Black Body Equations (more of a math prob)

In summary, the conversation discusses two equivalent equations for energy flux radiated from a black body, one dependent on frequency and the other on wavelength. The equations are equivalent through unit conversion and the use of modulus to account for the negative sign. Reference material is provided for further explanation.
  • #1
Rob Hal
13
0
Hi,

In one of my textbooks, I'm given these two equivalent equations for energy flux radiated from a black body, one dependent on frequency, the other on wavelength:

[tex] I(\lambda)d\lambda = \frac {2c^2h}{\lambda^5} \frac {1}{e^{(hc/KT\lambda)}-1}d\lambda[/tex]

and

[tex] I(\nu)d\nu = \frac {2h\nu^3}{c^2} \frac {1}{e^{(hv/KT)}-1}d\nu[/tex]

Now, I'm just trying figure out how are these equations equivalent? At first, I thought it was simple substitution of the relation [tex]\lambda = \frac{c}{\nu}[/tex], but that doesn't work... so I realize that you have to convert units by differentiating say in this case lambda, and I get the same equations with the exception of a negative sign which I can't see how it cancels.

To be quite honest, I'm just not sure I'm doing the converting right. I'm not exactly sure how to handle the left side of the equation in this case.

Any suggestions or references would be great... Thanks!
 
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  • #3
Excellent...I knew it was really elementary.
Thank you very much!
 
  • #4
the negative sign is due to the limit of the integral
if the lower limit of your lamda is 0 and upper limit is infinite,ie.
[tex] \int_{0}^\infty I(\lambda) d \lambda [/tex]

after you do the substitution, the lower limit of v will become infinite and the upper limit will be 0! ie,
[tex] -\int_{\infty}^0 I(v) dv [/tex]

in order to make the integral looks nicer, we eat the negative sign and flip the limit of the integral:
[tex] \int_{0}^\infty I(v) dv [/tex]

don't worry, your calculation is completely fine...

PS. the modulus is not a must... the web page just don't want to do this argument
 
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What are the Black Body Equations?

The Black Body Equations are a set of mathematical equations that describe the behavior of an idealized object that absorbs and emits all wavelengths of electromagnetic radiation, known as a black body. They are used in physics and astronomy to understand the properties of objects in space, such as stars and planets.

What is the Planck's Law in Black Body Equations?

Planck's Law is one of the three main equations in Black Body Equations. It describes the spectral radiance of a black body at a given temperature and wavelength. It states that the amount of energy emitted by a black body is directly proportional to the temperature and the wavelength of the radiation.

What is the Stefan-Boltzmann Law in Black Body Equations?

The Stefan-Boltzmann Law is another important equation in Black Body Equations. It relates the total energy emitted by a black body to its temperature. It states that the total energy emitted is proportional to the fourth power of the temperature. This means that as the temperature of a black body increases, the amount of energy it emits increases significantly.

What is Wien's Displacement Law in Black Body Equations?

Wien's Displacement Law is the third main equation in Black Body Equations. It describes the peak wavelength of radiation emitted by a black body at a given temperature. It states that the peak wavelength is inversely proportional to the temperature, meaning that as the temperature increases, the peak wavelength decreases.

How are Black Body Equations used in science?

Black Body Equations are used in various fields of science, including physics, astronomy, and thermodynamics. They are used to understand the properties of objects in space, such as stars and planets, and to calculate their temperatures and energy emissions. They are also used in engineering and materials science to study the properties of materials that absorb and emit radiation, such as solar panels and light bulbs.

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