Stefan-Boltzmann Constant: Deriving the Dimensional Formula

The Stefan-Boltzmann constant, denoted by the symbol $\sigma$ (sigma), is a physical constant that relates the total energy radiated per unit surface area of a black body across all wavelengths per unit time to the fourth power of its absolute temperature. This relationship is described by the Stefan-Boltzmann Law.

The Stefan-Boltzmann Law can be expressed as:

$\qquad I = \sigma T^4$

Where:

• $I$ is the emissive power (or intensity), which is the total energy radiated per unit surface area per unit time.
• $\sigma$ is the Stefan-Boltzmann constant.
• $T$ is the absolute temperature of the black body in Kelvin.

To find the dimensional formula of the Stefan-Boltzmann constant ($\sigma$), we can rearrange the equation:

$\qquad \sigma = \frac{I}{T^4}$

Dimensional Analysis of Emissive Power (I)

First, let's determine the dimensional formula for emissive power ($I$). Emissive power is defined as energy per unit area per unit time. Therefore, its formula is:

$\qquad I = \frac{\text{Energy}}{\text{Area} \times \text{Time}}$

Now, let's find the dimensions of each component:

• Energy (E): Energy has the dimensions of work, which is Force × Displacement. Force has dimensions of Mass × Acceleration ($[\text{M L T}^{-2}]$). So, the dimension of energy is $[\text{M L T}^{-2}] \times [\text{L}] = [\text{M L}^2 \text{T}^{-2}]$.
• Area (A): Area is a measure of two-dimensional space, so its dimension is $[\text{L}^2]$.
• Time (t): Time is a fundamental dimension, so its dimension is $[\text{T}]$.

Substituting these dimensions into the formula for $I$:

$\qquad [I] = \frac{[\text{M L}^2 \text{T}^{-2}]}{[\text{L}^2] \times [\text{T}]}$

$\qquad [I] = [\text{M L}^{(2-2)} \text{T}^{(-2-1)}]$

$\qquad [I] = [\text{M L}^0 \text{T}^{-3}]$

Dimensional Analysis of Temperature (T)

Temperature ($T$) is a fundamental quantity in the International System of Units (SI) and its dimension is represented by Kelvin (K).

$\qquad [\text{T}] = [\text{K}]$

Therefore, the dimension of $T^4$ is $[\text{K}^4]$.

Calculating the Dimensional Formula of Stefan-Boltzmann Constant ($\sigma$)

Now, we can substitute the dimensions of $I$ and $T^4$ back into the equation for $\sigma$:

$\qquad [\sigma] = \frac{[I]}{[T^4]}$

$\qquad [\sigma] = \frac{[\text{M L}^0 \text{T}^{-3}]}{[\text{K}^4]}$

$\qquad [\sigma] = [\text{M}^1 \text{L}^0 \text{T}^{-3} \text{K}^{-4}]$

Thus, the dimensional formula of the Stefan-Boltzmann constant is $\text{M}^1 \text{L}^0 \text{T}^{-3} \text{K}^{-4}$.

Comparing this with the given options:

The calculated dimensional formula matches option 1.