Wein’s Formula & Wein’s Laws

The problem of energy distribution in black body radiation stumped physicists for years. German physicist Max Planck finally cracked it. But he didn’t work in a vacuum.

Before Planck, Wilhelm Wien and the duo of Lord Rayleigh and James Jeans made critical contributions. Their work (and their failures) paved the way for Planck’s quantum revolution.

Wien’s Formula and Wien’s Laws

Wilhelm Wien tackled the black body radiation problem in 1893. Using classical thermodynamics, he derived a general formula for energy distribution and established two laws that still hold up experimentally today.

His radiation formula had a fatal flaw. But his displacement law and fifth power law? Those are still taught in physics courses worldwide.

Wien’s Radiation Formula

Wien showed that the amount of radiation \( E_\lambda d\lambda \) emitted by unit area of a black body per second at temperature \( T \) (in Kelvin), within the wavelength range \( \lambda \) to \( \lambda + d\lambda \), follows this formula:

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

Here, \( A \) is a constant and \( f(\lambda T) \) is some function of the product of wavelength and temperature. This was Wien’s general result from thermodynamics alone.

He went further. By making assumptions about the emission process, Wien proposed a specific form for that unknown function:

$$E_\lambda d\lambda = \frac{A}{\lambda^5} e^{-a/\lambda T} \, d\lambda$$

where \( a \) is another constant. This is Wien’s radiation formula (sometimes called Wien’s distribution law).

The Problem

Wien’s formula works beautifully at short wavelengths. The experimental data and the formula match almost perfectly in the ultraviolet region.

But at longer wavelengths (infrared region), it fails badly. The formula predicts less radiation than experiments actually measure. This wasn’t a small discrepancy. It was a fundamental breakdown.

Rayleigh and Jeans tried to fix this with their own formula. They succeeded at long wavelengths but predicted infinite energy at short wavelengths (the infamous “ultraviolet catastrophe”). Classical physics was stuck.

Wien’s Displacement Law

This one holds up. Wien proved that if \( \lambda_m \) is the wavelength where a black body emits maximum radiation at temperature \( T \), then:

$$\lambda_m T = b$$

where \( b \) is Wien’s displacement constant, equal to \( 2.898 \times 10^{-3} \) m·K.

The physical meaning is intuitive. As temperature increases, the peak wavelength shifts toward shorter wavelengths. Heat a metal and it glows red, then orange, then white, then blue. That’s Wien’s displacement law in action.

This is why we can estimate the surface temperature of stars by their color. Blue stars are hotter than red stars. The math works out precisely.

Wien’s Fifth Power Law

Wien also derived a relationship between the maximum spectral emissive power and temperature. If \( E_{\lambda_m} \) is the spectral emissive power at the peak wavelength \( \lambda_m \), then:

$$E_{\lambda_m} = k T^5$$

or equivalently:

$$\frac{E_{\lambda_m}}{T^5} = k$$

where \( k \) is a constant.

In plain terms: the maximum emissive power varies as the fifth power of absolute temperature. Double the temperature, and the peak emission intensity increases by a factor of 32.

This law is experimentally verified. Combined with the displacement law, it gives us powerful tools for analyzing thermal radiation.

What Experiments Actually Show

Here’s the scorecard for Wien’s contributions:

Wien’s Radiation Formula: Fails at long wavelengths. Historically important but not correct across the full spectrum.

Wien’s Displacement Law: Experimentally verified. Still used today in astrophysics, thermal imaging, and materials science.

Wien’s Fifth Power Law: Experimentally verified. A direct consequence of the displacement law and Planck’s law.

The failure of Wien’s radiation formula (and the Rayleigh-Jeans formula) forced physicists to accept that classical physics couldn’t explain black body radiation. Planck’s quantum hypothesis was the only way out.

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