Question

Derive Wien's law IpeakT=2.898 x 10-1°m.K by taking dp 27c2h as ehc/AkT-1

Answer 1

Answer is attached
Wien's displacement law The spectral distribution as a function of temperature is now to be examined more closely. It turns out that the maximum of the curve shifts with increasing temperature to ever shorter wavelengths. The dependence of this wavelength Imax on the temperature is given by the following equation. This equation is also known as Wien's displacement law. K Imax 2897,8 um T Wien's displacement law (18)

The Wien's displacement law can be obtained by determining the maxima of Planck's law. For this purpose, the function (1) must be derived with respects to the wavelength 1. By using the product rule and setting the derivative equal to zero, one gets: 40 2thc2 dig(~) ! d) 1 mit Ig(0) = (19) 15 exp hc AkBT - 1

hc exp ( hc kBTA 1 5 dl (0) di 2702 = kpTX [exp(wit) - 1 (20) 2 16 XP ( exp hc kRTA - 1 27C2 exp ( White hc kТА 5 - 0 (21) 16. (exp( voeten) - 1 exp xp ( hc kBTA 1 >0 =0 This equation will only be zero if the term in the round bracket becomes zero:

exp exp hc kBTA (hehet exp (whea) - 1 -5 = 0 (22) With the substitution given below, this equation can be simplified: X:= hc ХkBT X : exp(x) exp(x) – 1 5 = 0 (23) This equation can only be solved numerically, e.g. with the Newton's method. The result will be x = 4,9651. With this result the wavelength Imax can be determined as a

function of the temperature by rearranging the equation (23): hc hc xkB K Imax 0, 0028978 m K T 2897,8 um T = (24) XkBT T K Amar 2897,8 um T (25) The maximum of the spectral intensity can also be determined for the frequency form Is(f). For this the function Is(f) must be derived with respect to frequency f and setting the derivative equal to zero:

2897, K Amax 8 um T (25) The maximum of the spectral intensity can also be determined for the frequency form Is(f). For this the function Is(f) must be derived with respect to frequency f and setting the derivative equal to zero: dls (f) di Lo 0 T fmax = 5,879 · 1010 Hz (26) K