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Using Wien's formula, demonstrate that (a) the most probable radiation frequency omega_(pr) prop t, (b) the maximum spectral density of thermal radiation (u_(omega))_(max) prop T^(3), (c ) the radiosity M_(e) prop T^(4). |
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Answer» SOLUTION :(a)The most probable radiation frequency `omega_(P)` is the frequency for which `(d)/(d omega) u_(omega) = 3 omega^(2)F(omega//T)+(omega)/(T)^(3)F(omega//T) = 0` The maximum frequency is the ROOT other than `omega = 0` of this EQUATION. It is `omega=- (3TF(omega//T))/(F'(omega//T))` or `omega_(Pr) = x_(0)T` where `x_(0)` is the solution of the TRANSCENDENTAL equation `3F(x_(0)) +x_(0)F' (x_(0)) =0` (b) The maximum spectral DENSITY is the density corresponding to most probable frequency. It is `(u_(omega))_(max) = x_(0)^(3)F(x_(0))T^(3) alphaT^(3)` where `x_(0)` is defined above. (c ) The radiosity is `M_(e) = (c)/(4) underset(0)overset(oo)int omega^(3)F((omega)/(T)) d omega = T^(4) [(4)/(c) underset(0)overset(oo)int x^(3)F(x)dx]alpha T^(4)` |
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