# Black body radiation, Plancks' Law question?

• applestrudle
In summary, the question poses a scenario where a perfect black body cylinder with a length of 0.02m and a temperature of 2000K is emitting 100W of energy. The goal is to determine the diameter of the cylinder. The attempt at a solution involves using Planck's Law and Wien's displacement law to calculate the frequency of peak radiation and then using dimensional analysis to set up an equation to solve for the radius. However, it is suggested that the Stefan-Boltzmann law may also need to be used for a complete solution.
applestrudle

## Homework Statement

A cylinder of length 0.02m emits 100W and is at 2000K. It is a perfect black body. What is it's diameter?

## Homework Equations

Planck's Law:

B(λ,T) =

http://en.wikipedia.org/wiki/Planck's_law

solid radian = A/r^2 (not sure if needed or not)

## The Attempt at a Solution

surface area = 0.02 x 2 pi r + 2 pi r^2

solid radians of the cylinder = (pi r^2) / (r^2 + 0.01^2)

r is radius of the cylinder

frequency of peak radiation = 2.07 x 10^14

(I used Wien's displacement law to get the peak frequency)

B x surface area x solid radians x frequency = 100W

(I'm using dimensional analysis here, not really sure what's going on)

100 = 9.104 x 10^-10 x (0.04 pi r + 2 pi r^2) x (pi r^2)/(r^2 x 10^-6) x 2.07 x 10^14

which rearranges to...

5.303 x 10^-14 (r^2 + 1x10^-6) = 0.04 (pi)^2 r^3 + 2 (pi)^2 r^4

I can't solve for r :(

any help or explanation is much appreciated

Don't you have to use the Stefan-Boltzmann law ??

## 1. What is black body radiation?

Black body radiation refers to the electromagnetic radiation emitted by an object that absorbs all of the radiation it receives, regardless of the frequency or wavelength. This type of radiation is observed when an object is heated to a high temperature, such as a star or a hot piece of metal.

## 2. What is Planck's Law?

Planck's Law is a mathematical equation proposed by German physicist Max Planck that describes the spectral density of energy emitted by a black body at a given temperature. It is used to calculate the intensity of black body radiation at different wavelengths and temperatures.

## 3. How does temperature affect black body radiation?

As the temperature of a black body increases, the intensity of its radiation also increases and shifts towards shorter wavelengths. This is known as Wien's displacement law, which states that the peak wavelength of radiation emitted by a black body is inversely proportional to its temperature.

## 4. Why is black body radiation important?

Black body radiation is important because it provides a theoretical basis for understanding the behavior of light and matter. It is also used in many practical applications, such as in thermodynamics, astrophysics, and the development of technologies like incandescent light bulbs and solar cells.

## 5. How is black body radiation related to the concept of a "perfect absorber" and "perfect emitter"?

A black body is often referred to as a "perfect absorber" because it absorbs all of the radiation that is incident upon it. Similarly, it is also considered a "perfect emitter" because it emits radiation at all wavelengths and at the maximum possible intensity for a given temperature. This concept is used to describe the behavior of real objects and to analyze their emissions as if they were ideal black bodies.

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