Planck's Law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature \( T \). It provides the theoretical foundation for understanding thermal radiation and is a cornerstone of quantum mechanics.
The formula for Planck's Law is given by:
\[ I(\nu, T) = \frac{2h\nu^3}{c^2} \cdot \frac{1}{e^{\frac{h\nu}{kT}} - 1} \]
where:
- \( I(\nu, T) \) is the intensity of radiation at frequency \( \nu \) and temperature \( T \),
- \( h \) is Planck's constant (\( \approx 6.626 \times 10^{-34} , \text{J s} \)),
- \( \nu \) is the frequency of the radiation,
- \( c \) is the speed of light in a vacuum (\( \approx 3.00 \times 10^8 , \text{m/s} \)),
- \( k \) is Boltzmann's constant (\( \approx 1.381 \times 10^{-23} , \text{J/K} \)),
- \( T \) is the absolute temperature of the black body in kelvins.
The other two equations you mentioned are separate concepts:
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\( E = hf \): This is the energy of a photon, where \( E \) is the energy, \( h \) is Planck's constant, and \( f \) is the frequency of the photon. It quantifies the relationship between the energy of a quantum (like a photon) and its frequency.
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\( E = mc^2 \): This formula is Einstein's theory of relativity, which expresses the equivalence of mass \( m \) and energy \( E \), with \( c \) being the speed of light in a vacuum. It shows that mass can be converted into energy and vice versa.
In summary, Planck's Law relates to black body radiation, \( E = hf \) relates to the energy of photons, and \( E = mc^2 \) relates to mass-energy equivalence.
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