# How can you have a negative voltage?

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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If I define positive to be in one direction, then if it is in the opposite direction, what should I call it? – Steeven May 27 '15 at 11:05
Voltage is a scalar quantity (J/C)) – Bionic Person Jun 6 '15 at 7:17

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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You should consider the sign. If $\Delta V = V_B - V_A > 0$, it means the electric potential in B is higher than in A. If it is negative, it is the other way around. That's an important difference, no? – Dries May 27 '15 at 14:24
The difference is the same regardless of whether you add a constant to all terms to render them negative or positive – Joshua Lin May 28 '15 at 21:26
But voltages cannot cancel out if you neglect the signs? – Doeser Aug 26 '15 at 16:18

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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Let's say you want to know the electric potential difference r meters away from an electron sitting in a void of space. We will denote this potential difference as $V_e$.

The electric potential difference describes the difference in potential energy of a unit positive charge from one point in space to another and the work done on a unit positive charge to carry it from the same point to the other.

This gives the equation: $$V_a-V_b=k\frac{Q}{r_a}-k\frac{Q}{r_b}$$

$V_a$ is the final potential and $V_b$ is the initial. In this scenario, it is conventional to make $V_b$ the zero reference point at $r = \infty$, which makes $r_b = \infty$ and $V_b = 0$. Therefore, the equation becomes:

$$V_a=k\frac{Q}{r_a}$$

Because we are looking at the potential difference of an electron, $Q=-e$, where $e$ equals the elementary charge. So, the equation, in this case, is:

$$V_e=k\frac{-e}{r_a}$$

This equation gives a negative value no matter how far away the positive test charge is from the electron. This makes sense because a positive point charge at $r_b = \infty$ loses potential energy as it is brought toward the electron (downstream), and negative work is done.

If the electron is replaced with a proton, the potential difference, denoted by $V_p$, will become:$$V_p=k\frac{e}{r_a}$$

If you have a proton and an electron you will simply compute $V_e + V_p$, because electric potential is a scalar quantity.

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Hi Bionic Person and welcome to Physics.SE! Please see this help post to learn how to write your equations in a way nicer way i.e. in $\LaTeX$, in order to improve legibility. Thanks! – gonenc May 27 '15 at 12:21
Thanks. It looks very pretty now. – Bionic Person May 28 '15 at 21:21

"negative" voltage is relative! Potential is potential. If I have two batteries connected in series and measure volts from the point where they are connected, I will measure +V on one end and -V on the other. This is a basic electronics question and now you have a basic answer.

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