Solving EM Wave Emission from Cavity at T: Power per Unit Area

In summary, the problem states that a cavity at temperature T is emitting EM waves isotropically, with a frequency distribution given by Planck's Law. The goal is to find the value of d<S>/dw, which represents the effective poynting vector magnitude per unit solid angle. It is known that <S>=c<e> and the total solid angle for a sphere is 4 pi. The solution involves integrating over a solid angle of 2 pi, which results in a value of (c/4)<e>, as desired. However, there may be some confusion with the solid angle business, as the relation |<\vec{S}>|=c<e> only holds for plane waves propagating in a
  • #1
phystudent17
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0

Homework Statement


Basically, the problem states that a cavity at temperature T is emitting EM waves isotropically in all directions (with frequency distribution of Planck's Law). If the time averaged density is <e>, find the value of d<S>/dw where w is the solid angle and the quantity is the effective poynting vector magnitude per unit solid angle. Hence I am to show the power per unit area that passes in one direction (i.e. into solid angle of 2 pi) through any plane within the cavity is dP/dA= (c/4)<e>/ Note that the unit system is Gaussian. Basically, I am stuck at the first part of the problem.


Homework Equations



Some equations that I know are <S>=c<e>, the total solid angle for a sphere is 4 pi.

The Attempt at a Solution



I have a feeling the solution is really simple but I cannot get into the physics of it. Is d<S>/dw just <S>/ 4pi= (c/4 pi)<e>? But then integrating over a solid angle of 2 pi gives me (c/2)<e> which is off by a factor of 2. And I really don't get the solid angle business. Can someone point me in the right direction? Thanks.
 
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  • #2
"But then integrating over a solid angle of 2 pi gives me (c/2)<e> which is off by a factor of 2"

That's because you should be integrating over a solid angle of 4 pi, as you already know!
 
  • #3
but now i want the power per unit area passing through one direction and that has a solid angle of 2 pi not 4 pi. the qn requires me to show that integrating over the solid angle of 2 pi gives me (c/4)<e>
 
  • #4
I did misread your question, sorry about that. In any case, the relation |<\vec{S}>|=c<e> holds for plane waves propagating in a given direction. It's not a general relation.
 

1. How do you calculate the power per unit area for EM wave emission from a cavity at a given temperature?

The power per unit area for EM wave emission from a cavity at a given temperature can be calculated using the Stefan-Boltzmann law, which states that the power radiated per unit area from a blackbody is proportional to the fourth power of its absolute temperature. This can be expressed as P = σAT^4, where P is the power per unit area, σ is the Stefan-Boltzmann constant, A is the surface area of the cavity, and T is the temperature in Kelvin.

2. What is the significance of solving EM wave emission from a cavity at temperature?

Solving EM wave emission from a cavity at a given temperature is important in various fields such as physics, engineering, and astronomy. It helps in understanding the behavior of electromagnetic radiation and its interaction with matter, as well as in designing and optimizing various devices that use EM waves, such as antennas and sensors.

3. What factors affect the power per unit area of EM wave emission from a cavity?

The power per unit area of EM wave emission from a cavity is affected by several factors, including the temperature of the cavity, the surface area of the cavity, and the emissivity of the material used to construct the cavity. Other factors such as the geometry and shape of the cavity, as well as any external influences like surrounding temperature or radiation, can also affect the power per unit area.

4. How does the power per unit area of EM wave emission from a cavity change with temperature?

The power per unit area of EM wave emission from a cavity increases with an increase in temperature according to the Stefan-Boltzmann law. This means that as the temperature of the cavity increases, the power per unit area of EM wave emission will also increase, resulting in a higher intensity of radiation being emitted from the cavity.

5. Can the power per unit area of EM wave emission from a cavity be controlled?

Yes, the power per unit area of EM wave emission from a cavity can be controlled by adjusting the temperature, surface area, and emissivity of the cavity. By manipulating these factors, it is possible to increase or decrease the intensity of radiation emitted from the cavity, making it a valuable tool in various applications that require precise control of EM wave emission.

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