Wien's law (sunlight in, infrared out) tells you the color of an object's glow. This lesson answers the other question: how much energy does it give off? The answer climbs astonishingly fast with temperature.
Key idea
Each square meter of a surface at temperature $T$ (in kelvins) gives off $$E = \sigma T^4, \qquad \sigma = 5.67 \times 10^{-8}\ \text{W m}^{-2}\,\text{K}^{-4}$$ This is the Stefan–Boltzmann law for a perfect emitter (a "blackbody"). Double the kelvin temperature and the energy goes up $2^4 = 16$ times.
Why it works
Two effects multiply. A hotter surface sends out more light at every wavelength, and its peak moves to shorter wavelengths, where each bit of light carries more energy. Adding up the whole spectrum gives a fourth power. You can see it in a fire: embers that look only a little hotter than others glow far brighter. The ground, water and clouds are close to perfect emitters in the infrared, so $\sigma T^4$ describes them well.
Worked example: Earth's glow
How much infrared does a square meter of ground at 15 °C give off?
- Kelvins: $T = 15 + 273 = 288$ K.
- $T^4 = 288^4 \approx 6.88 \times 10^{9}$ K⁴.
- $E = 5.67 \times 10^{-8} \times 6.88 \times 10^{9} \approx 390$ W/m²: about as much as four bright 100-watt bulbs, from every square meter of ground, day and night.
Answer: About 390 W/m²
Worked example: Twice as hot
A hot plate at 600 K is compared with a cool plate at 300 K. How many times as much energy does each square meter of the hot plate give off?
- The ratio is $(T_2/T_1)^4 = (600/300)^4 = 2^4$.
- $2^4 = 16$. No need to compute either energy separately.
Answer: 16 times as much
Worked example: A warmer day
A sidewalk warms from 10 °C to 20 °C. By what factor does its infrared glow grow?
- Use kelvins: 283 K and 293 K. (Using Celsius would say "twice as hot", which is wrong.)
- Ratio: $(293/283)^4 \approx 1.035^4 \approx 1.15$.
- The glow grows about 15%, not 16 times.
Answer: About 1.15 times
Common mistake
Expecting doubling the temperature to double the glow. The energy goes as $T^4$, so doubling the kelvin temperature multiplies it by 16, and tripling it by 81. A second trap is to double the Celsius temperature: 10 °C to 20 °C is only a 3.5% rise in kelvins. Check: always put kelvins into $T^4$.
The fourth power is why Earth's temperature is so steady. If Earth warms a little, it gives off a lot more infrared, which pulls it back. A 1% rise in kelvin temperature raises the glow by about 4%. This is the strongest of all climate feedbacks, and it is a stabilizing one.