Statistical Physics

5.3 Black-Body Radiation 79Fig. 5.8. Light with wave vector k coming out of a hole of area S. Weplacethehole(thick circle) inthexy plane, and take the z-axis perpendicular to the hole. Theradiation (with wave vector k) inthecanted cylinder comes out of the hole duringatimeinterval∆tthe hole (z-axis) and k. Since the energy per unit volume is 〈n〉 ck/V ,theenergy coming out of the hole is S cos θc ∆t 〈n〉 ck/V , which is equivalent tocos θ 〈n〉 c 2 k/V per unit area per unit time.We sum over the values of k whose energy lies between ν and ν +dν toobtain K ν (T )dν We express k by use of polar coordinates. The number ofmodes in an infinitesimal volume in wave vector space defined by⎧⎨ k to k +dkθ to θ +dθ⎩0 ≤ φ ≤ 2πis [2V/(2π) 3 ]2π sin θ dθk 2 dk. This number is multiplied by the emitted energy〈n〉 c 2 k cos θ/V and integrated over the range 0 ≤ θ ≤ π/2. We then obtainK ν (T )dν, the energy in the interval k to k +dk:K ν (T )dν =∫ π/20dθ 〈n〉 c 2 k cos θ 12π 2 sin θk2 dk= 1(2π) 2 c2 k 3 〈n〉 dk. (5.24)Using the relation k =(2π/c)ν and dk =(2π/c)dν, weobtainK ν (T )= 2πh 1( ν)c 2 ν3 exp(hν/k B T ) − 1 ≡ ν3 F . (5.25)TThus we have obtained Wien’s displacement law, and also obtained the unknownfunction as F (x) =(2πh/c 2 )/[exp(hx/k B ) − 1].