I´m currently studying electrostatics and I´m having a hard time understanding why do the number of electric field lines is proportional to the intensity of electric field? I just don´t see it.
Can we formally derive a formula that relates the number of electric field lines $N$ and the intensity of the electric field $|\vec E|$?
Field lines are just something we use to visualize the direction of the electric field vector. It's a standard convention to draw them so that the density of the lines is proportional to the strength of the field. It's not a fact of nature, it's part of the definition of how you draw the lines.
If you have a system with a small number of point charges, then this convention becomes an analogy to Gauss' Law
$$\nabla{}\cdot\vec{E} = \frac{\rho}{\varepsilon_0}$$
In terms of field lines, this basically becomes the rule that field lines can only begin or end on a charge.
If you follow this rule and draw the field lines uniformly exiting or entering each charge, then you will naturally end up with the density of field lines proportional to the strength of the field.
As already explained in the other answer, field lines are "not a fact of nature". Indeed, the number of field lines is not a meaningful physical quantity, but only a useful tool to visualize the magnetic of electric fields. It is not a meaningful quantity because it is not measurable.
The equation you are looking for is just $$N=a |E|$$ where $N$ is the number of lines, $|E|$ is the field intensity and $a$ is a proportionality constant. The constant a is arbitrary, and you can basically decide how many lines to draw in order to make your plot/figure looking better. In this sense, they are not measurable and they are not a physical quantity, because one cannot answer unambiguously the question "How many field lines are in the point P?". The answer to this question is a matter of convention, because it depends on the constant $a$. However, the ratio between the number of field lines at two different point is equal to the ratio of the field intensity.
In summary, the number of lines is really just a useful way to visualize the field, they are not a physically well defined quantity.
