Black body radiation and the derivation of Stefan Boltzman

In summary, when converting to x as the variable of integration, dλ is replaced with dx since x is the new variable of integration. This is because dλ can be expressed as -(a/x^2)dx, where a is a constant, when taking the differential of the relation λ = a/x.
  • #1
dcarmichael
17
2
Homework Statement
The total intensity i(t) radiated from a blackbody is given by the integral from 0 to infinity of all wavelengths of the Planck distribution.But I keep seem to be getting the wrong answer. Could someone point out where I'm going wrong
Relevant Equations
Let l=lambda I(l,T)=(2Pihc^2)/l^5 *1/(e^(hc/lkT)-1)
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  • #2
What did you replace dλ with when converting to x as the variable of integration?
 
  • #3
haruspex said:
What did you replace dλ with when converting to x as the variable of integration?
I didnt indicate it but dλ is replaced with dx since x is new variable of integration
 
  • #4
dcarmichael said:
I didnt indicate it but dλ is replaced with dx since x is new variable of integration
##d\lambda \neq dx##

Note that you can write ##\lambda = \large \frac{a}{x}##, where ##a## is a constant. Taking the differential of both sides of this relation, you should get ##d\lambda = \boxed ?\, dx##. What goes inside the box?
 
  • #5
TSny said:
##d\lambda \neq dx##

Note that you can write ##\lambda = \large \frac{a}{x}##, where ##a## is a constant. Taking the differential of both sides of this relation, you should get ##d\lambda = \boxed ?\, dx##. What goes inside the box?
dλ= -(a/x^2)dx
 
  • #6
Ok. Go for it.
 

1. What is black body radiation?

Black body radiation refers to the electromagnetic radiation emitted by a hypothetical object that absorbs all incoming radiation and reflects none. This means that a black body absorbs and emits radiation at all wavelengths, making its emission spectrum continuous.

2. Who is Stefan Boltzman and what is his contribution to the study of black body radiation?

Stefan Boltzman was a physicist who developed the Stefan-Boltzman law, which relates the total energy radiated by a black body to its temperature. This law is an important tool for understanding the behavior of black body radiation and its role in thermodynamics.

3. How is the Stefan-Boltzman law derived?

The Stefan-Boltzman law is derived from the Planck radiation law, which describes the spectral density of electromagnetic radiation emitted by a black body at a given temperature. By integrating this law over all wavelengths, the Stefan-Boltzman law can be obtained.

4. What is the significance of the Stefan-Boltzman law?

The Stefan-Boltzman law is significant because it provides a way to calculate the amount of radiation emitted by a black body at a given temperature. This is important in many fields, including astrophysics, where it is used to understand the radiation emitted by stars and other celestial bodies.

5. How does the Stefan-Boltzman law relate to the concept of thermal radiation?

The Stefan-Boltzman law is a fundamental part of the study of thermal radiation, as it describes the relationship between the temperature of a black body and the amount of thermal radiation it radiates. This law is essential for understanding the transfer of heat and energy in various systems and processes.

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