Wien's displacement law follows from setting in Planck's spectral energy density

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Scientific programming for physicists

Scientific Computing for Physicists, Ch The numerical toolkit closed: backward error and the audit of a library routine

Problem

Wien's displacement law follows from setting

\mathrm{d}u_\lambda/\mathrm{d}\lambda = 0

in Planck's spectral energy density, which reduces to the transcendental equation

f(x) \equiv (x-5)e^{x} + 5 = 0,

where

x = hc/(\lambda k_{\rm B} T)

evaluated at the wavelength of peak emission,

h

is the Planck constant,

c

the speed of light and

k_{\rm B}

the Boltzmann constant. ((a)) Evaluate

f(4)

and

f(5)

, and hence give a bracket in which

f

changes sign. ((b)) Using Brent's method within that bracket, find the root

x_r

to six decimal places. ((c)) Using

h = 6.626070\times10^{-34}\,\mathrm{J\,s}

c = 2.997925\times10^{8}\,\mathrm{m\,s^{-1}}

and

k_{\rm B} = 1.380649\times10^{-23}\,\mathrm{J\,K^{-1}}

, compute Wien's displacement constant

b = hc/(k_{\rm B}x_r)

, defined by

\lambda_{\rm max}T = b