Ch 39: Particles Behaving as Waves

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 39: Particles Behaving as Waves

Problem 40

Chapter 39, Problem 40

The shortest visible wavelength is about $400$ nm. What is the temperature of an ideal radiator whose spectral emittance peaks at this wavelength?

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Identify the relationship between the peak wavelength and the temperature of an ideal radiator using Wien's Displacement Law: $ \lambda_{\text{max}} T = b $, where $ \lambda_{\text{max}} $ is the peak wavelength, $ T $ is the temperature, and $ b $ is Wien's constant ($ b = 2.897 \times 10^{-3} \ \text{m·K} $).

Convert the given wavelength $ \lambda_{\text{max}} = 400 \ \text{nm} $ into meters for consistency in SI units. Use the conversion $ 1 \ \text{nm} = 10^{-9} \ \text{m} $, so $ \lambda_{\text{max}} = 400 \times 10^{-9} \ \text{m} $.

Rearrange Wien's Displacement Law to solve for the temperature $ T $: $ T = \frac{b}{\lambda_{\text{max}}} $.

Substitute the values of $ b = 2.897 \times 10^{-3} \ \text{m·K} $ and $ \lambda_{\text{max}} = 400 \times 10^{-9} \ \text{m} $ into the equation $ T = \frac{b}{\lambda_{\text{max}}} $.

Simplify the expression to calculate the temperature $ T $ in Kelvin. Ensure the units cancel appropriately to confirm the result is in Kelvin.

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Key Concepts

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

Wien's Displacement Law

Wien's Displacement Law states that the wavelength at which the emission of a black body spectrum is maximized is inversely proportional to its absolute temperature. This means that as the temperature of an ideal radiator increases, the peak wavelength of emitted radiation shifts to shorter wavelengths. The law can be mathematically expressed as λ_max = b/T, where λ_max is the peak wavelength, T is the temperature in Kelvin, and b is Wien's displacement constant (approximately 2898 μm·K).

Black Body Radiation

Black body radiation refers to the theoretical spectrum of electromagnetic radiation emitted by an idealized perfect black body, which absorbs all incident radiation. The emitted radiation depends solely on the body's temperature and is described by Planck's law. This concept is crucial for understanding how real objects emit radiation and how their temperature can be inferred from the spectrum of light they emit.

Spectral Emittance

Spectral emittance is a measure of how much radiation a body emits at a specific wavelength compared to that of a perfect black body at the same temperature. It is a critical factor in determining the efficiency of thermal radiation from real materials. The peak spectral emittance at a given wavelength can be used to calculate the temperature of the radiator using Wien's Displacement Law, linking the physical properties of materials to their thermal behavior.

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The shortest visible wavelength is about 400400 nm. What is the temperature of an ideal radiator whose spectral emittance peaks at this wavelength?

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Key Concepts

Wien's Displacement Law

Black Body Radiation

Spectral Emittance

The shortest visible wavelength is about $400$ nm. What is the temperature of an ideal radiator whose spectral emittance peaks at this wavelength?