What is the kinetic energy gained by an electron due to acceleration through a potential difference of 1 V?

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A charged particle only gains energy by moving from one point to another at a different potential. If the electron doesn't move, then it doesn't gain any energy. So, the electron actually has to move from point A to point B to gain energy from the potential difference.

The following is a response to saying the electron gains "10 joules" of energy.

The kinetic energy gained by a charged particle when passing through a potential difference is equal to the charge of the particle multiplied by the potential difference. $$K = qV$$ where $K$ is the kinetic energy, $q$ is the charge of the particle, and $V$ is the voltage difference. An electron would only gain 10 joules of kinetic energy if it had a charge of 1 Coulomb, which is a very large charge. The actual charge of an electron is about 1.602$\times$10$^{-19}$ C. So, an electron passing through a 10V potential difference ends up with 1.602$\times$10$^{-18}$ joules of kinetic energy.

Because these numbers are so small, a different unit of energy is often used. This unit is called the electron-volt (eV), which is equal to the energy gained by an electron when it passes through a 1-volt potential difference. One electron-volt is equal to 1.602$\times$10$^{-19}$ J. This makes calculations easier because, to use your problem as an example, an electron passing through a 10V potential difference gains 10 eV of energy.