Problem 58 - Master Resource Book in JEE Main Physics

Question

What is dimensional formula of thermal conductivity? [UP SEE 2006] (a) \(\left[\mathrm{MLT}^{-1} \theta^{-1}\right]\) (b) [MLT \(\left.^{-3} \theta^{-1}\right]\) (c) \(\left[\mathrm{M}^{2} \mathrm{LT}^{-3} \theta^{-2}\right]\) (d) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2} \theta\right]\)

Step-by-Step Solution

Answer

The dimensional formula for thermal conductivity is \([MLT^{-3}\theta^{-1}]\), so option (b) is correct.

Step 1: Understanding Thermal Conductivity

Thermal conductivity (abla) is a measure of a material's ability to conduct heat. Its dimensional formula is derived from the relation of heat transfer: \[ Q = -k A \frac{dT}{dx} \cdot t \] where \( Q \) is the heat transferred, \( k \) is thermal conductivity, \( A \) is the area, \( dT \) is the temperature difference, \( dx \) is the thickness, and \( t \) is the time.

Step 2: Analyzing the Heat Transfer Formula

The formula for heat transfer can be expressed with dimensions as: \[ [Q] = [k] \times [A] \times [\frac{dT}{dx}] \times [t] \]. Here, \( [Q] = [ML^2T^{-2}] \), \( [A] = [L^2] \), \( [\frac{dT}{dx}] = [\theta L^{-1}] \), and \( [t] = [T] \).

Step 3: Deriving the Dimensional Formula for Thermal Conductivity

Using the equation \( [k] = \frac{[Q]}{[A][(dT/dx)][t]} \), we substitute the dimensions to find \[ [k] = \frac{[ML^2T^{-2}]}{[L^2][\theta L^{-1}][T]} \].

Step 4: Simplifying the Expression

Simplify the expression: \[ [k] = \frac{[ML^2T^{-2}]}{[L^2][\theta L^{-1}][T]} = [ML^3T^{-3}\theta^{-1}] \].

Key Concepts

Thermal Conductivity

Thermal conductivity, often denoted as "k," is a fundamental property that characterizes how well a material can conduct heat. This property is crucial, especially in fields like engineering and materials science. Imagine heat flowing through a metal spoon; how quickly the handle warms up indicates the material's thermal conductivity. **Understanding Thermal Conductivity**- **High thermal conductivity:** Materials like metals with high thermal conductivity allow heat to flow through them easily, making them good conductors.- **Low thermal conductivity:** Materials such as wood or plastic have low thermal conductivity, meaning they resist heat flow, often making them good insulators.If a material conducts heat efficiently, it has a higher thermal conductivity value. This measure is essential for applications such as building construction or designing thermal insulators. To understand thermal conductivity's dimensional formula, one can refer to the equation:\[ Q = -k A \frac{dT}{dx} \cdot t \]This is a fundamental relationship in heat transfer, showing that the rate of heat flow (\(Q\)) is proportional to the material's thermal conductivity (\(k\)), the area (\(A\)), the temperature gradient \(\frac{dT}{dx}\), and the time (\(t\)).

Heat Transfer

Heat transfer is the movement of thermal energy from one object or material to another. It happens via three primary modes: conduction, convection, and radiation. Let's break these down:- **Conduction:** This is the transfer of heat through direct contact between materials. For example, heat traveling through a metal rod when one end is heated.- **Convection:** This occurs when heat is transferred through a fluid (liquid or gas) moving from one location to another. Imagine water heating up in a pot and starting to swirl.- **Radiation:** This is the transfer of heat through electromagnetic waves, like the warmth felt from the sun. The dimension of heat transfer can be represented within the context of thermal conductivity. Using the formula\:\[ [Q] = [k] \times [A] \times [\frac{dT}{dx}] \times [t] \]We can see how each component plays a role in determining the overall transfer of heat. Here, \(Q\) is measured in energy terms, typically joules (\(ML^2T^{-2}\)), and it shows how dimensions interrelate within the context of thermal conductivity.

Dimensional Formula

The dimensional formula provides a way to express physical quantities in terms of their base dimensions, like mass (\(M\)), length (\(L\)), time (\(T\)), and temperature (\(\theta\)). This is essential for checking equations’ consistency and aiding dimensional analysis.**Deriving the Dimensional Formula**For thermal conductivity, the dimensional formula is derived as follows:Using the heat transfer equation:\[ [k] = \frac{[Q]}{[A][(dT/dx)][t]} \]By substituting the known dimensional equations:- \([Q] = [ML^2T^{-2}]\)- \([A] = [L^2]\)- \([\frac{dT}{dx}] = [\theta L^{-1}]\)- \([t] = [T]\)This results in:\[ [k] = \frac{[ML^2T^{-2}]}{[L^2][\theta L^{-1}][T]} \]After simplifying, we find:\[ [k] = [ML^3T^{-3}\theta^{-1}] \]This shows that thermal conductivity has dimensions of mass, length, time, and temperature, each playing a role in its measurement and understanding through dimensional analysis.