Direction: Consider a spherical body A of radius R which placed concentrically in a hollow enclosure H, of radius 4R as shown in the figure. The temperature of the body A and H are \[{{T}_{A}}\] and \[{{T}_{H}},\] respectively. |
Emissivity, transitivity and reflectivity of two bodies A and H are \[\text{(}{{e}_{A}},\text{ }{{e}_{H}})\text{ (}{{t}_{A}},\text{ }{{t}_{H}}\text{)}\] and \[({{r}_{A}},\text{ }{{r}_{H}})\] respectively. |
For answering following questions assume no absorption of the thermal energy by the space in-between the body and enclosure as well as outside the enclosure and all radiations to be emitted and absorbed normal to the surface. |
[Take \[\sigma \times \,4\pi {{R}^{2}}\,\times \,{{300}^{4}}=\,\beta J{{s}^{-1}}\]] |
A) The bodies lose their distinctiveness inside the enclosure and both of them emit the same radiation as that of the black body.
B) The rate of heat loss by A in both cases is the same and is equal to\[\beta \,J{{s}^{-1}}\].
C) The rates of heat loss by A in both the cases are different.
D) From this information we can calculate exact rate of heat loss by A in different cases.
Correct Answer: C
Solution :
[c] |
If \[\beta =\sigma \times 4\pi {{R}^{2}}\times \,{{300}^{4}},\] |
Then \[\sigma \times \,4\pi \,{{(4R)}^{2}}\times \,{{600}^{4}}\,=256\beta \,=\gamma \] |
Let \[{{a}_{H}}\,={{e}_{H}}=0.5\] |
and for A in 2nd case, \[{{e}_{A}}=\,{{a}_{A}}=\,0.5\] |
For 1st case, \[{{P}_{emitted}}\,=\,\beta \,J{{s}^{-1}}\] |
\[{{P}_{absorbed}}\,=\,\frac{\gamma }{2}\,+\frac{\beta }{2}\] |
Rate at which energy is lost, \[P=\left( \beta -\frac{\gamma }{2}-\frac{\beta }{2} \right)\,J{{s}^{-1}}\] |
For 2nd case, |
\[{{P}_{emitted}}=\left( \frac{\beta }{2}+\,\frac{\beta }{8}+\frac{\beta }{32}+.... \right)\,+\left( \frac{\gamma }{4}+\frac{\gamma }{16}+... \right)\] |
\[=\,\frac{2\beta }{3}+\,\frac{\gamma }{3}\] |
\[{{P}_{absorbed}}=\,\left( \frac{\beta }{8}+\,\frac{\beta }{32}+.... \right)\,+\,\left( \frac{\gamma }{4}+\,\frac{\gamma }{16}+... \right)\,\,=\frac{\beta }{6}+\frac{\gamma }{3}\] |
Rate at which heat is lost, \[P=\,\frac{\beta }{2}\]. |
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