# Black body radiation and the derivation of Stefan Boltzman

• dcarmichael
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.
dcarmichael
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)

What did you replace dλ with when converting to x as the variable of integration?

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

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?

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

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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