question_answer

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 \[\sigma \times \,4\pi {{R}^{2}}\,\times \,{{300}^{4}}=\,\beta J{{s}^{-1}}\]]

In above question, if body A has \[{{e}_{A}}=0.5,\] \[{{r}_{A}}=0.5\] and for H, \[{{e}_{H}}=0.5,\] \[r=0.5,\] then mark out the correct statement.

A) The rate at which A loses the energy is

B) The rate at which the spherical surface containing P receives the energy is zero

C) The rate at which the spherical surface containing Q receives the energy is\[\beta \].

D) All of the above

Correct Answer: D

Solution :

[d] Now, in this case, each of incidence, reflection and absorption take place. The rate of which energy has been lost by A is,  \[P=\,-\,[{{P}_{absorbed}}-\,{{P}_{emitted}}]\]             \[=-\,\left[ \frac{\beta }{8}+\,\frac{\beta }{32}+\,..... \right]\,+\,\left[ \frac{\beta }{2}\,+\frac{\beta }{8}+\,\frac{\beta }{32}+\,.... \right]\,=\,\frac{\beta }{2}\] The rate at which energy is received by P is,  \[{{P}_{1}}=0\] The rate at which energy is received by Q is, \[{{P}_{2}}=\,\left( \frac{\beta }{2}+\,\frac{\beta }{8}+\,... \right)\,+\,\left( \frac{\beta }{4}+\,\frac{\beta }{16}+.. \right)\]             \[=\,\frac{\beta }{2}\,\times \,\frac{4}{3}+\,\frac{\beta }{4}\,\,\times \,\frac{4}{3}=\beta \]