Problem 171 - Master Resource Book in JEE Main Physics

Question

A slab consists of two portions of different materials of same thickness and having the conductivities \(K_{1}\) and \(K_{2}\). The equivalent thermal conductivity of the slab is [Karnataka CET 2008] (a) \(K_{1}+K_{2}\) (b) \(\frac{K_{1} K_{2}}{K_{1}+K_{2}}\) (c) \(\frac{2 K_{1} K_{2}}{K_{1}+K_{2}}\) (d) \(\sqrt{K_{1}+K_{2}}\)

Step-by-Step Solution

Answer

The equivalent thermal conductivity is option (c): \( \frac{2 K_{1} K_{2}}{K_{1}+K_{2}} \).

Step 1: Identify the Problem

We need to find the equivalent thermal conductivity of a slab made up of two materials with different conductivities, both having the same thickness.

Step 2: Recall Thermal Conductivity Concept

When materials are in series, and have the same thickness, the equivalent thermal conductivity is given by the formula for two conductors in series: \[ K_{eq} = \frac{2 K_{1} K_{2}}{K_{1} + K_{2}} \] where \( K_{1} \) and \( K_{2} \) are the thermal conductivities of the two materials.

Step 3: Apply the Formula

Using the formula, we substitute \( K_{1} \) and \( K_{2} \) into the expression for equivalent conductivity: \[ K_{eq} = \frac{2 K_{1} K_{2}}{K_{1} + K_{2}} \].

Step 4: Identify the Correct Answer

From the given options, option (c) matches our derived formula for the equivalent thermal conductivity: \( \frac{2 K_{1} K_{2}}{K_{1}+K_{2}} \).

Key Concepts

Conductivity in Series

When we place two materials one after another in a series configuration, their thermal conductivities combine in a special way. Think of it like stacking two different types of sandwiches. Each layer has its own resistance to heat flow based on its material property. This resistance adds up as the heat tries to move through each layer. The thicker the resistance, the more difficult it is for heat to pass through. To calculate the total resistance to heat flow for a series, we use a specific formula that combines the conductivities of the materials. If the materials are of the same thickness, the formula simplifies, and it shows us how they share the load of conducting heat. This is captured in the formula for equivalent thermal conductivity for series: \[ K_{eq} = \frac{2 K_{1} K_{2}}{K_{1} + K_{2}} \] In this formula, \( K_1 \) and \( K_2 \) are the thermal conductivities of the first and second materials, respectively. The formula essentially tells us how to combine them to find the equivalent single-layer material that would have the same thermal conductivity as the two stacked together.

Equivalent Thermal Conductivity

Equivalent thermal conductivity is a concept used to simplify understanding complex systems of materials. Imagine you have a slab composed of different materials, like a layer cake. Instead of calculating heat transfer through each layer separately, equivalent conductivity allows you to consider the whole slab as if it were one single material. Why is this important?

• It provides a straightforward calculation for thermal resistance in layered systems.
• It simplifies comparison between different objects or materials.
• It helps engineers and scientists efficiently design insulation or conductive systems.

When the layers are of equal thickness, the method to find this equivalent conductivity involves the formula \[ K_{eq} = \frac{2 K_{1} K_{2}}{K_{1} + K_{2}} \] This formula gives us a clear and concise way to determine how well, overall, the slab can conduct heat just as if it were made of a single material.

Thermal Physics

Thermal physics is the study of heat transfer and distribution in physical systems. It's a branch that combines elements of thermodynamics and statistical mechanics to explain how heat moves and changes substances. Key components include:

• Understanding how energy is transferred between objects.
• How temperature differences drive heat flow.
• Calculating the rates of energy transfer.

Research in thermal physics helps improve energy efficiency in systems like heaters, fridges, and even designing better thermal insulation. By mastering how thermal conductivity works, along with associated concepts like equivalent thermal conductivity and conductivity in series, one can better control heat flow in practical applications. In essence, thermal physics isn't just about understanding heat—it's about optimizing and engineering systems to harness its power effectively.