If Electrical flux is no. of field lines passing through the area then why the formula is $E$ dot $A$ (Area)? I have been taught that flux is no. of field lines passing through the surface. But my question how  does the formula for calculating electric flux is matching with the above statements?
The formula for electric flux is $E \cdot S$. How does the field lines is getting by the dot product of $E$ and $S$ (area vector). May be i am not able to understand the relation between the field lines and field strength? Please someone explain?
 A: *

*Let me state just in case, that when we refer to a surface vector we refer to the orthogonal vector that points away from the surface:




*The actual formula for flux is not the cross product (which increases when things become perpendicular), but the dot product (which increases when things become parallel), which is obviously what happens here, since the flux is gonna become larger when the surface vector becomes parallel with the electric field:





*And finally, the most important, what actually is the number of "electric lines" is the flux itself, not the electric field. The field is more like the density of lines, which is the number of lines in an area of size 1 (that is why a bigger field means the strength is higher at that point, while a bigger flux can just be that you took into account a bigger area). That is why the flux is the field (density of lines [n of lines/area]) multiplied by the area! :).


(In this last picture, the electric field in the left panel, is higher than in the right panel, because you see way more lines per area unit, they are more "condensed" -> higher electric field.
But if the electric field lines in the left, when we do zoom out stop in that area and the rest is empty of lines, but in the right panel if we zoom out, there keeps appearing new ones at some point taking enough area the left panel will have higher total flux, even when in the centre it has a weaker electric field. )
