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Transform Planck's formula for space spectral density u_(omega) or radiation form the variable omega to the variables v (linear frequency) and lambda (wavelength). |
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Answer» Solution :We write `u_(omega)d omega = overset~(U)_(v)dv` where `omega = 2pi v` Then`overset~(u)_(v) = (2picancelh(2PIV)^(3))/(pi^(2)c^(3)) (1)/(e^(2picancelhv//kT)-1) = (16pi^(2)cancelhv^(3))/(c^(3))(1)/(e^(2picancelhv//kT)-1)` ALSO `-overset~(u)(lambda, T)d lambda = u_(omega)d omega` where `lambda = (2pic)/(omega)`, `d omega =- (2pic)/(lambda^(2))d lambda` `overset~(u)(lambda,T) = (2pic)/(lambda^(2))u((2pic)/(lambda),T)` `= (2pic)/(lambda^(2))((2pic)/(lambda))^(3)(cancelh)/(pi^(2)c^(3)) (1)/(e^(2cancelhc//lambdakT)-1) = (16pi^(2)c cancelh)/(lambda^(5))(1)/(e^(1picancelhc//lambdakT)-1)` |
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