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How can you have a negative voltage? I don't really understand the concept of negative voltage, how can it exist?

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Voltage is a difference of potential energy for electric charges, and potentials are defined from forces, so that $F=-\nabla V$, where $V$ is the potential and $F$ is the force. When you have determined the potential $V$ you can now add any constant you want, or any function $f$ that doesn't depend on coordinates, because $-\nabla V=-\nabla (V+f)$, as $f$ doesn't depend on coordinates. This last thing is what's known as setting an origin of potential energy, something you sure have heard about. Well, in that function $f$ where you set your origin for potentials, you can have negative voltages between two points.

Another way to see the voltage is the work you must do per unit charge to move that charge from one point to another, even here, when we're dealing with differences, you can also have negative voltages.

If you're talking about circuits, all of the above applies, you can set a potential difference $V$ by, let's say, a battery, then you can use some device so that the potential energy is even lower than the one set by the - sign of the battery, as you set the potential origin 0 for that battery, then that point will have a negative voltage.

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The simplest thing to do here is remember that only diferences in potential matter, and that means that we can add or subtract a constant from every voltage in a system without changing it's behavior, so we can render any negative number positive to make you feel better or just as easily render all the positive ones negative.

In short, it's just a number and you shouldn't fret about it having a sign.

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