Question

Wavelength corresponding to ${{E}_{max}}$ for the moon is 14 microns. Estimate the surface temperature of the moon, if $b=2.884 \times 10^{-3} \mathrm{MK}$.

Answer 1

We should know that the wavelength is the distance between two wave crests, which is the same as the distance between two troughs. The number of waves that pass-through a given point in one second is called the frequency, measured in units of cycles per second called Hertz. As the full spectrum of visible light travels through a prism, the wavelengths separate into the colours of the rainbow because each colour is a different wavelength. Violet has the shortest wavelength, at around 380 nanometres, and red has the longest wavelength, at around 700 nanometres. Gamma rays have the highest energies, the shortest wavelengths, and the highest frequencies. Radio waves, on the other hand, have the lowest energies, longest wavelengths, and lowest frequencies of any type of EM radiation.

Complete step by step answer
We should know that Wien's law, also called Wien's displacement law, is the relationship between the temperature of a blackbody (an ideal substance that emits and absorbs all frequencies of light) and the wavelength at which it emits the most amount of light. This relationship is called Wien's displacement law and is useful for determining the temperatures of hot radiant objects such as stars, and indeed for a determination of the temperature of any radiant object whose temperature is far above that of its surroundings. Wien's Law tells us that objects of different temperatures emit spectra that peak at different wavelengths. Hotter objects emit most of their radiation at shorter wavelengths; hence they will appear to be bluer.
It is given that:
Given, $\lambda_{m}=14$ micron $=14 \mu m=14 \times 10^{-6} \mathrm{m}$
and Wien constant, $b=2892 \times 10^{-6} \mathrm{mK}$
So now according to Wien's displacement law,
$\lambda_{m}=\dfrac{b}{T}$ or $T=\dfrac{b}{\lambda_{m}}=\dfrac{2892 \times 10^{-6}}{14 \times 10^{-6} \mathrm{m}}$
$\therefore \mathrm{T}=5^{\circ} \mathrm{C}$

Hence the surface temperature of moon is ${{5}^{{}^\circ }}C$.

Note We should know that black-body radiation has a characteristic, continuous frequency spectrum that depends only on the body's temperature, called the Planck spectrum or Planck's law. As the temperature increases past about 500 degrees Celsius, black bodies start to emit significant amounts of visible light. It occurs due to a process called thermal radiation. Thermal energy causes vibration of molecules or atoms, which in turn vibrates the charge distribution in the material, allowing radiation by the above mechanisms. That radiation, for a perfect absorber, follows the blackbody curve. The primary law governing blackbody radiation is the Planck Radiation Law, which governs the intensity of radiation emitted by unit surface area into a fixed direction from the blackbody as a function of wavelength for a fixed temperature. The mathematical function describing the shape is called the Planck function.
It should be known that blackbody radiation is a cornerstone in the study of quantum mechanics. This experiment is what led to the discovery of a field that would revolutionize physics and chemistry. Quantum mechanics gives a more complete understanding of the fundamental mechanisms at the sub-atomic level.