Stefan–Boltzmann Law
Total energy radiated by a black body scales with the fourth power of temperature.
Formula
Variables
Example
Double a star’s temperature and it radiates 2⁴ = 16 times more power per area.
How it works
The Stefan–Boltzmann law says that an ideal black body emits thermal radiation at a rate per unit area proportional to the fourth power of its absolute temperature, so hot objects radiate vastly more energy than cooler ones. The T^4 dependence comes from the physics of electromagnetic modes in thermal equilibrium: higher temperature means many more photons across a wider range of frequencies, so the total emitted power rises very rapidly. For real surfaces, the law is usually adjusted by an emissivity factor, but the basic scaling still captures how strongly temperature controls thermal radiation.
When to use it
Use this law when you need the radiated power from a hot object, especially in astrophysics, heat transfer, infrared emission, or furnace and thermal design problems. It is also useful when comparing how emission changes with temperature or estimating a body's surface temperature from its radiated flux.
Worked example, step by step
A furnace wall behaves approximately like a black body at 900 K. Find the emitted power per unit area. Step 1: Write the law: j = σT^4 Step 2: Substitute values: j = (5.67 × 10^-8 W m^-2 K^-4)(900 K)^4 Step 3: Compute the temperature term: 900^4 = 6.561 × 10^11 Step 4: Multiply: j = (5.67 × 10^-8)(6.561 × 10^11) Step 5: Final value: j ≈ 3.72 × 10^4 W/m^2 So the furnace emits about 37.2 kW of radiant power per square metre.
Common mistakes to avoid
- Using Celsius instead of kelvin for T
- Forgetting that the formula gives power per unit area, not total power
- Applying the black-body form to a real surface without accounting for emissivity
Did You Know?
A body's radiated power rises with the fourth power of temperature — double the temperature, 16× the energy.
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