Problem 63
Question
If wavelength of maximum intensity of radiation emitted by sun and moon are \(0.5 \times 10^{-6} \mathrm{~m}\) and \(10^{-4} \mathrm{~m}\) respectively, the ratio of their temperatures is (a) \(1: 100\) (b) \(1: 200\) (c) \(200: 1\) (d) \(400: 1\)
Step-by-Step Solution
Verified Answer
(c) 200:1.
1Step 1: Concept Understanding
This problem involves Wien's Displacement Law, which states that the product of the wavelength of maximum intensity (\( \lambda_{max} \)) and the temperature (\(T\)) is constant for a black body:\[ \lambda_{max} \times T = b \]where \(b\) is Wien's constant, approximately \(2.898 \times 10^{-3} \mathrm{~m~K}\).
2Step 2: Application of Wien's Law
For the Sun, the equation can be written as:\[ \lambda_{sun} \times T_{sun} = b \]Plug in the given wavelength of the Sun:\[ 0.5 \times 10^{-6} \times T_{sun} = b \]
3Step 3: Equation for the Moon
Similarly, we can write for the Moon:\[ \lambda_{moon} \times T_{moon} = b \]Use the given wavelength for the Moon:\[ 10^{-4} \times T_{moon} = b \]
4Step 4: Setting Equations Equal
Set the two equations for \(b\) from the Sun and the Moon equal to each other to find the ratio of temperatures:\[ 0.5 \times 10^{-6} \times T_{sun} = 10^{-4} \times T_{moon} \]
5Step 5: Ratio Calculation
Rearrange to solve for the ratio:\[ \frac{0.5 \times 10^{-6}}{10^{-4}} = \frac{T_{moon}}{T_{sun}} \]Calculating the left side gives:\[ \frac{0.5 \times 10^{-6}}{10^{-4}} = 0.005 \]So the temperature ratio is:\[ \frac{T_{moon}}{T_{sun}} = 0.005 \]or, inversely,\[ \frac{T_{sun}}{T_{moon}} = 200 \]Therefore, the ratio of \(T_{sun}\) to \(T_{moon}\) is \(200:1\).
Key Concepts
Black Body RadiationTemperature RatioWavelength of Maximum Intensity
Black Body Radiation
Black body radiation refers to the type of electromagnetic radiation emitted by an idealized surface that absorbs and re-emits all incident radiation with perfect efficiency. Such a surface is known as a "black body." In reality, a perfect black body does not exist, but many objects approximate this behavior.
A key feature of black body radiation is that the emitted radiation has a characteristic spectrum dependent solely on the temperature of the body. At higher temperatures, more energy is radiated at shorter wavelengths, which means higher frequency radiation is dominant. For example, as a black body heats up, it might glow red to white as the temperature increases.
The relation between the temperature of a black body and its emission spectrum is an important part of physics. This connection is mathematically described by Wien's Displacement Law, which helps in identifying the temperature of stars, including the sun, by looking at the color of light they emit. By understanding black body radiation, we can also gain insight into the underlying physical properties of many astronomical and everyday objects.
A key feature of black body radiation is that the emitted radiation has a characteristic spectrum dependent solely on the temperature of the body. At higher temperatures, more energy is radiated at shorter wavelengths, which means higher frequency radiation is dominant. For example, as a black body heats up, it might glow red to white as the temperature increases.
The relation between the temperature of a black body and its emission spectrum is an important part of physics. This connection is mathematically described by Wien's Displacement Law, which helps in identifying the temperature of stars, including the sun, by looking at the color of light they emit. By understanding black body radiation, we can also gain insight into the underlying physical properties of many astronomical and everyday objects.
Temperature Ratio
The temperature ratio concept connects the temperatures of two objects by comparing their respective values mathematically. It is particularly relevant when using Wien's Displacement Law to compare objects that emit black body radiation, such as stars or planets like the sun and moon.
To determine the temperature ratio using Wien's Displacement Law, we utilize the formula: \[ \lambda_{max} \times T = b \]
Here, \( \lambda_{max} \) represents the wavelength of maximum intensity, \( T \) represents the temperature, and \( b \) is a constant (Wien's constant). For two different black bodies, setting the products \( \lambda_{max} \times T \) equal for both gives us a way to find their temperature ratio.
By rearranging the terms appropriately, one can solve for the ratio of temperatures. This approach was demonstrated in the exercise, comparing the sun and moon. It provided a calculated temperature ratio of 200:1, indicating the sun is significantly hotter than the moon.
To determine the temperature ratio using Wien's Displacement Law, we utilize the formula: \[ \lambda_{max} \times T = b \]
Here, \( \lambda_{max} \) represents the wavelength of maximum intensity, \( T \) represents the temperature, and \( b \) is a constant (Wien's constant). For two different black bodies, setting the products \( \lambda_{max} \times T \) equal for both gives us a way to find their temperature ratio.
By rearranging the terms appropriately, one can solve for the ratio of temperatures. This approach was demonstrated in the exercise, comparing the sun and moon. It provided a calculated temperature ratio of 200:1, indicating the sun is significantly hotter than the moon.
Wavelength of Maximum Intensity
The wavelength of maximum intensity is the particular wavelength at which a black body emits the most energy. In other words, at this wavelength, the radiation output from the black body is at its peak. The concept is critical in understanding black body radiation spectra.
According to Wien's Displacement Law, there is an inverse relationship between the wavelength of maximum intensity and the temperature of the black body. Specifically, as the temperature increases, the peak wavelength decreases, resulting in a shift toward the blue end of the spectrum.
This principle allows astronomers and physicists to estimate the temperature of stars and other celestial bodies by observing the color of the light they emit. The exercise example shows how the wavelengths of maximum intensity for the sun and moon differ significantly—highlighting their temperature differences.
According to Wien's Displacement Law, there is an inverse relationship between the wavelength of maximum intensity and the temperature of the black body. Specifically, as the temperature increases, the peak wavelength decreases, resulting in a shift toward the blue end of the spectrum.
This principle allows astronomers and physicists to estimate the temperature of stars and other celestial bodies by observing the color of the light they emit. The exercise example shows how the wavelengths of maximum intensity for the sun and moon differ significantly—highlighting their temperature differences.
- For the sun: \( \lambda_{max} = 0.5 \times 10^{-6} \) m
- For the moon: \( \lambda_{max} = 10^{-4} \) m
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