1145_Physics ProblemsTechnical Physics

1145_Physics ProblemsTechnical Physics - 486 Introduction...

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486 Introduction to Quantum Physics Thus, λ max .. . T = ×⋅ × × 6 626 075 10 2 997 925 10 4 965 115 1 380 658 10 2897755 10 34 8 23 3 Js ms JK mK ej . This result is very close to Wien’s experimental value of max . T 2898 10 3 m K for this constant. P40.63 (a) Planck’s radiation law predicts maximum intensity at a wavelength max we find from dI d d d hc e hc e e hc kT hc e hc k T hc k T hc k T hc k T λλ πλ == R S T U V W =−− F H G I K J +− 02 1 1 1 2 5 1 25 1 2 2 26 1 B BB B B bg af or + = hce kTe e hc k T hc k T hc k T B B B B 7 2 6 1 5 1 0 which reduces to 51 hc ee hc k T hc k T B F H G I K J −= . Define x hc = B . Then we require 55 ex e xx . Numerical solution of this transcendental equation gives x = 4965 . to four digits. So max . = hc B , in agreement with Wien’s law. The intensity radiated over all wavelengths is IT dA B hc d e hc k T , 0 2 5 0 2 1 ∞∞ zz =+= B . Again, define x hc = B so = hc xk T B and d hc xkT dx =− 2 B . Then, AB hc x k T hcdx hcxkTe hc xdx e x x x += = =∞ 2 1 2 1 255 5 55 2 0 4 32 3 0 ππ B B B 4 . The integral is tabulated as π 4 15 , so (in agreement with Stefan’s law)
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