Understanding Planck’s Law

The Formula

Planck’s Radiation Law describes the spectral radiance of electromagnetic radiation emitted by a black body at a given absolute temperature $T$ and wavelength $\lambda$.

Where:

• $B(\lambda, T)$ = spectral radiance ($\text{W} \cdot \text{sr}^{-1} \cdot \text{m}^{-3}$)
• $h$ = Planck’s constant ($6.626 \times 10^{-34} \ \text{J}\cdot\text{s}$)
• $c$ = speed of light ($3.0 \times 10^{8} \ \text{m}/\text{s}$)
• $\lambda$ = wavelength ($\text{m}$)
• $k_B$ = Boltzmann constant ($1.381 \times 10^{-23} \ \text{J}/\text{K}$)
• $T$ = absolute temperature ($\text{K}$)

Usefulness in Physics

Planck’s Law is foundational to modern physics and represents the birth of quantum theory. Before Planck, classical physics could not accurately model the observed spectrum of blackbody radiation, leading to the “ultraviolet catastrophe.” Planck’s revolutionary idea—that energy is quantized (comes in discrete packets, or photons)—perfectly explained the blackbody curve.

Its applications are vast, including:

• Astrophysics: It is used to determine the temperature and energy output of stars, which are excellent approximations of blackbodies. [Image of blackbody radiation curve]
• Thermodynamics: It defines the total energy radiated by a blackbody, leading to the Stefan-Boltzmann Law.
• Cosmology: The Cosmic Microwave Background (CMB) radiation has a perfect blackbody spectrum defined by Planck’s Law, which is crucial evidence for the Big Bang theory.

How to Use the Calculator

1. Enter Wavelength ($\lambda$): Input the wavelength in meters ($\text{m}$). For example, for $500\ \text{nm}$, you would enter $5.0\text{e-}7$ (which is $5 \times 10^{-7}$).
2. Enter Temperature ($T$): Input the absolute temperature in Kelvin ($\text{K}$). For instance, the Sun’s surface temperature is about $5778\ \text{K}$.
3. Click “Calculate”: Press the blue “Calculate Spectral Radiance” button.
4. View Result: The spectral radiance $B(\lambda, T)$ in $\text{W} \cdot \text{sr}^{-1} \cdot \text{m}^{-3}$ will be displayed in scientific notation (rounded to 2 decimal places) for precision.