Planck’s Law of Radiation and Wein's Displacement Law

Planck’s Law of Radiation

The black body chamber is filled up not only with the radiation but also with simple harmonic oscillators of the molecular dimensions, energies of these dimensions are

$$ E = nh\nu $$

The oscillator cannot radiate or absolute energy continuously but an oscillator of frequency \(\nu\) can only radiate or absorbs energy in unit of quanta of magnitude \(h\nu \).

The average energy of Planck’s oscillators in terms of frequency with their corresponding temperature is
$$ \vec {E} = \frac {h\nu }{e^{h\nu/kT}-1} $$

The number of oscillators per unit volume in the frequency range \(\nu + \nu + d\nu \) is given by

$$N(\nu ) = \frac {8\pi v^2}{c^3}d\nu $$

Energy density is the range \(\nu\) to \(\nu + d\nu \) can be obtained by multiplying the average energy of Plank’s oscillators by the number of oscillators per unit volume in the same frequency range.

\begin{align*} E_{\nu }d_{\nu} &= \frac {8\pi\delta ^2}{c^3}.\frac {h\nu }{e^{h\nu / kT} -1}d\nu \\ &= \frac {8\pi h\delta ^3}{c^3}.\frac {1}{e^{h\nu / kT} - 1}d\nu \\ \text {In terms of wave length,} \\ \text {We know} \\ \nu &= \frac {c}{\lambda } \\ |d\nu | &= \left |-\frac {c}{\lambda }^2\right |d\lambda \\ E_{\lambda} d_{\lambda } &= \frac {8\pi h}{c^3} \left (\frac {c}{\lambda } \right )^3.\frac {1}{e^{h/\lambda kT}-1}d\lambda \\ \end{align*}

Wien’s Displacement Law

Wein’s displacement law states that the product of the wavelength corresponding to maximum energy\(\lambda _m\) and absolute temperature T is constant
$$\text {i.e.},\: \lambda _m T = \text {constant} $$

This constant is Wein’s displacement constant whose value as \(0.2896 \times 10^{-2}\: mK \). This law shows that with the increase of temperature, \(\lambda _m\) decreases.

Wein has also shown that the energy \(E_{\text {max}}\) is directly proportional to the fifth power of the absolute temperature.

\begin{align*} E_m &\propto T^5 \\ \text {or,}\: E_m &= \text {constant} \times T^5 \\ E_{\lambda } &= \lambda ^{-5}Af(\lambda T)\:\:\: \text {where A is constant}\\ \end{align*}

Deduction of Rayleigh – Jean law of spectral distribution of energy

The number of modes of vibration per unit volume in the frequency range \(\nu \:\text {and} \: \nu + d\nu \) is

$$N_{\nu }d\nu = \frac {8\pi \nu^2}{c^3}d\nu \dots (1) $$

Reyleigh and Jeans assumed that the law of equipartition of energy is applicable to radiation also, i.e. they considered the average energy of an oscillator as

$$\vec E = kT \dots (2) $$

The energy density i.e, energy per unit volume within the frequency \(\nu \) and \(\nu + d\nu \) is given by

\(U_{\nu }d\nu =\) the number of modes of vibration per unit volume in the frequency range \(\nu \:\text {and} \: \nu + d\nu \times \text {Average energy per mode of vibration.}\)

$$U_{\nu }d\nu = \frac {8\pi \nu ^2}{c^3}d\nu KT \dots (3) $$

This Reyleigh- Jean’s law in terms of frequency. Now in terms of wavelength

\begin{align*} U_{\lambda }d\lambda &= \frac {8\pi}{^3}\left (\frac {c}{\lambda ^2}\right )^2 \left (-\frac {c}{\lambda ^2}d\lambda \right ) KT \\ \text {Since the magnitude of}\\ i.e.\: d\nu &= \frac {c}{\lambda ^2}d\lambda \:\:\: [\because v =\frac {c}{\lambda }, \: dv = \frac {-c^2}{\lambda }d\lambda ] \\ \therefore U_{\lambda }d\lambda &= \frac {8\pi c^2}{c^3}\frac {c^2}{\lambda ^2}\frac{c}{\lambda ^2} d\lambda KT \\ U_{\lambda}d{\lambda } &= \frac {8\pi KT}{\lambda ^4}d\lambda \\ \end{align*}

This is Reyleigh - Jeans law in terms of wavelength.

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.