What Is the Stefan-Boltzmann Law and How Do You Use It?

The Stefan-Boltzmann law states that the total power radiated by an ideal black body is proportional to its surface area and the fourth power of its absolute temperature. In IB Physics topic B.1, for both SL and HL, use $L=\sigma AT^4$, with temperature measured in kelvin.

The Reasoning and Mechanism

Here, $L$ is the luminosity, or total radiated power, in watts; $A$ is emitting surface area in $\text{m}^2$; and $T$ is absolute temperature in kelvin. The Stefan-Boltzmann constant is $\sigma=5.67\times10^{-8}\ \text{W m}^{-2}\text{K}^{-4}$.

The fourth-power relationship is crucial. If temperature doubles while area remains constant, luminosity increases by $2^4=16$. Convert Celsius temperature using $T=\theta+273.15$ before substitution.

The law follows from the black-body spectrum: increasing temperature raises the intensity emitted at every wavelength and shifts the peak toward shorter wavelengths. Because power depends on $T^4$, even a small percentage change in temperature can produce a much larger percentage change in luminosity.

The common misconception is to insert Celsius directly into the equation. This produces a physically incorrect result because the relationship requires an absolute temperature scale.

For a spherical body such as a star, $A=4\pi r^2$, so

$$L=4\pi r^2\sigma T^4.$$

For a real surface with emissivity $e$, where $0\leq e\leq1$, multiply the black-body result by $e$.

Worked Example

A spherical star has radius $7.0\times10^8\ \text{m}$ and surface temperature $5800\ \text{K}$. First calculate its surface area implicitly by substituting $A=4\pi r^2$:

$$L=4\pi(7.0\times10^8)^2(5.67\times10^{-8})(5800)^4.$$

This gives

$$L=3.95\times10^{26}\ \text{W}\approx4.0\times10^{26}\ \text{W}.$$

This value is close to the Sun’s luminosity, which provides a useful order-of-magnitude check. A result many powers of ten away should prompt you to recheck the radius squared and temperature fourth power.

Exam Technique

For a Calculate question, show the equation, substitution, and final answer with units. Check whether the question gives area or radius, and distinguish total power $L$ from power per unit area, $L/A=\sigma T^4$. Because the data booklet supplies $\sigma$, focus on selecting and applying the correct equation rather than memorizing the constant.