Question

Dimensions of Stefan’s constant is:

• A. \( [ML^{-1}T^{-3}\theta^{-4}] \)
• B. \( [MT^{-3}\theta^{-4}] \)
• C. \( [M^2T^{-3}\theta^{-4}] \)
• D. \( [M^2T^{-2}\theta^{-4}] \)

Hint:

Stefan constant = power per unit area per \(T^4\), so no length term remains.

Answer 1

The Correct Option is B

To determine the dimensions of Stefan’s constant, we need to consider the Stefan-Boltzmann law, which states that the total energy radiated per unit surface area of a black body per unit time (also known as the emissive power, \( E \)) is proportional to the fourth power of the black body's absolute temperature (\( T \)). The formula is given by:

\(E = \sigma T^4\)

Where:

• \(E\) is the emissive power, which has dimensions of power per unit area \([ML^0T^{-3}]\).
• \(\sigma\) is the Stefan’s constant, which is what we need to find.
• \(T\) is the temperature with dimensions of \([\theta]\).

From the Stefan-Boltzmann law:

\([E] = [\sigma] [T]^4\)

We know:

• Dimensions of \( E \): \([ML^0T^{-3}]\)
• Dimensions of \( T^4 \): \([\theta^4]\)

\([ML^0T^{-3}] = [\sigma][\theta^4]\)

Solving for \([\sigma]\), we find:

\([\sigma] = [ML^0T^{-3}\theta^{-4}] = [MT^{-3}\theta^{-4}]\)

Thus, the correct dimensional formula for Stefan’s constant is:

\([MT^{-3}\theta^{-4}]\)

Therefore, the correct answer is \([MT^{-3}\theta^{-4}]\).

This aligns with the option given as the correct answer.