Proving I=(1/4)cE: Exploring Black Body Radiation

[sharomi]

I am looking to prove the relation
I=(1/4)cE
Where I is the radiation emittance which is the energy emitted by a black body per unit area per unit time and E is the radiation energy density (energy per wavelength summed over all wavelength/frequencies of electromagnetic radiation emitted).

I am familiar with the derivation for the intensity of a plane Electromagnetic wave: I=cE
which is just the amount of energy that flows through a unit area in a unit time and is easily understood as the total energy contained within a box of length C and unit width and height.

In this case i am trying to follow the same procedure but I don't get the above results (the relation can also be found at http://en.wikipedia.org/wiki/Planck's_law under overview except it's formulation is different: I=(1/4pi)cE

I encountered the topic in Alonso-Finn fundamental university physics in the discussion of Black body radiation and Planck's law.

If anyone can help me understand how to derive the relation at the top i'd appreciate it!

 

[Aaron Barnard]

The relation I=(1/4)cE comes from Planck's law, which states that the energy emitted by a black body per unit area per unit time at a given frequency is proportional to the frequency to the fourth power. Mathematically, this can be expressed as:

I = (2hv^3/c^2) * 1/4π * e^(-hv/KT),

where h is Planck's constant, v is the frequency of the radiation, c is the speed of light, T is the temperature of the black body, and K is Boltzmann's constant.

When you substitute in the values for h, v, c, and K, the equation simplifies to I=(1/4)cE.

 

[astrokreng]

Hello,

The relation I=(1/4)cE is known as the Stefan-Boltzmann law, which describes the relationship between the intensity of radiation emitted by a black body and its energy density. This law was derived by Josef Stefan in 1879 and later refined by Ludwig Boltzmann in 1884.

To understand how this relation is derived, we must first consider the properties of a black body. A black body is an idealized object that absorbs all radiation incident upon it and emits radiation at all wavelengths. It is also assumed to be in thermal equilibrium, meaning that the rate of energy absorbed must equal the rate of energy emitted.

Using these assumptions, we can consider a small volume element of the black body and apply the laws of thermodynamics to derive the Stefan-Boltzmann law. The energy emitted by this volume element can be expressed as:

dE = ρc dV

where dE is the energy emitted, ρ is the energy density, c is the speed of light, and dV is the volume element. This energy is emitted in all directions, so we must also consider the solid angle of emission, which is given by 4π.

Thus, the total energy emitted by the black body can be expressed as:

E = 4πρcV

where V is the total volume of the black body. We can also express the energy density ρ in terms of the temperature T of the black body and the Stefan-Boltzmann constant σ, which relates the energy density to the fourth power of the temperature:

ρ = σT^4

Substituting this into the equation for E, we get:

E = 4πσT^4V

Finally, we can express the intensity I as the energy emitted per unit area per unit time, which is given by:

I = E/AΔt

where A is the surface area of the black body and Δt is the time interval. Substituting the equation for E, we get:

I = 4πσT^4V/AΔt

Since the volume V is equal to the product of the surface area A and the thickness of the black body, we can rewrite this equation as:

I = 4πσT^4/Δt

Now, we know that the energy density is related to the intensity by the speed of light, c. So we can rewrite the equation as:

I =