Radiation and the Stefan-Boltzmann Law
Simple Explanation
Every object radiates thermal energy as electromagnetic waves, at a rate given by the Stefan-Boltzmann law: P=εσAT⁴ — a relationship that depends extremely steeply on temperature, since T is raised to the fourth power.
Why Do We Need It?
Radiation is the only heat transfer mechanism that works through empty space — it is how the Sun's energy reaches Earth, and it dominates heat loss from very hot objects.
Formula
The Stefan-Boltzmann Law
P = εσAT⁴
Every object radiates thermal energy at a rate that depends steeply on its absolute temperature (to the fourth power), scaled by its surface area and how effectively it radiates (its emissivity).
- P
- — power radiated, in watts (W)
- ε
- — emissivity of the surface (0 to 1, dimensionless) — 1 for a perfect radiator ('black body')
- σ
- — the Stefan-Boltzmann constant, 5.67×10⁻⁸ W/(m²·K⁴)
- A
- — surface area of the radiating object, in square metres (m²)
- T
- — absolute temperature of the object, in kelvin (K)
When to use it: Whenever the power an object radiates as heat (via electromagnetic radiation) needs to be found from its temperature and surface area.
Worked Example
Find the power radiated by a hot object
An object with emissivity 0.9 and surface area 0.5 m² is at a temperature of 400 K. Find the power it radiates.
Why Does This Work?
This law follows from how thermal radiation's intensity and spectrum both shift with temperature — as an object gets hotter, it doesn't just radiate somewhat more; the T⁴ dependence means even a modest temperature increase produces a dramatically larger radiated power.
Real-Life Example
Why a light bulb filament glows and radiates so much heat
An incandescent bulb's filament reaches temperatures around 2500-3000 K, far hotter than everyday objects.
Because radiated power depends on T⁴, that very high temperature makes the filament radiate enormously more power (as both light and heat) than the same filament would at room temperature.
Practice
An object with emissivity 0.8 and surface area 0.2 m² is at 300 K. Find the power it radiates. (300⁴=8.1×10⁹)
HardCommon mistake
Using Celsius instead of Kelvin for T — the Stefan-Boltzmann law only works with absolute temperature; using Celsius gives a completely wrong (and sometimes negative or nonsensical) result.
Quick Review
- P = εσAT⁴ — temperature must be in kelvin.
- Radiated power depends extremely steeply on temperature (fourth power).
- The only heat transfer mechanism that works through a vacuum.