Watch a lohar at his forge by the roadside. He pushes an iron rod into the coals and works the bellows. At first the rod looks the same, though it is already too hot to touch. Then it glows a dull red, then a bright orange, and at last it is yellow-white and almost painful to look at. The colour tells the smith when the iron is ready to hammer.
Radiation spread over wavelengths
The Stefan-Boltzmann law gives the total power of a black body. But that power is not shared equally among the wavelengths. Split the radiation with a prism and measure the power in each narrow band of wavelength. Plotting that power against the wavelength \(\lambda\) gives the black-body spectrum. This lesson uses Wien's constant \(b = 2.9 \times 10^{-3}\,\text{m K}\) and \(\sigma = 5.67 \times 10^{-8}\,\text{W/(m}^2\,\text{K}^4)\).
Before you move the slider below, predict. A black body is heated from 3000 K to 6000 K. What happens to the wavelength at which it radiates most strongly?
Three features show in every such curve. The curve starts near zero at very short wavelengths, rises to a single peak and then falls slowly in a long tail. A hotter body's curve lies above a cooler one at every wavelength, so it radiates more of every colour. And its peak lies at a shorter wavelength.
The area under the curve is the power radiated by each square metre at all wavelengths together. That is \(\sigma T^4\), so the area grows as the fourth power of the temperature. At 3000 K the curve in the figure hardly leaves the axis, and its area is only \(1/16\) of the area at 6000 K. If one body is at twice the absolute temperature of another, the areas under their curves are in the ratio \(16 : 1\).
The black-body spectrum is the graph of the power radiated per unit area per unit wavelength by a black body, against the wavelength. Its shape depends only on the temperature. The area under it is \(\sigma T^4\).
A graph shows the spectra of the same black body at two temperatures, \(T_1\) and \(T_2\). Curve 1 lies above curve 2 everywhere, and its peak is at a shorter wavelength. Which statements are true?
The peak slides with the temperature
In the figure the peak wavelength halves when the temperature doubles: 967 nm at 3000 K, 483 nm at 6000 K. The product of the peak wavelength and the absolute temperature stays the same. This is Wien's displacement law, found by Wilhelm Wien in 1893.
The wavelength \(\lambda_\text{max}\) at which a black body radiates most strongly is inversely proportional to its absolute temperature: \(\lambda_\text{max} T = b\). Wien's constant \(b\) is about \(2.9 \times 10^{-3}\,\text{m K}\).
The unit of \(b\) is metre kelvin, since it is a wavelength times a temperature. Wien's law turns a colour into a temperature. Find the peak of a body's spectrum, and \(T = b/\lambda_\text{max}\). A body whose spectrum peaks at \(1\,\mu\text{m}\) is at \(2.9 \times 10^{-3}/10^{-6} = 2900\,\text{K}\), about the temperature of a lamp filament.
Now the forge. At 800 K, iron radiates most strongly at \(2.9 \times 10^{-3}/800 = 3.6\,\mu\text{m}\), deep in the infrared. Only the short-wavelength edge of its spectrum reaches the visible band, at the red end, so the rod glows a dull red. At 1300 K the peak is at \(2.2\,\mu\text{m}\), and the curve spreads further into the visible, through orange and yellow. Hotter still, all the visible colours are strong together, and the iron looks nearly white.
The eye does not see the peak; it sees whatever part of the curve lies between 0.4 and 0.7 μm. For a dull red rod that part is a thin sliver at the red end of the tail. As the peak slides towards the visible band, the sliver thickens and takes in orange, yellow and then blue. The colour you see is the visible slice of a curve whose bulk you cannot see.
Find the peak wavelength of the radiation from (a) human skin at \(310\,\text{K}\) and (b) a lamp filament at \(2900\,\text{K}\). Name the region of each. Take \(b = 2.9 \times 10^{-3}\,\text{m K}\).
(a) \(\lambda_\text{max} = b/T\) \(= 2.9 \times 10^{-3}/310\) \(= 9.4 \times 10^{-6}\,\text{m}\) \(= 9.4\,\mu\text{m}\). This is far in the infrared, which is why a living body does not glow but a thermal camera sees it clearly.
(b) \(\lambda_\text{max} = 2.9 \times 10^{-3}/2900\) \(= 1.0 \times 10^{-6}\,\text{m}\) \(= 1.0\,\mu\text{m}\), just beyond the red, in the near infrared.
So even a lamp filament radiates most of its energy as infrared. An old filament bulb turns only a few per cent of its power into visible light and warms the room with the rest.
At what wavelength, in micrometres, does a black body at room temperature, \(300\,\text{K}\), radiate most strongly? Take \(b = 2.9 \times 10^{-3}\,\text{m K}\).
Two black bodies of equal area are at \(300\,\text{K}\) and \(600\,\text{K}\). What are the ratios of their radiated powers, and of their peak wavelengths, cooler to hotter?
The temperature of a star from its colour
No thermometer can reach a star. But a star radiates very nearly as a black body, so its spectrum tells its temperature. The Sun's spectrum peaks at about \(500\,\text{nm}\), in the green-blue. Wien's law gives \(T = 2.9 \times 10^{-3}/(5 \times 10^{-7})\) \(= 5800\,\text{K}\) for its surface. The Sun emits all colours strongly, so it looks white-yellow.
Look at Orion on a winter night. Betelgeuse, at his shoulder, is clearly orange-red. Its surface is at about 3500 K, and its peak, \(2.9 \times 10^{-3}/3500 = 830\,\text{nm}\), is in the near infrared, so red dominates what we see. Sirius, the brightest star in the sky, is at about 9900 K. Its peak is at \(290\,\text{nm}\), in the ultraviolet, so it looks blue-white.
A star's spectrum peaks at \(250\,\text{nm}\). Find its surface temperature, in kelvin. Take \(b = 2.9 \times 10^{-3}\,\text{m K}\).
Rank these by the wavelength at which they radiate most strongly, longest first.
Why does the spectrum have exactly this shape? Classical physics predicted that a hot body should radiate without limit at short wavelengths, which is plainly false. In 1900 Max Planck found the correct formula. He assumed that light energy is emitted in small packets, an idea that begins quantum physics and returns in Modern Physics.
In 1893 Wilhelm Wien, working at the national laboratory in Berlin, imagined the radiation in a cavity being slowly squeezed by a moving mirror. Using only thermodynamics and the Doppler effect, he showed that heating a cavity must shift its spectrum to shorter wavelengths in exact proportion. His displacement law was soon confirmed by careful measurements. It earned him the Nobel Prize in 1911.
Rescue teams after an earthquake, and border guards in Ladakh at night, use thermal cameras. A human body at 310 K radiates most strongly near \(9.4\,\mu\text{m}\), while cold rock and snow radiate far less. The camera's sensor is made for wavelengths from about 8 to 14 μm, right where warm bodies are brightest. A person shows up as a bright figure against a dark, cold background, with no light needed at all.
- Thinking a glowing body radiates most of its energy as visible light.Below about 4000 K the peak is in the infrared. Glowing iron and lamp filaments radiate mostly infrared.
- Using Celsius in \(\lambda_\text{max} T = b\).\(T\) is the absolute temperature, in kelvin.
- Reading the area under a spectrum as the peak height.The area is the total power per square metre, \(\sigma T^4\). The height is the power per unit wavelength at one wavelength.
- Mixing up nm and μm.\(1\,\text{nm} = 10^{-9}\,\text{m}\) and \(1\,\mu\text{m} = 10^{-6}\,\text{m}\), so \(500\,\text{nm} = 0.5\,\mu\text{m}\).