Energy of Photon Calculator With Temperature

Planck's Law for Spectral Energy Density:

\[ E = \frac{2 h f^3}{c^2} \frac{1}{\exp\left(\frac{h f}{k T}\right) - 1} \]

Average photon energy ≈ 2.7 kT for blackbody

Spectral Energy Density (E):

Average Photon Energy (≈2.7kT):

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1. What is Planck's Law?

Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature. It provides the energy of photons at different frequencies for a given temperature.

2. How Does the Calculator Work?

The calculator uses Planck's law equation:

\[ E = \frac{2 h f^3}{c^2} \frac{1}{\exp\left(\frac{h f}{k T}\right) - 1} \]

Where:

\( E \) — Spectral energy density (J·s/m³)
\( h \) — Planck's constant (6.626×10⁻³⁴ J·s)
\( f \) — Frequency (Hz)
\( c \) — Speed of light (299,792,458 m/s)
\( k \) — Boltzmann constant (1.381×10⁻²³ J/K)
\( T \) — Temperature (K)

Explanation: The equation describes how much energy is carried by photons at a specific frequency for a blackbody at temperature T. The average photon energy is approximately 2.7 times the thermal energy (kT).

3. Importance of Photon Energy Calculation

Details: Calculating photon energy is fundamental in thermodynamics, quantum mechanics, astrophysics, and many engineering applications including thermal imaging, solar energy, and lighting technologies.

4. Using the Calculator

Tips: Enter temperature in Kelvin and frequency in Hertz. The calculator will provide both the exact spectral energy density and the approximate average photon energy (2.7kT).

5. Frequently Asked Questions (FAQ)

Q1: What is a blackbody?
A: An idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence.

Q2: Why is average photon energy ≈2.7kT?
A: This comes from integrating Planck's law over all frequencies and dividing by total photon number. The exact factor is ζ(4)/ζ(3) ≈ 2.701.

Q3: What are typical applications?
A: Used in studying cosmic microwave background, star temperatures, thermal cameras, and understanding thermal radiation.

Q4: How does temperature affect photon energy?
A: Higher temperatures shift the spectrum toward higher frequencies (shorter wavelengths) and increase total radiated power (Stefan-Boltzmann law).

Q5: What's the difference between energy density and photon energy?
A: Energy density is energy per unit volume per unit frequency, while photon energy is the energy of individual photons at a given frequency.