Derive Stefan-Boltzmann law from Wien’s law.

• Tirain
Once you have that, you can plug it into Wien's law and start working from there.In summary, to derive Stefan-Boltzmann law from Wien's law, you can use the equation R(T) = -∞∫∞ R(λ,T)dλ and the relation ρ(λ,T)=(4/c)*R(λ,T). Additionally, you can use Wien's law p(λ,T) = f(λ,T)/^5 and Stefan's law R(T) = σT^4. To begin, determine the function f(\lambda,T) and plug it into Wien's law to start working on the derivation.
Tirain

Homework Statement

Derive Stefan-Boltzmann law from Wien’s law

Hint: you can use (without proof) R(T) = -∞∫∞ R(λ,T)dλ , ρ(λ,T)=(4/c)*R(λ,T)

Homework Equations

wien's law: p(λ,T) = f(λ,T)/^5
Stefan's law : R(T) = σT^4

The Attempt at a Solution

Honestly I am having trouble just starting this problem. I know I would be able to work it out if I knew where to begin.

Oh thanks!

Tirain said:

Homework Statement

Derive Stefan-Boltzmann law from Wien’s law

Hint: you can use (without proof) R(T) = -∞∫∞ R(λ,T)dλ , ρ(λ,T)=(4/c)*R(λ,T)

Homework Equations

wien's law: p(λ,T) = f(λ,T)/^5
Stefan's law : R(T) = σT^4

The Attempt at a Solution

Honestly I am having trouble just starting this problem. I know I would be able to work it out if I knew where to begin.

Well, the first step is: what is the function $f(\lambda,T)$ ?

To derive the Stefan-Boltzmann law from Wien's law, we can use the relationship between the spectral radiance (R) and the spectral energy density (ρ) given by ρ(λ,T)=(4/c)*R(λ,T). Since we know that Wien's law states that p(λ,T) = f(λ,T)/^5, we can substitute this into the equation for ρ(λ,T) to get ρ(λ,T) = (4/c)*R(λ,T) = (4/c)*f(λ,T)/^5.

Next, we can use the definition of the Stefan-Boltzmann law, R(T) = σT^4, and substitute this into the equation for R(λ,T) to get R(λ,T) = (4/c)*σT^4.

Now, we can integrate both sides of the equation with respect to wavelength (λ) from -∞ to ∞ to get ∫∞-∞ R(λ,T)dλ = (4/c)*∫∞-∞ σT^4dλ. By the definition of spectral radiance, the left side of the equation becomes R(T), and by the definition of the Stefan-Boltzmann law, the right side becomes R(T) = σT^4. Therefore, we have derived the Stefan-Boltzmann law from Wien's law.

1. What is Wien's law?

Wien's law, also known as the displacement law, is a general relationship between the temperature of an object and the wavelength at which it radiates the most energy. It states that the wavelength of maximum radiation is inversely proportional to the temperature of the object.

2. What is the Stefan-Boltzmann law?

The Stefan-Boltzmann law is a fundamental law of physics that relates the total energy emitted by a black body to its temperature. It states that the total emissive power of a black body is directly proportional to the fourth power of its absolute temperature.

3. How can Wien's law be used to derive the Stefan-Boltzmann law?

By combining Wien's law with the Planck radiation law, which describes the spectral energy density of a black body, we can derive the Stefan-Boltzmann law. This involves finding the wavelength of maximum radiation using Wien's law and then integrating the Planck radiation law over all wavelengths to find the total energy emitted, which is directly proportional to the fourth power of the temperature.

4. What are the assumptions made in deriving the Stefan-Boltzmann law from Wien's law?

In order to derive the Stefan-Boltzmann law from Wien's law, we assume that the object is a perfect black body, meaning it absorbs all radiation incident on it, and that its temperature is constant. Additionally, we assume that the object is in thermal equilibrium, meaning that the amount of energy it absorbs is equal to the amount it emits.

5. Why is it important to derive the Stefan-Boltzmann law from Wien's law?

Deriving the Stefan-Boltzmann law from Wien's law allows us to understand the relationship between temperature and the total energy emitted by an object at different wavelengths. This is important in many fields, including astrophysics and thermodynamics, as it allows us to make accurate predictions and calculations about the behavior of black bodies and other objects that emit thermal radiation.

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