For particles with mass, such as electrons, their momentum (p) and energy (E) are related through the relativistic energy-momentum equation:
E² = (pc)² + (mc²)²
In this equation, c represents the speed of light, m is the rest mass of the electron, p is the momentum of the electron, and E is its total energy (including rest energy).
When the electron is at rest (i.e., its speed is zero), the momentum term (pc) disappears, and the equation simplifies to:
E = mc²
This is the famous equation derived by Einstein, where the energy of an object at rest is given by its mass multiplied by the square of the speed of light.
When the electron is in motion with a speed less than the speed of light, both momentum and energy contribute to the equation. The momentum term, pc, accounts for the relativistic increase in momentum as the electron's speed approaches the speed of light. The rest mass term, mc², represents the energy contribution from the electron's rest mass.
The equation shows that as the electron's speed increases, its momentum increases, and this contributes to its overall energy. However, it's important to note that the total energy of the electron is always greater than its rest energy, even at speeds below the speed of light.
At speeds much lower than the speed of light (non-relativistic speeds), the equation can be approximated by the classical formula for kinetic energy:
E ≈ mc² + (1/2)mv²
In this approximation, the rest mass energy (mc²) and the non-relativistic kinetic energy (1/2)mv² contribute to the total energy of the electron, where v is its velocity.
In summary, for electrons in motion with speeds less than the speed of light, both momentum and energy contribute to their total energy, following the relativistic energy-momentum equation.