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Most of the definitions on flux and flux density, show a plot consisting of a positive charge emanating a field, and describe that as the number of field lines decrease, the field strength decreases. My question is, If an electric field (which is a vector field) is defined at every point in space, how does the density fall as we travel away, as there is a vector associated with every point in our "unit area" where ever it's considered. So how does the density "fall"?

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    $\begingroup$ don't think of it as field lines, interpret the equation as it is; at each point of the surface you're trying to calculate the flux through, you will be taking the component of the vector at that point perlendicular to the infinitesimal surface around that same point (this is why it's called flux, because it looks like it's going through the surface) and sum all of that. $\endgroup$ – Luyw Jul 18 '19 at 14:21
  • $\begingroup$ "What" is precisely going through the surface? $\endgroup$ – Aravindh Vasu Jul 18 '19 at 15:28
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    $\begingroup$ nothing, it's just imagining that if you consider the vector at one point to be the velocity of a particle, for example, then it is going to go through that surface in the direction of the component of the vector that is normal to it (the surface), since the tangential component is just responsible for its sideways motion. $\endgroup$ – Luyw Jul 18 '19 at 15:31
  • $\begingroup$ So, the definition of flux (it's the number of field lines passing through the given surface) on the bases of electric field lines is inconsistent? $\endgroup$ – Aravindh Vasu Jul 18 '19 at 15:34
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    $\begingroup$ It does not seem right to me to call it like that, I have just read in wikipedia something contradictory; it is stated that the number of field lines is the flux density and in another place that the flux is the number of field lines. Confusing. $\endgroup$ – Luyw Jul 18 '19 at 15:44
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Field lines are a vizualization tool. Each point on the line represents the direction of the field at that point. The lines are usually drawn so that their density corresponds to the field amplitude.

However, field lines are not particularly useful for calculations. Yes, a vector field is defined with a direction and amplitude at each point in space. The field amplitude falls off with distance from its sources. For a point charge, the field amplitude falls off as $1/r^2$. Fields from more complicated sources are found by superposition.

Electric flux is not generally measured in lines, but is caclulated by integrating over a surface the field amplitudes crossing perpendicular to the surface. Dividing by the area gives average flux per unit area, and taking the limit as the area goes to zero determines the local flux density.

Edit: Added to address comment regarding Volume II, Chapter 4 of the Feynman Lectures.

The number of electric field lines drawn from a charge ($q/\epsilon_0$) represents field strength. As Feynman says, '... strength of the electric field will be represented by the “density” of the lines.' You can count lines, but to quantify flux, you must know the field strength their density represents.

Feynman ends the chapter with the following quote.

"The field-line picture has its uses, however, so we might still like to draw the picture for a pair of equal (and opposite) charges. If we calculate the fields from Eq. (4.13) and the potentials from (4.24), we can draw the field lines and equipotentials. Figure 4-13 shows the result. But we first had to solve the problem mathematically!"

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  • $\begingroup$ I read Feynman texts recently, and encountered a sentence asserting "The number which leave a charge $q$ must be equal to $\frac{ q}{\epsilon}$" Pg No: 4-11 the Last line (in Filed Lines; equipotential surfaces) $\endgroup$ – Aravindh Vasu Jul 26 '19 at 0:20
  • $\begingroup$ @AravindhVasu: Nothing in my answer contradicted Feynman. I wish I could say it as well as he did. See my edit added to address your comment. $\endgroup$ – amateurAstro Jul 27 '19 at 1:38
  • $\begingroup$ Nah, I'm not saying your answer is contradicting, I'm just asking, do the field lines(which is a visualisation tool, yeah I understand) have a specific number or not? Yeah they do, as we wanted to relate the density of field lines with the field strength, we wanted them to have a number.how to arrive at that number? $\endgroup$ – Aravindh Vasu Jul 27 '19 at 1:46
  • $\begingroup$ The number of field lines is arbitrary. I do not know of any standard way to choose a number of lines for a given charge. The main point is that the density of lines indicates field strength. Farther from the sources, the lines separate as the field amplitude decreases. $\endgroup$ – amateurAstro Jul 27 '19 at 1:53
  • $\begingroup$ Okay cool, sorry for being redundant, but that $\frac{q}{\epsilon}$ has been bugging me for a long time now. Maybe they wanted the density of field lines to give the exact value of flux at any given space. $\endgroup$ – Aravindh Vasu Jul 27 '19 at 1:55
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Using electric field lines is just a very efficient method to represent a lot of information about the field in a compact diagram it does not mean that field lines actually exist .

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