Planck's Law and Black Body Radiation

Planck's Law

• The energy $E$ radiated by a black body is not continuous
• Energy is emitted or absorbed in discrete packets called quanta
• Energy of a quantum of radiation is proportional to the frequency $v$
• $E = h\nu$
• $h$ is Planck's constant ($h = 6.626 \times 10^{-34} J.s$)

Planck's Radiation Law

• Gives the energy density of electromagnetic radiation emitted by a black body in thermal equilibrium at a definite temperature $T$
• $u(\nu,T) = \frac{8\pi h\nu^3}{c^3} \frac{1}{e^{\frac{h\nu}{kT}}-1}$
• $u(\nu,T)d\nu$ is the energy per unit volume in the frequency range from $\nu$ to $\nu + d\nu$
• $T$: Absolute temperature of the black body
• $k$: Boltzmann constant ($k = 1.381 \times 10^{-23} J/K$)
• $c$: Speed of light ($c = 3.0 \times 10^8 m/s$)

Wien's Displacement Law

• $\lambda_{max} = \frac{b}{T}$
• $\lambda_{max}$: Wavelength of maximum intensity
• $b$: Wien's displacement constant ($b = 2.898 \times 10^{-3} m.K$)
• $T$: Absolute temperature of the black body

Stefan-Boltzmann Law

• Total energy radiated per unit surface area of a black body across all wavelengths per unit time
• Directly proportional to the fourth power of the black body's absolute temperature $T$
• $j^* = \sigma T^4$
• $j^*$: Total energy radiated per unit surface area
• $\sigma$: Stefan-Boltzmann constant ($\sigma = 5.670 \times 10^{-8} W m^{-2} K^{-4}$)
• $T$: Absolute temperature of the black body