# What Are the Differences Between Black Body Radiation Formulas?

• sss1
sss1
Homework Statement
Below
Relevant Equations
In the pictures
Im getting confused between the differences of all of these formulas.
I googled spectral radiance black body and all of the first four pictures came up. They represent the intensity of radiation at a particular wavelength right, or the y-axis of the black body radiation curve? So if I integrate this formula I should get the total intensity? Or the total area under the black body radiation curve? One of the pictures has frequency as the variable instead of wavelength tho? Is it finding the same thing but for when I'm given frequency instead of wavelength? And somehow the rest of the three pictures all have different numerators...?
And the last formula, which finds the radiated power for a specific wavelength, why does it have a delta lambda in it? Kinda confused on where it comes about. I understand that spectral radiancy has units Watts/m^3, so it makes sense to have A and delta lambda, because that has units m^3. But why not have lambda instead of delta lambda? And also if i integrated that formula it will give me Stefan Boltzmann's law? The total power radiated?

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sss1 said:
Homework Statement: Below
Relevant Equations: In the pictures

Im getting confused between the differences of all of these formulas.
I googled spectral radiance black body and all of the first four pictures came up. They represent the intensity of radiation at a particular wavelength right, or the y-axis of the black body radiation curve? So if I integrate this formula I should get the total intensity? Or the total area under the black body radiation curve? One of the pictures has frequency as the variable instead of wavelength tho? Is it finding the same thing but for when I'm given frequency instead of wavelength? And somehow the rest of the three pictures all have different numerators...?
And the last formula, which finds the radiated power for a specific wavelength, why does it have a delta lambda in it? Kinda confused on where it comes about. I understand that spectral radiancy has units Watts/m^3, so it makes sense to have A and delta lambda, because that has units m^3. But why not have lambda instead of delta lambda? And also if i integrated that formula it will give me Stefan Boltzmann's law? The total power radiated?

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Unfortunately, the way you have posted those images they cannot be clicked to reveal the whole text. If you cannot figure out how to do it, post the links.

haruspex said:
Unfortunately, the way you have posted those images they cannot be clicked to reveal the whole text. If you cannot figure out how to do it, post the links.
Does it work now?

haruspex said:
I had a look at the table and tried calculating the spectral radiancy using both frequency and wavelength, but got different answers?
I used these two formulas.
The wavelength I used was 966e-9m, and so the frequency should be (3e8)/(966e-9) Hz?

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sss1 said:
I had a look at the table and tried calculating the spectral radiancy using both frequency and wavelength, but got different answers?
I used these two formulas.
The wavelength I used was 966e-9m, and so the frequency should be (3e8)/(966e-9) Hz?
##B_\nu## is the spectral emissive power per unit area, per unit solid angle and per unit frequency. ##B_\lambda## is per unit wavelength.
I.e. ##B_\nu d\nu## is the total spectral emissive power per unit area, per unit solid angle for the frequency range ##(\nu,\nu+d\nu)##, etc.
Hence ##B_\nu =B_\lambda|\frac{d\lambda}{d\nu}|=B_\lambda\frac{c}{\nu^2}##.
If you make that substitution, and ##\nu=\frac c{\lambda}##, you should see one equation turn into the other.

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sss1

## What is a black body in physics?

A black body is an idealized physical object that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. It emits radiation at a characteristic spectrum that only depends on its temperature, described by Planck's law.

## What is Planck's law and how does it relate to black body radiation?

Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature. It states that the intensity of radiation emitted at a certain frequency is proportional to the frequency cubed and inversely proportional to the exponential of the frequency over temperature minus one.

## How do Wien's displacement law and the Stefan-Boltzmann law relate to black body radiation?

Wien's displacement law states that the wavelength at which the emission of a black body spectrum is maximized is inversely proportional to the temperature. The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a black body is proportional to the fourth power of its temperature.

## Why do different black body formulae seem contradictory?

The confusion often arises because different black body formulae, such as Planck's law, Wien's law, and the Stefan-Boltzmann law, describe different aspects of black body radiation. Planck's law gives the detailed spectral distribution, Wien's law gives the peak wavelength, and the Stefan-Boltzmann law gives the total emitted power. They are consistent with each other but apply to different measurements.

## How can I apply black body radiation laws to real-world objects?

Real-world objects are not perfect black bodies, but many can be approximated as such for practical purposes. To apply black body radiation laws, you need to know the temperature of the object and its emissivity, which is a measure of how closely the object approximates a black body. With these, you can use Planck's law, Wien's law, and the Stefan-Boltzmann law to estimate the radiation properties of the object.

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