Planck Blackbody Radiation Formula The spectral energy density of blackbody radiation as a function of frequency is given by Planck's radiation law: \[ u(\nu)d\nu = \frac{8\pi h\nu^3}{c^3} \frac{1}{e^{h\nu/kT}-1} \, d\nu \] where \(h = 6.626 \times 10^{-34} \, \text{J·s}\) is Planck's constant \(u(\nu)\) is the energy density of radiation per unit frequency interval \(\nu\) is the frequency \(T\) is the temperature \(k\) is the Boltzmann constant In these notes we will derive the above formula in a step by step manner. Waves in a Box Consider an electromagnetic wave travelling with the speed of light in some arbitrary direction represented by the position coordinate \(x\). If the wavelength of the wave is \(\lambda\), the amplitude of the wave along the \(x\)-direction can be written as \[ A(x) = A_0 \sin \left(\frac{2\pi x}{\lambda}\right) \] This expression can also be written in terms of the wave number \(k\): \[ A(x) = A_0 \sin(kx) \] where the ...
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