PX156 - A3a - blackbody models

1D blackbody

a box of length, $L$
the electric field:

\vec{E} (0, t) = \vec{E} (L, t) = 0

E_{n} \sim \sin (\frac{n π}{L}) \sin (ω_{n} t), n = 1, 2, 3

allowed standing waves, "modes":

λ_{n} = \frac{2 L}{n} ⟹ λ_{n} << 2 L f o r n >> 1

δ n = \frac{d n}{d λ} (- δ λ) = \frac{2 L}{λ^{2}} δ λ

energy in modes with wavelengths between $λ \to λ - δ λ :$

e (λ, T) = \frac{2 L}{λ^{2}} k_{B} T δ λ

3D blackbody

a cube of length, $L$

e (λ, T) = \frac{8 π L^{3}}{λ^{4}} k_{B} T δ λ

converting $e (λ, T)$ to $I (λ)$ gives the rayleigh-jeans law:

I (λ) = \frac{2 π c k_{B} T}{λ^{4}}

not found experimentally, as it lead to the 'ultraviolet catastrophe'