The determination that photons have zero rest mass stems from both theoretical considerations and experimental evidence. The theory of electromagnetism, formulated by James Clerk Maxwell in the 19th century, provided a framework for understanding electromagnetic waves, including light. Maxwell's equations predicted the existence of electromagnetic waves that propagate at the speed of light. These waves were later identified as photons, the quanta of electromagnetic radiation.
From a theoretical standpoint, it was recognized that if photons had rest mass, their speed would be affected by that mass according to Einstein's theory of special relativity. However, experiments conducted in the early 20th century consistently measured the speed of light in a vacuum to be a constant, independent of the motion of the source or the observer. This experimental evidence strongly suggested that photons have zero rest mass.
The most famous formula related to the energy of a photon is given by Einstein's equation: E = mc². Here, E represents the energy of the photon, m represents its mass (in this case, the rest mass), and c represents the speed of light in a vacuum. Since the speed of light is a constant, if the rest mass (m) were non-zero, it would imply that the energy of a photon depends on its velocity, which contradicts experimental observations.
However, it is important to note that while the rest mass of photons is considered to be zero, they do possess energy and momentum due to their motion and frequency. This energy and momentum are related to the frequency of the photon through the formula E = hf, where h is the Planck constant and f is the frequency of the photon.
Regarding the non-zero but extremely small mass of photons, current empirical measurements provide an upper limit for the rest mass of the photon. Experiments involving the scattering of photons off other particles or the measurement of the gravitational effects of photons have placed constraints on the rest mass of photons. These experiments have shown that the rest mass of the photon must be extremely small, less than 10^-18 electron volts (eV), or effectively zero for practical purposes.