electric-fieldselectricityelectrostatics
When calculating the electric flux over a certain area using the formula: $$\Phi=\int_s \vec E \cdot d \vec A$$
Why does the electric field vector have to be parallel to the area vector? In other words, why is only the field perpendicular to the area considered while calculating the flux? I don't quite understand the concept behind how if the electric field vectors are not passing through the area perpendicularly it results in a less value for the electric flux.
When the notions of electric and magnetic fields were conceptualized, they imagined that there was an invisible fluid being pushed around by charges, and they leveraged some of the equations and terminology of fluid mechanics.
The modern understanding of fields has largely gotten rid of this picture, but some colorful langauge like "electric flux" remains. If you want to picture positive charge as "amount of fluid added to region per unit time" and negative charge as "amount of fluid removed from region per unit time", you can, but this thinking only gets you so far. Safer to just think of it as an abstract mathematical definition.
Consider the flux through a tiny segment of a sphere. Since the electric field is parallel to the normal of the surface at all points, the flux is simply the electric field at that distance multiplied by the area of the element.
Now imagine tilting the top of the cone by an angle $\theta$ so that the corners still lie on the conical section, as seen below:
The area increases by a factor $\frac{1}{\cos\theta}$, however the electric field vector in the normal direction $E_n$ is decreased by a factor of $\cos\theta$. Therefore the flux through this surface is unchanged since flux is the product of the normal electric field component and the area.
Now imagine splitting the cube up into lots of these conical sections. Clearly the tilting of the top surfaces of these sections due to the fact it being a cube rather than a sphere does not affect the flux flowing through each area element. Therefore the total flux flowing through the cube is the same as a sphere.
Note that this was a simplified adaptation from a chapter of The Feynman Lectures on Physics which explains why the images do not quite match my explanations since I was just talking about the top surface of the conical section being tilted. Feynman explains the effect of the flux through a closed surface in a more complete way.
Best Answer
Definition:
Flux by definition is the amount of quantity going out or entering a surface.
Intuition:
In the above diagram, the black line represents the surface for which the flux is being calculated and the red lines represent the direction of the flow of a quantity.
In the above diagram, the quantity represented by the red lines are moving parallel to the surface. The quantity is not leaving the surface nor is some quantity entering the surface. Therefore, the flux is zero.
In the above diagram, the quantity represented by the red lines is leaving — or entering depending on your perspective — the surface. Therefore, there is a net flux through the surface.
Mathematical definition:
A vector dot product gives you the projection of a vector along another vector. The area vector is defined as the area in magnitude whose direction is normal to the surface.
Consider the following:
$$\phi = \vec{a}.\vec{b}$$
The above equation gives the amount of $\vec{a}$ that is along the direction of $\vec{b}$ times the vector ${b}$. It is equivalent to taking the scalar projection of $\vec{a}$ and multiplying it with the magnitude of $\vec{b}$.
As the flux by definition is numerically equal to the amount of quantity leaving the surface, we are concerned with the quantity passing perpendicularly through the surface. In this situation, the dot product helps us implicitly mention the above fact.
$$\phi = \int\vec{E}.d\vec{A}$$