Unraveling Wien's Law: Exploring Total Emissive Power

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Thanks again!In summary, the problem statement involves showing the total emissive power using Wien's law, as well as the relationship between the maximum wavelength and temperature. The attempt at a solution involved integrating Wien's law, but this proved difficult without a specific form for the function f(λT). However, by using a substitution and recognizing that the integral is a definite integral, the solution can be expressed as a constant times T^4, providing a hint for how to proceed with the problem.
  • #1
RedMech
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1. The problem statement:

Using Wien's law ρ(λ,T)=f(λ,T)/λ^5, show the following:

(a) The total emissive power is given by R = aT4 (the Stefan-Boltzmann law),
where a is a constant.
(b) The wavelength λmax at which ρ(λ,T) - or R(λ,T) - has its maximum is such that λ*T = b (Wien's displacement law), where b is a constant.

2. Homework Equations :

Wien's radiation law:
ρ(λ,T)=f(λ,T)/λ^5
ρ(λ,T)=c1/(λ^5*exp{c2/λT})

3. The Attempt at a Solution :

So I tried integrating Wien's equation from zero to infinity
ρ(total)dλ=c/4∫ρ(λ,T)dλ=c/4∫[f(λ,T)/λ^5]dλ. But I got nowhere.

Then I used the full expression of wien's law and tried the integration again
ρ(total)dλ=c/4∫[c1/(λ^5*exp{c2/λT})]dλ
I still didn't know what to do. So please help.
 
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  • #2


RedMech said:
1. The problem statement:

Using Wien's law ρ(λ,T)=f(λ,T)/λ^5, show the following:

(a) The total emissive power is given by R = aT4 (the Stefan-Boltzmann law),
where a is a constant.
(b) The wavelength λmax at which ρ(λ,T) - or R(λ,T) - has its maximum is such that λ*T = b (Wien's displacement law), where b is a constant.

2. Homework Equations :

Wien's radiation law:
ρ(λ,T)=f(λ,T)/λ^5
ρ(λ,T)=c1/(λ^5*exp{c2/λT})

3. The Attempt at a Solution :

So I tried integrating Wien's equation from zero to infinity
ρ(total)dλ=c/4∫ρ(λ,T)dλ=c/4∫[f(λ,T)/λ^5]dλ. But I got nowhere.
Without an explicit form for f(λ,T), you can't integrate this, as you probably realized.

Then I used the full expression of wien's law and tried the integration again
ρ(total)dλ=c/4∫[c1/(λ^5*exp{c2/λT})]dλ
I still didn't know what to do. So please help.
This approach should work. How did you try to integrate this? I'd try a substitution like u=1/λ and see where it goes.
 
  • #3


vela said:
This approach should work. How did you try to integrate this? I'd try a substitution like u=1/λ and see where it goes.

I substituted x=c2/λT for the sake of the exponential term.
dx=[-c2/λ^2T]dλ. The integral has become w=(c1*c*T^4)/4c2^4∫[x^3/e^x]dx (Please note that for c1 and c2, the 1 and 2 are subscripts of c. The independent c is the speed of light)

How is this equation looking?
 
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  • #4


Do you recognize that integral? Think gamma function. In any case, it's a definite integral, so it's just some number.
 
  • #5


vela said:
Do you recognize that integral? Think gamma function. In any case, it's a definite integral, so it's just some number.

I'll compute the integral and then leave the final expression for my instructor. Thanks a million for your help.
 
  • #6


RedMech said:
1. The problem statement:

Using Wien's law ρ(λ,T)=f(λ,T)/λ^5, show the following:

(a) The total emissive power is given by R = aT4 (the Stefan-Boltzmann law),
where a is a constant.
...

I tried integrating Wien's equation from zero to infinity
ρ(total)dλ=c/4∫ρ(λ,T)dλ=c/4∫[f(λ,T)/λ^5]dλ. But I got nowhere.

Wien's law is actually ρ(λ,T)=f(λT)/λ5 where f is an undetermined function of the product of λ and T. Using this, see if you can get the integral to yield a constant times T4.
 
  • #7


TSny said:
Wien's law is actually ρ(λ,T)=f(λT)/λ5 where f is an undetermined function of the . Using this, see if you can get the integral to yield a constant times T4.

@TSny, I was wondering if you might be able to give me a small hint in regards to how to proceed with this problem only using the ρ(λ,T)=f(λT)/λ5 form of Wien's law. I tried integration by parts but that just led to a more convoluted expression. I see that you underlined the phrase "product of λ and T" but I'm still not sure how to handle the f(λT) term in the integral.
 
  • #8


That calls for a substitution (change of variable) which would throw out of the integral exactly T to the power of 4.
 
  • #9


dextercioby said:
That calls for a substitution (change of variable) which would throw out of the integral exactly T to the power of 4.

Thank you dextercioby, my mistake was in assuming that I need to find the unknown function f(λT). I was able to figure out the answer based on your hint.
 

Related to Unraveling Wien's Law: Exploring Total Emissive Power

1. What is Wien's Law and how does it relate to total emissive power?

Wien's Law states that the wavelength at which an object emits the most light is inversely proportional to its temperature. This means that objects with higher temperatures emit shorter wavelengths of light, while cooler objects emit longer wavelengths. Total emissive power is the total amount of energy emitted by an object at all wavelengths, and it is directly related to an object's temperature and emissivity.

2. How is total emissive power measured?

Total emissive power can be measured using specialized instruments such as spectrophotometers or infrared cameras. These instruments measure the intensity of light emitted by an object at different wavelengths and can calculate the total amount of energy being emitted.

3. What factors affect the total emissive power of an object?

The total emissive power of an object is primarily affected by its temperature and emissivity. Other factors that may also play a role include the object's surface area, shape, and composition. Objects with higher temperatures and higher emissivity values will have a higher total emissive power.

4. How is Wien's Law used in practical applications?

Wien's Law is used in various practical applications, such as in thermal imaging and remote temperature sensing. By measuring the wavelength of light emitted by an object, we can determine its temperature and use this information in fields like astronomy, meteorology, and materials science.

5. Can Wien's Law be applied to all objects?

Wien's Law is a fundamental law of physics and can be applied to all objects that emit thermal radiation, including stars, planets, and man-made objects. However, some objects may not follow this law exactly due to factors like surface texture and complex compositions. In these cases, other laws and principles may need to be considered in addition to Wien's Law.

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