Planck’s Radiation Law and Solar Constant

Wein’s law and Rayleigh-jeans law from Planck’s law of radiation

According to plank’s Law we have,

$$E_{\lambda }d\lambda = \frac {8\pi hc}{\lambda ^5}\frac {1}{e^{\frac {hc}{\lambda KT} }-1} d\lambda $$

For shorter wave length \( e^{h}{\lambda KT} \) becomes large compared to unity and hence the Plank’s law reduces to

\begin{align*} E_{\lambda }d\lambda &= \frac {8\pi hc}{\lambda ^5}\frac {1}{e^{\frac {hc}{\lambda KT}}}d\lambda \\ \therefore E_{\lambda }d\lambda &= \frac {8\pi hce^{-\frac {h}{\lambda KT}}}{\lambda^5}d\lambda \\ \end{align*}

which is Wien’s law.

For longerwavelength,

\begin{align*} \therefore E_{\lambda }d\lambda &= \frac {8\pi hc}{\lambda ^5}\frac {1}{1 + \frac {hc}{\lambda KT}+\dots -1}d\lambda \\ E_{\lambda }d\lambda &= \frac {8\pi KT}{\lambda ^4}d\lambda \end{align*}
which is Rayleigh Jeans law. It is clear that Wien’s Law holds for shorter wavelength while Rayleigh Jeans law holds true for longer wavelength.

Deduction of Stefan’s constant from Planck’s Radiation Law

Total radiant energy in unit volume of an isothermal enclosure is

$$ E = \int _0^{\infty } E_{\lambda }\: d\lambda = \int \frac {8\pi hc}{\lambda ^5(e^{hc/ \lambda KT}- 1)}\: d\lambda \dots (1) $$

The total radiant energy in unit volume of an isothermal enclosure is also given by

\begin{align*} E &= AT^4 \dots (2) \\\end{align*}

where A is constant and T is the absolute temperature of enclosure

\begin{align*} \\ AT^4 &= 8\pi hc \int _0^{\infty }\frac {d\lambda }{\lambda ^5} \left (e^{\frac {hc}{\lambda KT}-1}\right )\dots (3) \\ \text {Putting,}\: x= \frac {hc}{\lambda KT} \\ \text {or,}\: \lambda &= \frac {hc}{xKT} \\ \therefore d\lambda &= \frac {-hdx}{x^2KT} \\ \end{align*}

Substituting these value in equation (3) we get \begin{align*}\ \\AT^4 &= 8\pi hc \int _0^{\infty } \frac {\frac {hc}{x^2KT}dx}{\left (\frac {xc}{xKT}\right )^5(e^x -1)} \\ &= \frac {8\pi K^4T^4}{h^3c^3}\int _0^{\infty } \frac {x^3dx}{e^x – 1} \\ \therefore AT^4 &= \frac {8\pi K^4T^4\pi ^4}{15h^3c^3}\:\left [\because \int _0^{\infty }\frac {x^3}{e^x – 1} = \frac {\pi ^4}{15}\right ] \\ \therefore A &= \frac {8\pi K^4\pi ^4}{15h^3c^3} \dots (4) \\ \end{align*}

\begin{align*} \text {Sine Stefan’s constant is given by,} \\ \sigma &= \frac {Ac}{4} \\ \therefore \sigma &= \frac {8\pi ^5 K^4}{15h^3c^3}\frac c4 = \frac {2\pi^5K^4}{15h^3c^2} \\ \end{align*}

Wien’s Constant (b)

According to Wien’s law, Wein’s constant
$$ b = \lambda _{m}T$$

Where \(\lambda _m\) is the wavelength for which the emitted energy is maximum. We have a relation

$$\lambda _m = \frac {hc}{4.965 KT}$$

$$ or, \: \lambda _mT = \frac {hc}{4.965} = 0.2896\: \text {cmK}$$

Solar Constant

Solar constant is defined as the amount of solar energy received per minute by unit area of a perfectly black body placed at a mean distance of the earth from the sun, in absence of the atmosphere and at right angle to the direction of sun’s rays. It is denoted by ‘S’.

Let E be the emissive power of sun, r be its radius. Then the radiant energy emitted by the sun per minute

$$4\pi r^2E\times 60 $$

S be the solar constant, R be the man distance between sun and earth, then the radiant energy absorbed by a perfectly black body having radius R as shown in the figure above is

\begin{align*}&= 4\pi R^2S \dots (2) \\ \text {In the absence of atmosphere,} \\ 4\pi R^2 S &= 4\pi r^2 E \times 60 \dots (3) \\ \text {According to Stefan’s law,}\\ E &= \sigma T^4 \dots (4) \\ \text {From equation}\: (3) and (4)\: \text {we get,} \\ 4\pi R^2 S &= 4\pi r^2 \sigma T^4 \times 60 \\ \text {or,}\: T^4 &= \left ( \frac Rr\right )^2 \frac {S}{60} \times \frac {1}{\sigma } \\ \therefore T &= \left [\left (\frac Rr\right )^2 \frac {S}{60}\times \frac {1}{\sigma } \right ]^{1/4} \dots (5) \\ \end{align*}

Equation (5) represents absolute temperature of sun in terms of Stefan’s constant and solar constant.

Bibliography

S.S. Singhal, J.P. Agarwal, Satya Prakash. heat and thermodynamics and statistical physics. pragati prakashan, 2010.

—. Heat and Thermodynamics and Statistical Physics. Pragati Prakashan, 2010.

Vatsyayan, Dr. Rakesh Ranjan. Refresher Course in Physics. kathmandu: Surya Book Traders, 2015.