Animated Solution for Physics - Properties of Solids and Liquids: The temperature $\theta$ at the junction of two insulating sheets, having thermal resistances $R_1$ and $R_2$ as well as top and bottom temperatures ${\theta }_1$ and ${\theta }_2$ (as shown in figure) is given by

Have you ever noticed how different branches of physics often rhyme with each other? The problem of finding the junction temperature between two insulating sheets is a perfect example of this. It is mathematically identical to finding the voltage between two resistors in a series electrical circuit!

Let's dive into the beautiful mechanics of steady-state heat transfer and see how we can derive the junction temperature using a simple, elegant principle.

Heat Current

Imagine two insulating sheets stacked on top of each other. The top sheet has a thermal resistance of $R_1$, and the bottom sheet has a thermal resistance of $R_2$. The top surface is maintained at a hot temperature ${\theta }_1$, and the bottom surface is kept at a cooler temperature ${\theta }_2$. We want to find the temperature $\theta$ exactly at the junction where the two sheets meet.

In thermodynamics, when a system reaches a steady state, it means the temperature profile is no longer changing with time. Because heat is not accumulating anywhere, the rate at which heat flows through the top sheet must be exactly equal to the rate at which it flows through the bottom sheet.

This rate of heat flow is called the heat current ($Q$). Just like electrical current ($I=\frac{\Delta V}{R}$), heat current is driven by a temperature difference and opposed by thermal resistance:

Since the sheets are in series, the heat current $Q$ is constant throughout the entire stack. We can write the heat current for each sheet individually and set them equal to each other.

For the top sheet, the heat flows from ${\theta }_1$ down to the junction $\theta$:

For the bottom sheet, the heat continues from the junction $\theta$ down to ${\theta }_2$:

Equating the two currents gives us our master equation:

Now, it is just a matter of isolating our unknown junction temperature, $\theta$. Let's cross-multiply to clear the denominators:

Expanding the brackets on both sides:

We want to group all the terms containing $\theta$ on one side. Let's move $-R_2\theta$ to the right side and $-R_1{\theta }_2$ to the left side:

Factoring out $\theta$ on the right side:

Finally, dividing by the sum of the resistances, we arrive at our answer:

Look closely at the final expression. Does it look familiar?

It is exactly the section formula from coordinate geometry, or the formula for the center of mass! The junction temperature $\theta$ is a weighted average of the boundary temperatures ${\theta }_1$ and ${\theta }_2$.

However, there is a beautiful twist: the weights are crossed. The temperature ${\theta }_1$ is weighted by the resistance of the other sheet ($R_2$), and ${\theta }_2$ is weighted by $R_1$. This makes physical sense: if $R_1$ is very small (a good conductor), the junction temperature will be very close to ${\theta }_1$. The math perfectly captures the physical reality!