Temperature of black bodies

In summary, we have used the Stefan-Boltzmann law to calculate the temperatures of a sphere and a disc at a distance a from the sun. The ratio of their temperatures is 0.84, with the sphere having a slightly lower temperature than the disc.
  • #1
thephysicsman
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Homework Statement



Two black bodies, a sphere and a disc, are situated at the same distance a from the sun. Calculate the ratio of their temperatures.

Homework Equations



Stefan-Boltzmann law:

[tex]U = \sigma T^4[/tex]

The Attempt at a Solution



At a distance a, the irradiance from the sun is

[tex]U = \frac{r_{sun}^2}{a^2}\cdot \sigma T_{sun}^4 [/tex]

The sphere absorbs

[tex]\pi r^2 \frac{r_{sun}^2}{a^2}\sigma T_{sun}^4[/tex]

and emits

[tex]4\pi r^2 \sigma T_{sphere}^4[/tex]

Equating the absorption and the emission, I get

[tex]T_{sphere}=T_{sun}\sqrt{\frac{r_{sun}}{2a}}[/tex]

The disc absorbs the same as the disc, and emits

[tex]2\pi r^2 \sigma T_{sphere}^4[/tex]

Equating the absorption and emission:

[tex]T_{disc}=T_{sun}\sqrt{\frac{r_{sun}}{a}}\cdot \frac{1}{2^{1/4}}[/tex]

[tex]\frac{T_{sphere}}{T_{disc}}=0.84[/tex]

Correct?
 
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  • #2


Your solution looks correct to me! You have properly used the Stefan-Boltzmann law to calculate the temperatures of the two black bodies. The ratio of their temperatures is indeed 0.84, meaning that the sphere would have a slightly lower temperature than the disc. Great job!
 

1. What is a black body?

A black body is an idealized object that absorbs all incoming electromagnetic radiation without reflecting or transmitting any of it. It is often used as a theoretical model to study thermal radiation.

2. Why is the temperature of black bodies important?

The temperature of black bodies plays a crucial role in understanding thermal radiation and the behavior of objects at different temperatures. It is also a key concept in thermodynamics and can be used to measure the temperature of other objects.

3. How is the temperature of black bodies measured?

The temperature of black bodies is typically measured using a thermocouple or a pyrometer, which measures the intensity of thermal radiation emitted by the object. These measurements can then be converted to temperature using the Stefan-Boltzmann law.

4. What is the relation between temperature and the color of a black body?

The color of a black body is directly related to its temperature. As the temperature of a black body increases, the color shifts from red to orange to yellow and eventually to white. This is due to the change in the spectrum of thermal radiation emitted by the object.

5. Are there any real-life examples of black bodies?

While there are no perfect black bodies in nature, many objects can closely approximate this idealized model. For example, stars and planets can be considered black bodies as they emit thermal radiation and absorb almost all incoming radiation. In addition, materials such as carbon black and graphite can also behave as black bodies.

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