# Wein’s Formula & Wein’s Laws

Various physicists tried to explain the problem of energy distribution in black body radiation, and finally, German Physicist **Max Planck** successfully solved the problem.

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.

**Table of Contents**

## 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 λ_{m} 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.