|MadSci Network: Physics|
The equation KE = m v^2 / 2 for the kinetic energy of the electron is only a good approximation when v/c is small (where c is the speed of light). The exact equation, which comes from the Special Theory of Relativity and has been carefully checked by many experiments, is KE = m c^2 ( 1/sqrt(1 - (v/c)^2) - 1) . Examining this equation, we see that the KE can become arbitrarily large, even though v/c is always less than 1. Actually, when you add energy to the electrons in an atom they become unbound, and they escape. When the electrons are bound very close to the nucleus of the atom and moving very fast, their total energy is very small, because the attractive (and therefore negative) potential energy between the electron and the nucleus overwhelms the kinetic energy. The thing which prevents the electrons from falling downhill into the nucleus is Heisenberg's Quantum Uncertainty Principle. So you can see that the speed of light does not limit the energy of the electrons.
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