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Fuel cells use a chemical reaction, rather than combustion (burning a fuel), to produce electricity in a process that is the reverse of electrolysis. In electrolysis, and electric current applied to water produces hydrogen and oxygen. By reversing this process, hydrogen and oxygen are combined in the fuel cell to produce electricity and water.
Fuel cells are really a family of technologies; there are several major types of fuel cells, differentiated by the type of electrolyte they use. To illustrate how a fuel cell works, we will use the PEM fuel cell as an example.

Hydrogen (fuel) is fed into the anode of the fuel cell. Oxygen (from air) is fed into the cathode side. Encouraged by a catalyst, electrons are stripped from the hydrogen atom. Freed of the electrons, the protons pass through the electrolyte, while the electrons are forced to take a different path to the cathode. As the electrons travel their separate path, they create an electric current that can be utilized. At the cathode, another catalyst rejoins the hydrogen atom, which then combines with the oxygen to create a molecule of water.