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Spectral Energy Distribution Curve of Black Body Radiations

Spectral Energy Distribution Curve of Black Body Radiations :- A black body is an ideal body which absorbs all incident radiation and emits maximum possible radiation at every temperature. The radiation it emits depends only on its temperature, not on the material. The plot of spectral emissive power (energy radiated per unit time, per unit area, per unit wavelength or frequency interval) versus wavelength (or frequency) at a fixed temperature is called the Spectral Energy Distribution Curve.

Experimental Observations (Lummer & Pringsheim, 1899–1900)

Heinrich Lummer and Ernst Pringsheim carried out careful experiments with a nearly perfect black body (hollow cavity with small hole). They measured the intensity distribution of radiation and observed that :

(1). For a given temperature, intensity rises from zero at λ = 0, reaches a maximum at some wavelength λ_max, and then falls again for large λ.

(2). As temperature increases,

The peak of the curve (λ_max) shifts towards shorter wavelength (towards blue).
The total area under the curve (i.e. total emitted energy) increases rapidly.

(3). These observations led to Wien’s displacement law :

$\displaystyle {{\lambda }_{max}}T=constant$

(4). The experimental curves looked like smooth bell-shaped distributions.

(5). At a particular temperature the area enclosed between the spectral energy curve shows the spectral emissive power of the body.

$\displaystyle Area=\int\limits_{0}^{\infty }{{{E}_{\lambda }}}d\lambda =E=\sigma {{T}^{4}}$

(6). The spectral emissive power at the wavelength of maximum emission grows as the fifth power of temperature.

$\displaystyle {{E}_{{{\lambda }_{m}}}}\propto {{T}^{5}}$

[This can be derived from the Planck’s Law :

$\displaystyle {{E}_{\lambda ,b}}(\lambda ,T)=\frac{2\pi h{{c}^{2}}}{{{\lambda }^{5}}}\frac{1}{{{e}^{{}^{hc}\!\!\diagup\!\!{}_{\lambda {{k}_{B}}T}\;}}-1}$

At the peak wavelength

$\displaystyle {{E}_{{{\lambda }_{m}}}}=\frac{2\pi h{{c}^{2}}}{\lambda _{m}^{5}}\frac{1}{{{e}^{{}^{hc}\!\!\diagup\!\!{}_{{{\lambda }_{m}}{{k}_{B}}T}\;}}-1}$

Since $\displaystyle {{\lambda }_{m}}T=b\Rightarrow {{\lambda }_{m}}=\frac{b}{T}$ ,

$\displaystyle \therefore {{E}_{{{\lambda }_{m}}}}=\frac{2\pi h{{c}^{2}}}{{{\left( \frac{b}{T} \right)}^{5}}}\frac{1}{{{e}^{{}^{hc}\!\!\diagup\!\!{}_{\left( b/T \right){{k}_{B}}T}\;}}-1}$

$\displaystyle \Rightarrow {{E}_{{{\lambda }_{m}}}}=\frac{2\pi h{{c}^{2}}{{T}^{5}}}{{{b}^{5}}}\frac{1}{{{e}^{{}^{hc}\!\!\diagup\!\!{}_{b{{k}_{B}}}\;}}-1}$

$\displaystyle \Rightarrow {{E}_{{{\lambda }_{m}}}}=\left( \frac{2\pi h{{c}^{2}}}{{{b}^{5}}}\frac{1}{{{e}^{{}^{hc}\!\!\diagup\!\!{}_{b{{k}_{B}}}\;}}-1} \right){{T}^{5}}$

$\displaystyle \Rightarrow {{E}_{{{\lambda }_{m}}}}\propto {{T}^{5}}$ ]

Spectral Energy Distribution Curve of Black Body Radiations

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