Consider length to have dimension $[L]$ and temperature to have dimension $[K]$.
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 $\alpha$.
- Substitute Dimensions: Replace each quantity with its dimensional representation: $ [\alpha] = \frac{[\Delta L]}{[L_0] [\Delta T]} $ $ [\alpha] = \frac{[L]}{[L] [K]} $
- 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]} $
- 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.
Important Questions from Dimensional formulae and dimensional equations
- Considering the Lorentz force $\vec{F} = q(\vec{v} \times \vec{B})$, where $F$ is force, $q$ is electric charge, and $v$ is velocity, what is the dimensional formula for magnetic flux density $B$?
Given that the energy stored in an inductor is expressed as $U_L = \frac{1}{2}LI^2$ and the power dissipated in a resistor is $P = I^2R$, where $L$ is inductance, $R$ is resistance, and $I$ is current, determine the dimension of the ratio $\frac{L}{R}$.
The dimensions of energy are:
If force $[F]$, acceleration $[A]$ and time $[T]$ are chosen as the fundamental physical quantities. Find the dimensions of pressure.
- The characteristic impedance of free space, $Z_0$, is given by the expression $Z_0 = \sqrt{\frac{\mu_0}{\epsilon_0}}$. If $\mu_0$ represents the magnetic permeability and $\epsilon_0$ represents the electric permittivity, what are the dimensions of $Z_0$?
