# Stefan Boltzmann Law Calculator

Created by Bogna Szyk
Last updated: Feb 02, 2023

This Stefan Boltzmann law calculator describes the power radiated from a body at a high temperature. With its help, you can calculate, for example, how much thermal energy is emitted from an asphalt road in the middle of the summer, or what the temperature of the Sun is.

If you want to learn what is the thermal radiation formula, simply keep on reading!

## Stefan Boltzmann law

The Stefan Boltzmann law states that the power radiated from a black body (an ideal emitter) is proportional to its temperature, raised to the fourth power. For every other type of substance, the power is proportional to both T⁴ and the emissivity of the body.

Our thermal radiation calculator uses the following formula:

$\scriptsize P = \sigma \times \epsilon \times A \times T^4$

where:

• $P$ is the power of body's thermal radiation.
• $\sigma$ is the Stefan Boltzmann constant, equal to $5.670367 \times 10^{-8}$. This constant is also used in our ideal gas law calculator.
• $\epsilon$ is the emissivity of the substance, expressed as the ratio of the thermal radiation from a surface to the radiation from a black body at the same temperature. The values of emissivity range from 0 (full reflection) to 1 (black body).
• $A$ is the surface area of the body.
• $T$ is the temperature of the body, expressed in Kelvins.

## Example: what is the temperature of the Sun?

You can use this Stefan Boltzmann law calculator to figure out what is the temperature of any celestial body - for example, of the Sun. To do it, you need to follow these steps:

1. Knowing the radius of Sun, $R = 6.963 \times 10^8 \text{ m}$, calculate its surface. If you assume that Sun is a perfect sphere, then its area is equal to
\scriptsize \begin{aligned} \qquad A &= 4\pi R^2 \\ &= 4\pi (6.963 \times 10^8)^2 \\ &= 6.093 \times 10^{18} \text{ m}^2 \end{aligned}
1. Then, find out what is the emissivity of the Sun. You can assume that it is a perfect black body with emissivity equal to 1.

2. You can measure how much solar power (on average) does the Earth receive. This value is called the solar constant and is equal to $S = 1367 \text{ W}/\text{m}^2$. The power emitted by the Sun is dispersed uniformly around it, what means that if you multiply this solar constant by the area of a sphere of radius equal to the distance between Earth and Sun $D$, you will obtain

\scriptsize \begin{aligned} \qquad P &= S \times a \\ &= S \times 4\pi D^2 \\ &= 1367 \times 4\pi (1.496 \times 10^{11})^2 \\ &= 3.845 \times 10^{26} W \end{aligned}
1. Then, all you have to do is input these values into the Stefan Boltzmann equation to find the temperature of the Sun:
\scriptsize \begin{aligned} \qquad P &= \sigma \times \epsilon \times A \times T^4 \\ T &= \left(\frac{P}{\sigma \times \epsilon \times A}\right) ^ {1/4} \\ T &= \!\left(\frac{3.845 \times 10^{26}}{5.67 \!\!\times\!\! 10^{-8} \times \!1\! \times 6.09\!\!\times\!\!10^{18}}\right) ^ {1/4} \\ T &= 5776 \text{ K} \end{aligned}
1. The temperature of the Sun is equal to $5776\text{ K}$. Don't believe it? Check it with the Stefan Boltzmann law calculator!
Bogna Szyk
Area
Temperature
°C
Material
Black body
Emissivity
W
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