Wein’s Formula & Wein’s Laws

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Various workers tried to explain the problem of energy distribution in black body radiation and finally the problem was successfully solved by German Physicist Max Planck. Before him, German Physicist Wilhelm Wein and British Physicist Lord Rayleigh & James Jean have tackled this problem and have given important laws. In fact, the work of their scientists paved the way for Planck to give his famous theory of radiation.

In this series of articles, I shall be discussing the various laws, special concentration on Planck’s law, concerning the black body in the brief.
This article is very first of the series & in I shall discuss briefly about Wein’s Laws. Other two useful topics will be discussed later.

Wein’s Formula & Wein’s Laws

The problem of black body radiation was first theoretically tackled by Wein in 1893. Besides giving a general formula for the energy distribution in the blackbody radiation, he gave following important and useful laws.

Wein’s Radiation Formula

Using well-known principles of classical thermodynamics, Wein showed that the amount of radiation $E_\lambda d \lambda$ emitted by unit area of a black-body per second at a temperature of T Kelvin in the wavelength range λ & λ+dλ is given by the formula,

$$E_\lambda d \lambda = \frac{A}{\lambda^5} f (\lambda, T) d \lambda$$

Where $A$ is a constant & $f(\lambda, T)$ is the function of the product of λ and T. This is the Wein’s formula.

Wein also obtained an expression for unknown function f(λ, T) and finally gave the relation

$$E_\lambda d \lambda = \frac{A}{\lambda^5} e^{-a/\lambda T} d \lambda$$ where a is another constant. This is the famous Wein’s radiation formula.

Drawback

Wein’s formula holds fairly good for the distribution of energy in the lower wavelength range, but fails on higher wavelength side.

Wein’s Displacement Law

Wein also showed that if λm is the wavelength at which the amount of radiation or emissive power of the black body is maximum at a temperature T, then $ \lambda_m T$= constant=K=$2.90 \times 10^{-3} m-K$.

This equation shows that as T increases, $\lambda_m$ shifts towards shorter wavelength side. Due to this reason, it is referred to as the Wein’s Displacement Law.

Wein’s fifth power formula

Wein also showed that if $E_{\lambda_m}$ is the value of spectral emissive power of a black body at temperature $T$ kelvin corresponding to wavelength $\lambda_m$, then $$E_{\lambda_m} \times T^{-5} =Constant=k$$

This is known as the Wein’s fifth power law and can be stated as:
$E_{{\lambda}_m}$ varies inversely as the fifth power of absolute temperature.
Experimentally, Wein’s Radiation Formula is not true but the remaining two are.

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