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January 6

Planck's Law & Thiesen

In its 1901 article, Planck's writes about Wien displacement and Stephan-Boltzmann laws that Thiesen deduced the formula:

E∙dλ = ϑ⁵ ψ(λϑ)∙dλ

I couldn't find Thiesen's article for this formula. Can someone explain to me how we arrive at this equation? Malypaet (talk) 13:31, 6 January 2024 (UTC)[reply]

Here's Thiesen's paper. He does not derive it explicitely though; in fact, he seems to attribute the derivation to Wien himself, so this equation should be equivalent to the original strong version of Wien's displacement law. It's easy enough to see that the equation satisfies the Stefan-Boltzmann law (substitute

x=\lambda T

, that cancels one power of

T

, giving

\int E\,d\lambda \propto T^{4}

. The familiar form of the displacement law follows trivially from the dependence of

\psi

on the product

\lambda T

. --Wrongfilter (talk) 14:10, 6 January 2024 (UTC)[reply]

Thanks, that's exactly what I was looking for. On the other hand, as the Stephan-Boltzman law is expressed in power, here we can find the power in the form:

B_λ(λ,T)∙dλ=(ϑ⁵ ψ(λϑ))/Δt∙dλ

With Δt for the SI unit of time (/Δt is in c/4π). Is it the correct writing ?

Malypaet (talk) 18:49, 6 January 2024 (UTC)[reply]

In fact the "strong version" is the "Wien’s fifth power law:

"According to this law, the maximum energy of emitted radiation Em is directly proportional to the fifth power of absolute temperature i.e.

𝐸_𝑚 ∝ 𝑇⁵ or 𝐸_𝑚 = 𝐾 𝑇⁵." [1]

I love this kind of text speaking about power but writing energy of emitted radiation.

Why not "energy of emitted radiation per unit of time", that is power ?

I know that with E_m/Δt = (K T⁵)/Δt can be simplified in E_m = K T⁵, but time is always present in nature.

But here you can also write E_m/Δt = K_p T⁵ as Δt is a constant of 1s !

Right ? Malypaet (talk) 22:26, 6 January 2024 (UTC)[reply]

In fact Wien's distribution law can be write

E∙dλ = C₁/λ⁵∙ψ(λϑ)∙dλ

and with wien's displacement law : λ_m=b/ϑ

with λ=λ_m => E∙dλ =(C₁/b)ϑ⁵∙ψ(λϑ)∙dλ

Thiesen derived it implicitely (probably) Malypaet (talk) 23:09, 7 January 2024 (UTC)[reply]