 1 $m^2$ and thickness 0.2 cm. Calculate the temperature outside. Thermal conductivity of glass is 0.2 cal/m C^o s and mechanical equivalent of heat is 4.2 J/cal.

If $\mathrm{\theta }$ is the temperature of outside, heat passing per second through the glass window,

$\frac{\mathrm{dQ}}{\mathrm{dt}}=\mathrm{KA}\frac{\left({\mathrm{\theta }}_1-{\mathrm{\theta }}_2\right)}{\mathrm{L}}=\frac{0.2\times 1\times (20-\mathrm{\theta })\mathrm{cal}}{0.2\times {10}^{-2}}=100(20-\mathrm{\theta })$

And heat produced per second by the heater in the room

$\mathrm{P}=\frac{{\mathrm{V}}^2}{\mathrm{R}}\frac{\mathrm{J}}{\mathrm{s}}=\frac{{\mathrm{V}}^2}{\mathrm{RJ}}\frac{\mathrm{cal}}{\mathrm{s}}=\frac{200\times 200}{20\times 4.2}=476.2\frac{\text{ cal }}{\mathrm{s}}$

Now as the temperature of the room is constant, the heat produced per second by heater must be equal to the heat conducted through the glass window.

$$\begin{gathered} 100(20-\mathrm{\theta })=476.2 \\ \Rightarrow \mathrm{\theta }=15{.24}^{\circ }\mathrm{C} \end{gathered}$$