Ch 39: Particles Behaving as Waves

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 39: Particles Behaving as Waves

Problem 41

Chapter 38, Problem 41

Two stars, both of which behave like ideal blackbodies, radiate the same total energy per second. The cooler one has a surface temperature $T$ and a diameter $3.0$ times that of the hotter star.
(a) What is the temperature of the hotter star in terms of $T$ ?
(b) What is the ratio of the peak-intensity wavelength of the hot star to the peak-intensity wavelength of the cool star?

Verified step by step guidance

Step 1: Use the Stefan-Boltzmann law to relate the total energy radiated per second (luminosity) of a blackbody to its surface area and temperature. The formula is: $ L = \sigma A T^4 $, where $ L $ is the luminosity, $ \sigma $ is the Stefan-Boltzmann constant, $ A $ is the surface area, and $ T $ is the temperature.

Step 2: Express the surface area $ A $ of a star in terms of its diameter $ d $. Since the star is spherical, $ A = 4 \pi r^2 $, and $ r = \frac{d}{2} $. Substituting, $ A = \pi d^2 $.

Step 3: Set the luminosities of the two stars equal, as they radiate the same total energy per second. For the cooler star, $ L = \sigma (\pi d_c^2) T_c^4 $, and for the hotter star, $ L = \sigma (\pi d_h^2) T_h^4 $. Equating these gives: $ \pi d_c^2 T_c^4 = \pi d_h^2 T_h^4 $.

Step 4: Substitute the given relationship between the diameters: $ d_c = 3.0 d_h $. This simplifies the equation to $ (3.0 d_h)^2 T_c^4 = d_h^2 T_h^4 $. Cancel $ d_h^2 $ on both sides and solve for $ T_h $ in terms of $ T_c $: $ T_h = T_c / 3^{1/2} $.

Step 5: Use Wien's displacement law to find the ratio of the peak-intensity wavelengths. The law states $ \lambda_{\text{peak}} T = b $, where $ \lambda_{\text{peak}} $ is the peak wavelength, $ T $ is the temperature, and $ b $ is a constant. For the two stars, $ \lambda_{\text{peak,hot}} / \lambda_{\text{peak,cool}} = T_c / T_h $. Substitute $ T_h = T_c / 3^{1/2} $ to find the ratio: $ \lambda_{\text{peak,hot}} / \lambda_{\text{peak,cool}} = 3^{1/2} $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law states that the total energy radiated per unit surface area of a black body is proportional to the fourth power of its absolute temperature (E ∝ T^4). This principle is crucial for understanding how the energy output of the two stars relates to their temperatures and sizes.

Wien's Displacement Law

Wien's Displacement Law describes the relationship between the temperature of a black body and the wavelength at which it emits radiation most intensely. Specifically, it states that the peak wavelength is inversely proportional to the temperature (λ_max ∝ 1/T). This law is essential for determining the ratio of peak-intensity wavelengths for the two stars.

Blackbody Radiation

Blackbody radiation refers to the electromagnetic radiation emitted by an idealized perfect black body in thermal equilibrium. It is characterized by a continuous spectrum that depends solely on the body's temperature, making it fundamental for analyzing the thermal properties and energy emissions of the stars in the question.

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