Problem 40 - Master Resource Book in JEE Main Physics

Question

Two rods of equal length and area of cross-section are kept parallel and lagged between temperatures \(20^{\circ} \mathrm{C}\) and \(80^{\circ} \mathrm{C}\). The ratio of the effective thermal conductivity to that of the first rod is \(\left[\right.\) the ratio \(\left.\left(\frac{K_{1}}{K_{2}}\right)=\frac{3}{4}\right]\) (a) \(7: 4\) (b) \(7: 6\) (c) \(4: 7\) (d) \(7: 8\)

Step-by-Step Solution

Answer

The correct answer is (b) 7:6.

Step 1: Understand the Problem

We have two rods with equal length and cross-sectional area. They are subjected to temperatures of \(20^{\circ}\)C and \(80^{\circ}\)C. We need to find the effective thermal conductivity of the system given the ratio \(\frac{K_1}{K_2} = \frac{3}{4}\).

Step 2: Find the Formula for Effective Thermal Conductivity

When two rods are in parallel, the effective thermal conductivity \(K_{\text{eff}}\) is given by:\[K_{\text{eff}} = \frac{K_1 + K_2}{2}\]

Step 3: Substitute Given Values into the Formula

Substitute the known ratio \(\frac{K_1}{K_2} = \frac{3}{4}\) into the effective conductivity formula: Let \(K_1 = 3x\) and \(K_2 = 4x\).\[K_{\text{eff}} = \frac{3x + 4x}{2} = \frac{7x}{2}\]

Step 4: Find Ratio of Effective to First Rod's Conductivity

To find the ratio of the effective thermal conductivity to the first rod's thermal conductivity:\[\frac{K_{\text{eff}}}{K_1} = \frac{\frac{7x}{2}}{3x} = \frac{7}{6}\]

Step 5: Identify the Correct Answer

The ratio of the effective thermal conductivity to the conductivity of the first rod is \(\frac{7}{6}\). Therefore, the correct multiple-choice option is \((b) \; 7:6\).

Key Concepts

Parallel Conduction

In heat transfer, parallel conduction occurs when two or more conductive paths exist side by side. This setup is akin to setting two thermal resistors in parallel, allowing heat to flow more efficiently between two endpoints. Imagine two rods situated next to each other, both experiencing a temperature difference between their ends. Those rods act like parallel paths for heat to travel. Conduction, basically, is the movement of heat through a material—when objects are set parallel, each path contributes to the total conduction. Thus, heat can spread out and travels through each rod based on their individual thermal conductivities.

To understand how it works: - Each rod or path contributes according to its thermal conductivity. - Since the lengths and cross-sections are equal, we're focusing solely on the conductivity property itself. Therefore, in the case of the rods with equal dimensions but different material thermal conductivities, each rod will let through an amount of heat proportional to its conductivity. The key concept here is how these conduits combine to create an overall effect, which we call **parallel conduction**.

Effective Thermal Conductivity

Effective thermal conductivity is an important concept when dealing with composite materials or systems with multiple parallel components like the two rods in our exercise. It represents an average thermal conductivity, a sort of summary value that encapsulates how well heat can transfer through a combined system.

For systems arranged in parallel, determining the effective thermal conductivity is straightforward. We use the formula:\[ K_{\text{eff}} = \frac{K_1 + K_2}{2} \]This formula essentially adds the individual rods' thermal conductivities and then averages them.

The effectiveness of this average is crucial in engineering because it gives a single, representative value of how the system performs as a whole, beyond individual rod characteristics. Using this concept, industries can easily gauge how a change in one component (like replacing a rod with a material of different thermal property) could affect the overall heat conduction.

Thermal Conductivity Ratio

The thermal conductivity ratio between materials is essential for understanding how different substances or components will conduct heat relative to each other. In our exercise, the given ratio, \( \frac{K_1}{K_2} = \frac{3}{4} \), tells us that rod 1 has a lesser conductivity compared to rod 2 by a factor of three to four.

To simplify calculations, translating those conductivities into expressions like \( K_1 = 3x \) and \( K_2 = 4x \) allows us to straightforwardly apply formulas and derive meaningful results. This provides an intuitive way to think about the problem, breaking the complexity into manageable chunks.- Ratios can help predict system behavior if components are altered.- Even without detailed numerical values, ratios offer insight into the comparative efficiency of materials.Calculating the ratio of effective thermal conductivity to one of the rods gives a direct measure of improvement or reduction in heat conduction performance due to the system arrangement. Here, it was calculated that this ratio was \( \frac{7}{6} \), demonstrating that overall, the parallel setup slightly outperforms the individual conductivity of the first rod.