Chapter 39: Problem 58
Light from an ideal spherical blackbody 15.0 cm in diameter is analyzed by using a diffraction grating that has 3850 lines/cm. When you shine this light through the grating, you observe that the peak-intensity wavelength forms a first-order bright fringe at \(\pm\)14.4\(^\circ\) from the central bright fringe. (a) What is the temperature of the blackbody? (b) How long will it take this sphere to radiate 12.0 MJ of energy at constant temperature?
Short Answer
Step by step solution
Identify the Diffraction Grating Equation
Calculate the Wavelength
Use Wien's Law to Find the Temperature
Calculate Power Using Stefan-Boltzmann Law
Find Time to Radiate 12.0 MJ
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Diffraction Grating
When light hits a diffraction grating, it bends at specific angles depending on the wavelength. This bending causes the light to produce what is known as diffraction lines or diffraction maxima. The equation governing this is:
- \[ d \sin \theta = m \lambda \]
- \( d \) represents the distance between the grating lines (grating spacing).
- \( \theta \) is the angle at which the maximum occurs.
- \( m \) is the order of the maximum (e.g., first, second order), typically an integer.
- \( \lambda \) is the wavelength of the light.
Wien's Displacement Law
The formula for Wien's displacement law is:
- \[ \lambda_{max} T = b \]
- \( \lambda_{max} \) is the peak wavelength in meters.
- \( T \) is the absolute temperature in Kelvin.
- \( b \) is Wien’s displacement constant \( 2.898 \times 10^{-3} \text{ m} \cdot \text{K} \).
Stefan-Boltzmann Law
Mathematically, it is represented as:
- \[ P = \sigma A T^4 \]
- \( P \) is the total power emitted by the blackbody.
- \( \sigma \) is the Stefan-Boltzmann constant \( 5.67 \times 10^{-8} \text{ W/m}^2 \cdot \text{K}^4 \).
- \( A \) is the surface area of the blackbody.
- \( T \) is the temperature in Kelvin.