Problem 36 - Master Resource Book in JEE Main Physics

Question

Coefficient of thermal conductivity has the dimensions (a) \(\left[\mathrm{ML}^{-1} \mathrm{~T}^{3} \mathrm{~K}^{3}\right]\) (b) \(\left[\mathrm{ML}^{-1} \mathrm{~T}^{-3} \mathrm{~K}^{-1}\right]\) (c) \(\left[\mathrm{MLT}^{-3} \mathrm{~K}^{-1}\right]\) (d) \(\left[\mathrm{MLT}^{-3} \mathrm{~K}\right]\)

Step-by-Step Solution

Answer

The correct answer is (b) \([\text{ML}^{-1} \text{T}^{-3} \text{K}^{-1}]\).

Step 1: Understanding the Problem

The task is to find the dimensions of the coefficient of thermal conductivity. Thermal conductivity is a measure of a material's ability to conduct heat. The formula for calculating the heat conducted is given by Fourier's law, which is: \[ \frac{Q}{t} = k \cdot A \cdot \frac{\Delta T}{L} \] where \( Q \) is the heat transferred, \( t \) is time, \( k \) is the thermal conductivity, \( A \) is the area, \( \Delta T \) is the temperature difference, and \( L \) is the thickness of the material.

Step 2: Dimensional Analysis of the Equation

To find the dimensions of \( k \), we start by finding the dimensions of each term in Fourier's law:1. \( Q \) (heat) has dimensions of energy, \([ ext{ML}^2 ext{T}^{-2} ]\) because it's energy and in Joules2. \( t \) has dimensions of \([ ext{T} ]\)3. \( A \) has dimensions of area \([ ext{L}^2 ]\)4. \( \Delta T \) is temperature with dimensions \([ ext{K} ]\)5. \( L \) which is thickness has dimensions \([ ext{L} ]\).

Step 3: Equating Dimensional Formulae

From the equation \( \frac{Q}{t} = k \cdot A \cdot \frac{\Delta T}{L} \), we isolate \( k \):\[ k = \frac{Q}{t \cdot A \cdot \frac{\Delta T}{L}} = \frac{Q \times L}{t \times A \times \Delta T} \]Substituting dimensions:\[ \left[ \text{ML}^2 \text{T}^{-2} \right] \left[ \text{L} \right] / \left( \left[ \text{T} \right] \left[ \text{L}^2 \right] \left[ \text{K} \right] \right) \]

Step 4: Simplifying the Dimensions

Combining and simplifying the dimensions further we get:\[ \left[ \text{ML}^3 \text{T}^{-2} \right] / \left[ \text{L}^2 \text{T} \text{K} \right] = \left[ \text{ML}^{3-2} \text{T}^{-2-1} \text{K}^{-1} \right] = \left[ \text{ML}^1 \text{T}^{-3} \text{K}^{-1} \right] \]

Step 5: Matching with the Options

The correct dimensional formula — \( \left[ \text{ML}^{1} \text{T}^{-3} \text{K}^{-1} \right] \) — matches with option (b) in the given choices. Thus, the correct answer is (b).

Key Concepts

Dimensional Analysis

Dimensional analysis is a powerful technique used to determine the dimensions of various physical quantities. This method helps to check the consistency of physical equations and assists in deriving the dimensional formula of quantities like thermal conductivity.

In the context of thermal conductivity, the goal is to express this property in terms of fundamental dimensions such as mass \(M\), length \(L\), time \(T\), and temperature \(K\). By analyzing the equation given by Fourier's law, we are able to determine the dimensions of thermal conductivity by using the known dimensions of the interacting quantities.

Dimensional analysis primarily involves breaking down complex dimensions into simpler terms. By equating and manipulating these dimensions, we can often predict the relationships between different physical quantities.

Fourier's Law

Fourier's law is fundamental when discussing heat transfer through a medium. It provides a quantitative form of expressing how heat flows from hot to cold areas within a substance.

The law is expressed as: \\[ \frac{Q}{t} = k \cdot A \cdot \frac{\Delta T}{L} \]
Here:

• \( Q \) is the heat transfer rate, measured in Joules,
• \( t \) is the time over which heat transfer occurs,
• \( k \) is the thermal conductivity of the material,
• \( A \) is the cross-sectional area through which heat flows,
• \( \Delta T \) is the temperature difference across the material,
• \( L \) is the thickness or length the heat travels through.

Fourier's law is essential in understanding how quickly and efficiently heat can be transferred through materials with varying thermal conductivities.

Heat Transfer

Heat transfer is a process by which heat energy moves from a region of higher temperature to a region of lower temperature. It can occur in various ways:

• Conduction: Direct transfer of heat through a material.
• Convection: Heat transfer by the movement of fluid (liquid or gas).
• Radiation: Transfer of heat through electromagnetic waves without involving particles.

Conduction is what Fourier's law primarily addresses. It involves molecular diversity, where more energetic molecules transfer some of their energy to less energetic neighbors.

Understanding how heat transfer works is crucial in fields such as engineering and environmental science as it enables us to design better systems for heating and cooling.

Dimensions of Physical Quantities

Dimensions of physical quantities define the measure of a quantity in terms of basic fundamental units. These basic dimensions are:

• Mass \( [M] \)
• Length \( [L] \)
• Time \( [T] \)
• Temperature \( [K] \)

In dimensional analysis, these fundamental units serve as building blocks for complex constructs such as velocity, acceleration, force, and thermal conductivity.

For instance, the dimension of velocity is expressed as \[ [LT^{-1}] \], combining length and time to express motion. By using these basic dimensions, we can decompose or understand how complex equations and formulas in physics relate and maintain dimensional consistency throughout physical computations and simulations.