Understanding the Linear Expansion Formula

The question asks us to find the dimensional formula for the coefficient of linear expansion, denoted by $\alpha$. We are given the relationship that describes the change in length $(\Delta L)$ of a material based on its original length $(L_0)$, the coefficient of linear expansion $(\alpha)$, and the change in temperature $(\Delta T)$. The formula provided is:

$ \Delta L = L_0 \alpha \Delta T $

To find the dimensions of $\alpha$, we need to understand the dimensions of the other quantities involved.

Dimensions of Known Quantities

We are given the following information about the dimensions:

• The dimension of length $(\Delta L)$ is [L].
• The dimension of original length $(L_0)$ is [L].
• The dimension of temperature change $(\Delta T)$ is [K] (representing Kelvin).

Deriving the Dimensional Formula for $\alpha$

We can rearrange the given formula to solve for $\alpha$:

$ \alpha = \frac{\Delta L}{L_0 \Delta T} $

Now, let's substitute the dimensions of each term into this equation. Let $[\alpha]$ represent the dimensional formula of $\alpha$.

1. Substitute Dimensions: Replace each quantity with its dimensional representation: $ [\alpha] = \frac{[\Delta L]}{[L_0] [\Delta T]} $ $ [\alpha] = \frac{[L]}{[L] [K]} $
2. Simplify the Expression: The dimension of length $[L]$ appears in both the numerator and the denominator, so they cancel each other out: $ [\alpha] = \frac{1}{[K]} $
3. Express in Standard Form: We can write this as $[K^{-1}]$. To express the full dimensional formula including dimensions for mass $[M]$, length $[L]$, and time $[T]$, we assign them a power of zero, as they do not appear in the expression for $\alpha$: $ [\alpha] = [M^0 L^0 T^0 K^{-1}] $

Conclusion on Coefficient of Linear Expansion Dimensions

The dimensional formula for the coefficient of linear expansion $(\alpha)$ is therefore $[M^0 L^0 T^0 K^{-1}]$. This indicates that $\alpha$ is independent of mass, length, and time, but depends inversely on temperature.