More Precisely 3.2 - More About the Radiation Laws

More About the Radiation Laws

As mentioned in Section 3.4, Wien’s law relates the temperature T of an object to the wavelength λ_max at which it emits the most radiation. (The Greek letter λ—lambda—is conventionally used to denote wavelength.) Mathematically, if we measure T in kelvins and λ_max in centimeters, we find that

We could also convert Wien’s law into an equivalent statement about frequencyƒ, using the relationƒ = c/λ, but the law is most commonly stated in terms of wavelength and is probably easier to remember that way.

EXAMPLE: For a blackbody with the same temperature as the surface of the Sun, approximately 6000 K, the wavelength of maximum intensity is (0.29/6000) cm, or 480 nm, corresponding to the yellow-green part of the visible spectrum. A cooler star with a temperature of 3000 K has a peak wavelength of (0.29/3000) cm = 970 nm, just longward of the red end of the visible spectrum, in the near infrared. The blackbody curve of a star with a temperature of 12,000 K peaks at 242 nm, in the near ultraviolet, and so on.

Stellar temperatures are typically measured in thousands of kelvins, so it is helpful to rewrite the above equation in more "stellar" units:

We can also give Stefan’s law a more precise mathematical formulation. With T measured in kelvins, the total amount of energy emitted per square meter of its surface per second (a quantity known as the energy flux, F) is given by

The constant σ (the Greek letter sigma) is known as the Stefan-Boltzmann constant, or often just Stefan’s constant, after Josef Stefan, the Austrian scientist who formulated the equation.

The SI unit of energy is the joule (J). Probably more familiar is the closely related unit called thewatt (W), which measures power—the rate at which energy is emitted or expended by an object. One watt is the emission of one joule per second. For example, a 100-W lightbulb emits energy (mostly in the form of infrared and visible light) at a rate of 100 J/s. In these units, the Stefan-Boltzmann constant has the value σ = 5.67 x 10^-8 W/m² • K⁴.

EXAMPLE: Notice just how rapidly the energy flux increases with increasing temperature. A piece of metal in a furnace, when at a temperature of 3000 K, radiates energy at a rate of about 5.67 x10^-8 W/m² • K⁴ x (0.01m)² x (3000 K) ⁴ = 460 W for every square centimeter of its surface area. Doubling its temperature to 6000 K, the surface temperature of the Sun (so that it becomes yellow-hot, by Wien’s law) increases the energy emitted by a factor of 16 (four "doublings"), to 7.3kilowatts (7,300 W) per square centimeter.

Notice also that the law relates to energy emitted per unit area. The flame of a blowtorch is considerably hotter than a bonfire, but the bonfire emits far more energy in total, because it is much larger.

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