Animated Solution for Physics - Properties of Solids and Liquids: Three rods made of the same material and having the same cross-section have been joined as shown in the figure. Each rod is of the same length. The left and right ends are kept at $0^{\circ }\text{C}$ and $90^{\circ }\text{C}$ respectively. The temperature of junction of the three rods will be

Imagine you are standing at a Y-shaped intersection where three identical metallic highways meet. Two of these highways bring in a scorching heat from a source at $90^{\circ }\text{C}$, while the third highway acts as an escape route, dumping heat into a freezing sink at $0^{\circ }\text{C}$. The question is: what is the temperature at the exact center where these three roads meet?

This is a classic problem of steady-state heat conduction. It might look like a complex thermodynamic puzzle, but it is actually governed by a beautifully simple principle of balance.

In physics, nature loves equilibrium. When heat flows through a material, it doesn't just pile up randomly. In a steady state, the temperature at any given point remains perfectly constant over time. This means that whatever heat energy enters a junction must exactly equal the heat energy leaving it.

Does this sound familiar? It should! This is the exact same logic behind Kirchhoff's Current Law in electricity, which states that the sum of currents entering a node equals the sum of currents leaving it. Here, instead of electrical current, we are dealing with heat current ($H$).

Mathematically, the heat current through a rod is given by:

where $\Delta T$ is the temperature difference across the rod, and $R$ is the thermal resistance. The thermal resistance itself depends on the material's properties and geometry: $R=\frac{L}{KA}$, where $L$ is the length, $A$ is the cross-sectional area, and $K$ is the thermal conductivity.

The problem gives us a massive shortcut: all three rods are made of the same material, have the same cross-section, and are of the same length. This means their thermal resistances are identical!

Let's call the unknown temperature at the junction $\theta$. Heat will naturally flow from the hotter $90^{\circ }\text{C}$ ends towards the junction, and then from the junction towards the colder $0^{\circ }\text{C}$ end.

Applying our continuity principle ($H_{\text{in}}=H_{\text{out}}$), we can write:

Substituting our formula for heat current, we get:

Because the thermal resistance $R$ is the same for every rod, it acts as a common denominator. We can elegantly multiply the entire equation by $R$ to cancel it out completely. This leaves us with a straightforward linear equation:

Let's group the terms to solve for $\theta$:

Moving the $-2\theta$ to the right side of the equation:

Finally, dividing by 3 gives us the junction temperature:

The junction stabilizes at exactly $60^{\circ }\text{C}$. This problem beautifully demonstrates how complex physical systems often boil down to simple conservation laws. If the rods had been made of different materials or had different lengths, the $R$ values wouldn't have canceled out, but the core philosophy—that the heat entering must equal the heat leaving—would remain absolutely unchanged. Master this principle, and you can solve any thermal circuit thrown your way!