Problem 93 - Master Resource Book in JEE Main Physics

Question

Four rods of identical cross-sectional area and made from the same metal form the sides of square. The temperature of two diagonally opposite points are \(T\) and \(\sqrt{2} T\) respectively in the steady state. Assuming that only heat conduction takes place, what will be the temperature difference between other two points? (a) \(\frac{\sqrt{2}+1}{2} T\) (b) \(\frac{2}{\sqrt{2}+1} T\) (c) 0 (d) None of these

Step-by-Step Solution

Answer

(c) 0

Step 1: Understand the Problem

In this problem, we have a square with four rods, each made from the same metal and having the same cross-sectional area. Heat conduction occurs between the rods. Given the temperatures at two diagonally opposite points, we need to find the temperature difference at the other two points.

Step 2: Establish Conditions of Steady State

In the steady state, the rate of heat flow into any point in the system is equal to the rate of heat flow out. Therefore, the temperatures at opposite points establish a thermal equilibrium where continuous heat exchange balances them out.

Step 3: Analyze Diagonal Temperature Difference

We have a square with temperatures at diagonally opposite points as \( T \) and \( \sqrt{2}T \). Due to symmetry in the steady state, the middle of each diagonal will have a uniform average temperature between these two points.

Step 4: Calculate Midpoint Temperature of Diagonals

The temperatures at the midpoints of two diagonals (\(T\) and \(\sqrt{2}T\)) will be the arithmetic mean: \( \frac{T + \sqrt{2}T}{2} = \frac{(1+\sqrt{2})T}{2} \).

Step 5: Determine Temperature at Other Points

In a symmetric setup with heat conduction only through identical rods, if the system reaches equilibrium, the temperature difference between any two adjacent points (sides, not across the diagonal) should be 0 due to symmetry.

Key Concepts

Steady State Heat Transfer

When we talk about steady state heat transfer, we're essentially discussing a situation where the temperature distribution in an object doesn't change over time. This means that any point within the object has a constant temperature, as long as the external conditions remain the same.
Let's imagine a rod at thermal equilibrium. Heat enters one end and exits the other, but the amount of heat coming in equals the amount going out. The entire length of the rod maintains a uniform gradient of temperatures that won't change unless something external impacts it.

• Heat transfer at steady state happens without any heat accumulation.
• Temperature distribution remains constant over time.
• Incoming and outgoing heat rates are balanced.

In the problem above, the four rods around the square are in steady-state heat transfer. This means whatever heat flows into the system from a hot spot flows out through a cold spot, keeping all internal points at fixed temperatures.

Thermal Equilibrium

The concept of thermal equilibrium is crucial in understanding heat transfer systems. It describes a scenario where all parts of the system reach a uniform temperature, with no net flow of thermal energy between any two parts.
In a practical sense, if two or more objects or systems are in thermal contact and they have reached the same temperature, they are in thermal equilibrium. No heat energy will migrate from one object to another; instead, they will maintain this stable state.

• No temperature difference remains to drive heat exchange.
• Energy transfer ceases when thermal equilibrium is reached.
• The state is stable unless disturbed by external influences.

For the rods in our exercise, reaching thermal equilibrium means each point within the rods has the same temperature over time. The system is balanced, as the heat flow into any vertex of the square equals the heat flowing out, leading to no net temperature changes across the square.

Symmetry in Heat Distribution

Symmetry plays a significant role in simplifying the analysis of heat conduction problems. In systems where components are arranged symmetrically, the heat distribution can also exhibit symmetry, making it easier to predict temperature patterns and differences.
Consider our square system of rods. Due to symmetry, the midpoint of each diagonal in the square reaches the average temperature of the temperatures at the ends of that diagonal. This is because the heat has equal pathways to travel in either direction down the rod.

• Symmetric structures often lead to symmetric temperature distributions.
• Midpoints of symmetrically aligned segments share a mean temperature.
• Symmetry helps in predicting equilibriums without detailed calculations.

In the given exercise, symmetry indicates that the temperature difference between the other two points, which aren't diagonally opposite, should be zero. Thus, they must share the same temperature due to the balanced distribution of heat across all rods.