# Wien's displacement law's proof

• jonathanpun
In summary, when proofing Wien's Law in an assignment, it is important to differentiate the frequency dependent energy density function and set it equal to zero in order to solve for the value of a. However, this value may not correspond to the correct value for the Wien's constant b, as found through a different method using a wavelength dependent Planck's energy density equation. This is due to the different functional forms of intensity vs wavelength and intensity vs frequency. Ultimately, the value obtained for b will depend on the method used and there is no deeper explanation for the discrepancy between the two values.
jonathanpun
When I was doing my assignment, I need to proof the Wien's Law.
The question given frequency dependent energy density function. So differentiate it respect to frequency v. Equate it with zero, and solve. i solved the value a=2.82144 = hv/KT=hc/(lambda)KT, i cannot get a correct value for the Wien's constant b = 2.898*10^-3, i only get 5.102*10^-3
But if i convert the frequency dependent Planck's energy density equation into wavelength dependent, and differentiate, solve. I get a*exp(a)-5*exp(a)+5=0 => a=4.96511=hc/(lambda)KT. Then i get a correct value for b.

So my question is why the Max. wavelength seems not corresponding to Max. frequency?

I have read the wikipedia about the law. but i don;t understand why it take ""the value 4 in this equation (midway between 3 and 5) yields a "compromise" wavelength-frequency-neutral peak, which is given for x = 3.92069039487...""

And it said "Because the spectrum from Planck's law of black body radiation takes a different shape in the frequency domain from that of the wavelength domain, the frequency location of the peak emission does not correspond to the peak wavelength using the simple relationship between frequency, wavelength, and the speed of light."

So what is the connection of the two value b obtained by two method?

Well, you just did the math. The frequency peak is different from the wavelength peak because the functional form of intensity vs wavelength is different from the functional form of intensity vs frequency, so when you find the maxima they are in slightly different places. I don't think there is any deeper explanation.

Since Wien's displacement law is stated in terms of wavelength, this is the correct function to use. I found this surprising too when I first encountered it, but the mathematics speaks for itself.

## 1. What is Wien's displacement law?

Wien's displacement law is a principle in physics that describes the relationship between the wavelength of a blackbody's peak emission and its temperature. It states that as the temperature of a blackbody increases, the wavelength at which it emits the maximum amount of radiation decreases.

## 2. How was Wien's displacement law discovered?

Wien's displacement law was discovered by the German physicist Wilhelm Wien in 1893. He studied the spectrum of a blackbody radiator and found that the wavelength of maximum radiation shifted towards shorter wavelengths as the temperature increased.

## 3. What is the mathematical expression for Wien's displacement law?

The mathematical expression for Wien's displacement law is λ_max = b/T, where λ_max is the wavelength of maximum radiation, b is the Wien displacement constant (approximately equal to 2.898 x 10^-3 m K), and T is the absolute temperature of the blackbody in Kelvin.

## 4. How is Wien's displacement law related to Planck's law?

Wien's displacement law is a special case of Planck's law, which describes the spectral energy density of a blackbody at a given temperature. Planck's law also takes into account the full spectrum of radiation emitted by a blackbody, while Wien's displacement law only describes the peak emission wavelength.

## 5. Can Wien's displacement law be applied to objects other than blackbodies?

Yes, Wien's displacement law can be applied to any object that emits thermal radiation, including stars, planets, and even humans. However, the object must be in thermal equilibrium and have a well-defined temperature in order for the law to accurately predict the peak emission wavelength.

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