How is the energy of an electron related to the voltage across its gap?

The energy of an electron is directly related to the voltage across its gap, meaning that as the voltage increases, the energy of the electron also increases proportionately. This principle is grounded in the concept that when a voltage is applied, it creates an electric field that accelerates the electrons. The work done on the electrons by the electric field is quantified in terms of energy, which corresponds to the charge of the electron multiplied by the voltage.

For example, if the voltage is doubled, the energy gained by the electron also doubles. This relationship is critical in various applications in physics and engineering, such as in particle accelerators and electronic devices, where controlling electron energy through voltage settings is fundamental to their operation. Thus, the direct proportionality signifies that higher voltage leads to higher energy levels for the electrons involved.