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: aR 2 /(R 2 + br 2 ), and find the numerical coefficients a and b.
Text Solution
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Let the surrounding temperature be 
. The rate of energy loss of the black body before being surrounded by the spherical shell is
The energy loss per unit time by the black body after being surrounded by the shell is 
, where 
is temperature of the shell.
The energy loss per unit time by the shell is
Since 
, we obtain
Hence 
, i.e., 
and 
.
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