Understanding the Stefan-Boltzmann Constant ($\sigma$) Value

The question asks for the theoretical value of the Stefan-Boltzmann constant, denoted by the Greek letter sigma ($\sigma$). This constant is fundamental in physics, particularly in thermodynamics and radiative heat transfer.

The Stefan-Boltzmann Law

The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a black body—which is the amount of energy emitted by the body—is directly proportional to the fourth power of the body's thermodynamic temperature. A black body is an idealized object that absorbs all incident electromagnetic radiation.

The law can be expressed mathematically as:

Where:

• $I$ represents the radiant emittance (power emitted per unit surface area) in Watts per square meter ($\text{Wm}^{-2}$).
• $P$ is the total power radiated in Watts (W).
• $A$ is the surface area of the black body in square meters ($\text{m}^2$).
• $\sigma$ is the Stefan-Boltzmann constant.
• $T$ is the absolute thermodynamic temperature in Kelvin (K).

Theoretical Value of $\sigma$

The Stefan-Boltzmann constant ($\sigma$) can be derived from other fundamental constants, including the speed of light ($c$), the Planck constant ($h$), the Boltzmann constant ($k_B$), and the speed of light ($c$):

The theoretical value calculated using this formula is approximately:

Analyzing the Options

Let's compare the theoretical value with the given options:

• Option 1: $5.91 \times 10^7 \text{ Wm}^{-2}\text{K}^{-4}$ - This value has the wrong exponent and magnitude.
• Option 2: $5.17 \times 10^9 \text{ Wm}^{-2}\text{K}^{-4}$ - This value has the wrong exponent and magnitude.
• Option 3: $5.35 \times 10^{10} \text{ Wm}^{-2}\text{K}^{-4}$ - This value has the wrong exponent and magnitude.
• Option 4: $5.67 \times 10^{-8} \text{ Wm}^{-2}\text{K}^{-4}$ - This value matches the accepted theoretical value of the Stefan-Boltzmann constant, including its units.

Therefore, the correct theoretical value of the Stefan-Boltzmann constant ($\sigma$) among the choices provided is $5.67 \times 10^{-8} \text{ Wm}^{-2}\text{K}^{-4}$.