What Is the Role of Voltage in Battery Functionality?

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I have a few questions about voltage. In my physics class we learned that voltage was the difference in electrical between two points. The way that it was explained was the change in potential energy moving an electron from one point to another in an electric field. This makes sense to me but I don't understand how this applies to something like a battery. How does one end of a battery have more electrical potential than the other end?
 
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