Proving Wien's Law Includes Stefan-Boltzmann & Wien's Displacement Laws

[O_o]

Hi,

I'm supposed to prove that Wien's Law: $$P(\lambda,T) = \frac{f(\lambda T)}{\lambda^5}$$ includes Stefan-Botlzmann's Law $$R(T) = \sigma T^4$$ and Wien's Displacement Law: $$\lambda_{max} T = b$$

For Wien's Displacement Law:

I know that I would have to find when $$P(\lambda ,T)$$ graphed against $$\lambda$$ has a slope of 0. So I think I need to find the derivative with respect to $$\lambda$$. But the only two equations for $$P(\lambda,T)$$ I have are $$P(\lambda,T) = \frac{f(\lambda T)}{\lambda^5}$$ and $$P(\lambda,T) = \frac{8\pi kT}{\lambda^4}$$

So if I take the derivative of $$P(\lambda,T) = \frac{8\pi kT}{\lambda^4}$$ with respect to $$\lambda$$ I have
$$8\pi kT} * (-4) * \lambda^{-5} = 0$$ Where I'm guessing that everything except $$\lambda$$ is being held constant and I don't know what to do from there.

Any hints or corrections of things I said would be appreciated. Thanks.

 

[James R]

I think you need to start with Planck's equation, which may be part of the f() function, above. I don't think your expression for P in terms of T and lambda is correct (although it might represent Wien's guess). That function has no peak - it gives an "ultraviolet catastrophe" as lambda goes to zero - there is no lambda_max.

 

[quicknote]

Hello,

Thank you for your question. Proving Wien's Law includes Stefan-Boltzmann's Law and Wien's Displacement Law involves understanding the relationships between these laws and how they are derived. Let's start with Wien's Law:

P(\lambda,T) = \frac{f(\lambda T)}{\lambda^5}

This equation describes the spectral radiance emitted by a blackbody at a specific temperature (T) and wavelength (\lambda). The function f(\lambda T) represents the spectral distribution of the blackbody radiation, which is dependent on both the temperature and wavelength. This equation is derived from Planck's law, which states that the spectral distribution of blackbody radiation is proportional to the temperature and wavelength. Therefore, Wien's Law already includes the relationship between temperature and wavelength.

Now, let's look at Stefan-Boltzmann's Law:

R(T) = \sigma T^4

This equation describes the total radiative power emitted by a blackbody at a specific temperature (T). The constant \sigma is known as the Stefan-Boltzmann constant and is equal to 5.67 x 10^-8 W m^-2 K^-4. This law is derived from the Stefan-Boltzmann law, which states that the total radiative power emitted by a blackbody is proportional to the fourth power of its temperature. Therefore, Stefan-Boltzmann's Law is already included in Wien's Law, as it is represented by the function f(\lambda T) in the numerator.

Lastly, let's consider Wien's Displacement Law:

\lambda_{max} T = b

This equation describes the relationship between the peak wavelength (\lambda_{max}) of blackbody radiation and its temperature (T). The constant b is known as Wien's displacement constant and is equal to 2.8978 x 10^-3 m K. This law is derived from Wien's Law, where the peak wavelength occurs at the maximum of the spectral distribution. Therefore, Wien's Displacement Law is also included in Wien's Law, as it is represented by the factor \lambda^{-5} in the denominator.

In conclusion, Wien's Law already includes both Stefan-Boltzmann's Law and Wien's Displacement Law. These laws are all interrelated and derived from the fundamental principles of blackbody radiation. I hope this helps clarify your understanding of these laws. If you have any further questions, please let me know.