Understanding Stefan Boltzmann Constant Units

The question asks for the units in which the Stefan-Boltzmann constant ($\sigma$) is expressed. To find the units of the Stefan-Boltzmann constant, we need to look at the equation it is used in, which is the Stefan-Boltzmann law.

The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit solid angle is directly proportional to the fourth power of the black body's absolute temperature. The formula is typically written as:

$$\frac{\dot{Q}}{A} = \epsilon \sigma T^4$$

Where:

• $\dot{Q}$ is the total heat energy radiated per unit time (power). Its unit is Watts ($\mathrm{W}$).
• $A$ is the surface area. Its unit is square meters ($\mathrm{m}^2$).
• $\frac{\dot{Q}}{A}$ is the radiant heat power per unit area (radiant flux density). Its unit is Watts per square meter ($\mathrm{W/m}^2$).
• $\epsilon$ (epsilon) is the emissivity of the object. It is a dimensionless quantity, ranging from 0 to 1.
• $\sigma$ (sigma) is the Stefan-Boltzmann constant. We want to find its units.
• $T$ is the absolute temperature of the surface. Its unit is Kelvin ($\mathrm{K}$).

To find the units of $\sigma$, we can rearrange the formula to solve for $\sigma$:

$$\sigma = \frac{\dot{Q}/A}{\epsilon T^4}$$

Now, let's substitute the units of each term into this rearranged equation:

• Units of $(\dot{Q}/A)$ are $\mathrm{W/m}^2$.
• Units of $\epsilon$ are dimensionless (no units).
• Units of $T$ are $\mathrm{K}$, so units of $T^4$ are $\mathrm{K}^4$.

Substituting these units into the expression for $\sigma$:

$$\text{Units of } \sigma = \frac{\mathrm{W/m}^2}{\text{dimensionless} \times \mathrm{K}^4}$$

$$\text{Units of } \sigma = \frac{\mathrm{W/m}^2}{\mathrm{K}^4}$$

This can also be written as $\mathrm{W} / (\mathrm{m}^{2} \mathrm{~K}^{4})$ or $\mathrm{W} \mathrm{m}^{-2} \mathrm{~K}^{-4}$.

Now, let's compare this derived unit with the given options:

• Option 1: $\mathrm{W} / \mathrm{m}^{2} \mathrm{~K}^{2}$ - Incorrect, the temperature is raised to the power of 4.
• Option 2: $\mathrm{W} / \mathrm{m}^{2} \mathrm{~K}$ - Incorrect, the temperature is raised to the power of 4.
• Option 3: $\mathrm{W} / \mathrm{m}^{2} \mathrm{~K}^{4}$ - Correct, this matches the units we derived.
• Option 4: $\mathrm{Wm}^{2} \mathrm{~K}^{2}$ - Incorrect, $\mathrm{m}^2$ and $\mathrm{K}^2$ are in the numerator.

Therefore, the Stefan Boltzmann constant is expressed in the unit $\mathrm{W} / \mathrm{m}^{2} \mathrm{~K}^{4}$.

Revision Table: Stefan Boltzmann Law

Additional Information on Stefan Boltzmann Constant and Blackbody Radiation

The Stefan-Boltzmann law is fundamental in understanding thermal radiation. It applies precisely to ideal black bodies, which are theoretical objects that absorb all incident electromagnetic radiation and emit thermal radiation maximally. Real objects have an emissivity ($\epsilon$) between 0 and 1, which accounts for how effectively they radiate compared to a black body.

The value of the Stefan-Boltzmann constant ($\sigma$) is approximately $5.670374419 \times 10^{-8} \mathrm{~W/m}^2 \mathrm{~K}^4$. This constant plays a crucial role in calculating heat transfer by radiation, analyzing the energy output of stars, and understanding climate physics.

Understanding the units helps in verifying calculations and ensuring dimensional consistency in physics problems. The unit $\mathrm{W/m}^2 \mathrm{~K}^4$ clearly shows that the constant relates radiant power per unit area to the fourth power of absolute temperature.