I need to test my understanding regarding the physical meaning of the electric field flux. First of all, the electric flux is found by the surface integral of the field dot product a differential surface. So what I understand is by doing this dot product we are capturing the normal component of the field to the surface that is enclosing, for example, the source of this field which is a charge and by doing this surface integral we are collecting the normal components of the field on this surface. So as a result, the flux can be considered as the field lines that are penetrating a surface which can be considered as a flow of the electric field (If the field was as a flow). Please correct me if I am wrong.
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$\begingroup$ I think that's good enough.Of course you can always go deeper but the idea is that one. $\endgroup$– FGSUZCommented Feb 26, 2019 at 18:51
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$\begingroup$ You may be interested in reading this from just yesterday: physics.stackexchange.com/q/462920 $\endgroup$– G. SmithCommented Feb 26, 2019 at 19:07
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Yes, what you are understanding is correct. I should point out that it is the nature of electrostatics that makes flux a physically important quantity. In particular, the flux, as defined in the way you defined (i.e., the surface integral of the vector inner-product of the field at a point and the area element of the considered surface), turns out the same for all the closed surfaces that enclose the same charged particle(s). And this is in direct correspondence (up-to a subtle interjection of the principle of superposition) with the $\frac{1}{r^2}$ dependence of the electric field, i.e., the Coloumb law (in a universe with $3$ spatial dimensions). If the empirical fact about the electric fields had been that they fall off as, say, $\frac{1}{r^{3}}$--nobody would be bothered to define a flux as it would not be indicative of any useful physical quantity. But, thankfully, in our universe, it does indicate an important physical quantity, namely, the charge enclosed within the surface over which we are finding the flux.
Finally, I should point out that the "flow" picture of the flux--while incredibly seductive and useful to a point--ultimately breaks down in serious ways if you stretch it to an extreme mechanistic version. This is best explained in the chapter on Electrostatics in the Feynman Lectures on Physics (See, section 4-5).