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Definition from Wikipedia:

An electric potential (also called the electric field potential, potential drop or the electrostatic potential) is the amount of work needed to move a unit of positive charge from a reference point to a specific point inside the field without producing an acceleration.

If that is the definition of a voltage. Shouldn't the formula to calculate it be $V = Fe \cdot d$. Why do we divide it by $q$. Is it because it is a positive test charge as explained above?

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The force exerted on a charge $q$ in a uniform electric field $E$ is

$$F=qE$$

The work done on moving the charge a distance $d$ in the field is

$$W=qEd$$

The potential difference, or voltage, between two points is defined as the work per unit charge to move the charge between the two points, or

$$V=Ed$$

So your formula is for work, not voltage.

ADDENDUM:

To answer your follow up comment:

Why do a need this idea of work per charge, we didn't have that when we explaind gravity, right?

In classical mechanics, the "gravitational potential" at a location is equal to the work per unit mass that would be done by the force of gravity if an object were moved from a specific location to a fixed reference location. It is therefore analogous to the "electrical potential" with mass fulfilling the same role as charge.

On the other hand, "Electrical potential energy" = $qEd$ is analogous to "gravitational potential energy" = $mgh$.

Or to put it another way, electrical potential is not the same thing as electrical potential energy and gravitational potential is not the same thing as gravitational potential energy.

Hope this helps.

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  • $\begingroup$ Why do a need this idea of work per charge, we didn't have that when we explaind gravity, right? $\endgroup$ – Hilea Apr 15 at 12:32
  • $\begingroup$ See the ADDENDUM to my answer. Hope that helps. $\endgroup$ – Bob D Apr 15 at 12:58
  • $\begingroup$ ... the gravitational potential near the surface of the Earth is $V_g=gh$, meaning that height is analogous to electric potential. The analogy is useful for understanding electric potential. $\endgroup$ – garyp Apr 15 at 13:03
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This is because, the definition is in terms of unit charge.

$$V=\frac{W}{Q}$$

That is the formula. Voltage is the work done per unit charge.

So if you write down a fraction - say the speed of an object, you would say distance divided by time. So, let us say it is something like $3m/s = \frac{6}{2}$. That is, $6m$ in $2s$. But that is $3m$ in one second.

That's the analogy.

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