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Question: The light from the sun is found to have a maximum intensity near the wavelength of \(470\;{\text{nm}...

The light from the sun is found to have a maximum intensity near the wavelength of 470  nm470\;{\text{nm}}. Assuming the surface of the sun as a black body, the temperature of the sun is …………………..
[ Wien’s constant b=2.898×10−3  mKb = 2.898 \times {10^{ - 3}}\;{\text{mK}}]
A. 5800  K5800\;{\text{K}}
B. 6050  K6050\;{\text{K}}
C. 6166  K6166\;{\text{K}}
D. 6500  K6500\;{\text{K}}

Explanation

Solution

Hint The above problem is based on Wien's displacement law. This law gives the relationship between the maximum wavelengths of the light that can be emitted from the source. The product of the wavelength of the light and temperature of the light is constant and this constant is known as the Wien’s constant.

Complete step by step answer
Given: The wavelength of the light for maximum intensity is λmax⁡=470  nm=470  nm×10−9  m1  nm=4.70×10−7  m{\lambda _{\max }} = 470\;{\text{nm}} = 470\;{\text{nm}} \times \dfrac{{{{10}^{ - 9}}\;{\text{m}}}}{{1\;{\text{nm}}}} = 4.70 \times {10^{ - 7}}\;{\text{m}}.
The Wien’s constant is b=2.898×10−3  mKb = 2.898 \times {10^{ - 3}}\;{\text{mK}}.
Apply the Wien’s displacement law to calculate the formula for temperature of the sun.
λmax⁡T=b{\lambda _{\max }}T = b
Substitute 2.898×10−3  mK2.898 \times {10^{ - 3}}\;{\text{mK}} for b and 4.70×10−7  m4.70 \times {10^{ - 7}}\;{\text{m}} for λmax⁡{\lambda _{\max }} in the above expression to find the temperature of the sun.
(4.70×10−7  m)T=2.898×10−3  mK\left( {4.70 \times {{10}^{ - 7}}\;{\text{m}}} \right)T = 2.898 \times {10^{ - 3}}\;{\text{mK}}
T=6165.95  KT = 6165.95\;{\text{K}}
T≈6166  KT \approx 6166\;{\text{K}}

Thus, the temperature of the sun is 6166  K6166\;{\text{K}} and the option (C) is the correct answer.

Additional information The thermal radiation emitted by a body becomes visible at high temperatures. The maximum temperature of the radiation becomes observable for shorter wavelengths. The temperature of the body is inversely proportional to the wavelength of the emitted thermal radiation. The area under the curve between the wavelength and temperature of the body gives the total emissive power of the body.

Note Conversion of temperature in Kelvin from degree Celsius and unit of wavelength in meters is necessary before applying the Wien’s displacement law. The body emits the higher thermal radiation for a smaller wavelength of the light.