A spherical black body of radius r at absolute temperature T...

A spherical black body of radius $r$ at absolute temperature $T$ is surrounded by a thin spherical and concentric shell of radius $R$, black on both sides. Show that the factor by which this radiation shield reduces the rate of cooling of the body (consider space between spheres evacuated, with no thermal conduction losses) is given by the following expression: $a R^2 /\left(R^2+b r^2\right)$, and find the numerical coefficients $a$ and $b$.

Key Concepts

Blackbody Radiation

This concept refers to an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. A blackbody also emits radiation in a characteristic spectrum that depends only on its temperature, making it a fundamental object of study in thermodynamics and quantum mechanics.

Stefan-Boltzmann Law

This law quantifies the power radiated from a blackbody in terms of its temperature; specifically, the total energy radiated per unit surface area is proportional to the fourth power of the absolute temperature. It is crucial for calculating the radiative heat loss from bodies and is fundamental in problems involving thermal radiation.

Radiative Heat Exchange in Enclosures

In configurations where radiative transfer occurs between surfaces, such as a body enclosed by a concentric shell, the net radiation exchange must take into account the properties of both surfaces (emissivity, absorptivity) and the geometric arrangement. This concept involves determining how much radiation emitted by one surface is absorbed or reflected by the other, which is key to understanding the cooling reduction provided by shields.

Geometric Configuration and View Factors

The geometric arrangement of radiating surfaces significantly influences the net radiative exchange. View factors (or configuration factors) quantify the fraction of radiation leaving one surface that directly strikes another. In the case of concentric spheres, symmetry simplifies the determination of these factors, allowing one to relate the radiative exchange area to the sphere radii and derive factors like the numerical coefficients in the expression for the shielding effect.

Transcript

00:01 In this question we have given a spherical ball and it is kept in another holo spherical shell that is b and we have given that the surface of a and the inner surface of b emits as a black body and we have to find the rate of change of temperature of a and that of b.

00:20 So first of all i am writing the given data for this cution.

00:26 So for a spherical shell the area of this shell that is a, a spherical shell that is a, is given by 20 semi -square and we have given the value of heat capacities that is ms for a it is 42 jr per degree celsius and we have given the temperature of a that is 100 degrees celsius or i can say 373 kelvin similarly for b spherical shell we have given the area of this shell that is 80 semi -semi square and heat capacity that is ms is equal to 82 jule per degree celsius and we have given the temperature of this shell that is 20 degrees celsius or i can say in kelvin it is 293 kelvin now it is told in cution that kb is low and it's poor it is a poor conductor and ka is high thus a will absorb no heat and conduct all.

01:43 So it is given that kb is low so it is poor conductor and we have given that ka is high it will conduct all so we can see that ms for a into rate of change of temperature that is d -t -by -dt for a will be equals to radiation heat transfer that is row into of this is a surface area for a and temperature difference that is 373 to the power 4 minus 293 to the power 4 now put all the data in this equation so we get d theta by dt for a that is 6 into 10 to the power minus 8 this can be written as row or i can write in sigma that is boltzman coefficient so 6 into 10 to the power minus 8 into this is 373 to the power 4 minus 293 to the power 4 upon 42 and the area is 20 semi -square that is 20 into 10 to the power minus 4...

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