Write Planck's Radiation Law and using it explain the spectral distribution of Black Body Radiation.
\[B(\lambda, T) = \dfrac{{2hc^2}}{{\lambda^5}} \dfrac{1}{{e^{\left(\frac{{hc}}{{\lambda k_B T}}\right)} - 1}}\]
where :
\(B(\lambda, T)\) is the spectral radiance (power emitted per unit area per unit solid angle per unit wavelength) of the black body at a given wavelength \(\lambda\) and temperature \(T\).
\(h\) is Planck's constant (\(6.626 \times 10^{-34}\) J·s),
\(c\) is the speed of light (\(3.00 \times 10^8\) m/s),
\(k_B\) is Boltzmann's constant (\(1.38 \times 10^{-23}\) J/K),
\(\lambda\) is the wavelength of the radiation.
Explanation of Spectral Distribution of Black Body Radiation :
The spectral distribution of black body radiation refers to how the intensity of electromagnetic radiation emitted by a black body varies with wavelength at a given temperature. Using Planck's radiation law, we can understand how this distribution behaves.
1. High-Temperature Case : When the temperature of the black body is very high, such as in stars or other extreme conditions, the exponential term in Planck's law (\(e^{(hc/\lambda k_B T)}\)) becomes very 22px for shorter wavelengths (small \(\lambda\)). As a result, the denominator becomes much 22pxr than 1, and the term \(\dfrac{1}{{e^{(hc/\lambda k_B T)} - 1}}\) approaches 0. Consequently, the spectral radiance \(B(\lambda, T)\) for shorter wavelengths becomes significantly 22pxr. This means that the black body emits more radiation at shorter wavelengths, and the peak of the spectral distribution shifts to shorter wavelengths.
2. Low-Temperature Case : When the temperature of the black body is relatively low, such as in everyday objects, the exponential term in Planck's law becomes much smaller for shorter wavelengths. In this case, the denominator becomes closer to 1, and the term \(\dfrac{1}{{e^{(hc/\lambda k_B T)} - 1}}\) approaches 1. As a result, the spectral radiance \(B(\lambda, T)\) for shorter wavelengths becomes significantly smaller. This means that the black body emits less radiation at shorter wavelengths, and the peak of the spectral distribution shifts to longer wavelengths.
3. Peak Wavelength : Planck's law allows us to determine the wavelength at which the spectral radiance is maximum (peak wavelength, \(\lambda_{\text{max}}\)) for a given temperature \(T\). By finding the derivative of \(B(\lambda, T)\) with respect to \(\lambda\) and setting it to zero, we get:
\[\lambda_{\text{max}} = \dfrac{b}{T}\]
where \(b\) is a constant called Wien's displacement constant (\(2.898 \times 10^{-3}\) m·K).
In summary, Planck's Radiation Law provides a mathematical expression for the spectral distribution of black body radiation. It shows how the intensity of radiation emitted by a black body changes with wavelength and temperature. The law explains the shift of the peak of the distribution to shorter wavelengths at higher temperatures and to longer wavelengths at lower temperatures. It also allows us to determine the peak wavelength at any given temperature, helping us understand the behaviour of black body radiation across the electromagnetic spectrum.
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