Sunday, February 13, 2011

Mastering Physics: ± Blackbody Radiation and Continuous Spectra

A recurring theme throughout the study of quantum physics has been the omnipresence of Planck's constant h. This universal constant of the microscopic world first made its appearance in 1900 in the study of the radiation of so-called blackbodies.
A blackbody is a substance that absorbs radiation of all wavelengths and radiates in a continuous spectrum at all wavelengths. It is given the name blackbody because an object that absorbs light at all wavelengths appears black to the human eye.
By the end of the 19th century, several properties of blackbody radiation had been established. First, the total intensity I (the average rate of radiation of energy per unit surface area) emitted from a blackbody was shown to be proportional to the fourth power of its temperature:
I = \sigma T^4.
This is called the Stefan-Boltzmann law for a blackbody. The constant of proportionality sigma is known as the Stefan-Boltzmann constant and was determined to be \sigma = 5.67 \times 10^{-8}\; {\rm W/(m^2 \cdot K^4)}. It had also been discovered that the wavelength at which the radiation intensity was maximum varied inversely with temperature. This result, known as the Wien displacement law, is written
\lambda_{\rm m} T = 2.90 \times 10^{-3}\; \rm m \cdot K,
where lambda_m is the wavelength with the greatest radiated intensity.One aspect of blackbody radiation that remained unexplained was the full wavelength dependence of the intensity of the radiation, I(lambda). In 1900, largely through trial and error, Max Planck formulated the following equation that successfully explained the wavelength dependence of the intensity:
I(\lambda) = \frac{2 \pi h c^2}{\lambda^5(e^{hc/ \lambda k_{\rm B} T} - 1)},
where h is Planck's constant, c is the speed of light in vacuum, and k_B is Boltzmann's constant. Planck justified his law by claiming that different modes of electromagnetic oscillations within the cavity could only emit radiation in increments of energy equal to Planck's constant h multiplied by the frequency f. At first, Planck did not believe in that idea himself, but the revolutionary concept of quantization (or "clumping") of energy paved the way for the "quantum revolution" in physics.

Part A
Consider a blackbody that radiates with an intensity I_1 at a room temperature of 300\; \rm K. At what intensity I_2 will this blackbody radiate when it is at a temperature of 400\; \rm K?
Express your answer in terms of I_1.

  I_2  = 3.16I_{1}

Part B
At what wavelength lambda_m would the intensity of blackbody radiation be at a maximum when the blackbody is at 2900\; \rm K?
Express your answer in meters to two significant figures.

  lambda_m  = 1.00×10−6
  \rm m

Part C
An astronomer is trying to estimate the surface temperature of a star with a radius of 5.0 \times 10^8\; \rm m by modeling it as an ideal blackbody. The astronomer has measured the intensity of radiation due to the star at a distance of 2.5 \times 10^{13}\; \rm m and found it to be equal to 0.055\; \rm W/m^2. Given this information, what is the temperature of the surface of the star?
Express your answer in kelvins to two significant digits.

  T  = 7000
  \rm K