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I know that the electric potential in any given point (considering only a point charge) in a region is defined to be:

[tex]\varphi = \frac{q}{4\pi \varepsilon_{0} d}[/tex]

I know too that the electric potential generally can be defined as:

[tex]\varphi = - \int_{P1}^{P2} \overrightarrow{E} \cdot \overrightarrow{ds}[/tex]

The line integral of any path between P1 and P2 blah blah blah, you already know...

But what I really want to know is: how do derive the first simplified equation from this second general equation?

I got some progress but I still having trouble:

(Considering a region in space with only a point charge).

Because of symmetry we know that through any point I choose passes exact one electric field line and this line passes through the point charge too. So the electric field line is always parallel to the segment that connects point charge and the point I choose. Thus the line integral with the dot product inside can be replaced by the electric field strength multiplied by the distance between the point charge and the point I choose. Is that right?

So, what about the minus before the integral? Why it's in the first equation? I'm I thinking wrong about something? If someone could help me I would be grateful...

If someone has an analogy to why the electric potential decreases with the distance (and not the square of the distance) I would be grateful too...

Thank you,

Rafael Andreatta