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Why does number of field line be a 'finite ' coming out from a charge $Q$ (that is $Q/\epsilon$) where there are 'infinite' number of point around the charge $Q$ each which has a specific value of Electric field strength.

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    $\begingroup$ I don't recall seeing a statement of Gauss's law talking about field lines. Could you repeat it here? $\endgroup$
    – JiK
    Commented Aug 16, 2016 at 10:40

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Let me recapitulate the concept. The field lines are continuous to be drawn in such a way that,

  • the tangent drawn at any point will be parallel to the field at that point.

  • the density field lines in a region (NOT at a point) is proportional to the strength of field in that region.

Clearly these statements do not mention anything about the field strength at a point. So let us note couple of points here. Also more importantly, THEY ARE NOT REAL. Yes this is just a visualization tool created by Faraday.

  1. Field lines concept is a qualitative one. So total number of field lines is not that important. The important point is the field line density. We fix the number of field lines while drawing them. Because we can not draw infinite number of field lines.

  2. A more quantitative idea would be solid angle. The total solid angle is always $4\pi$ so people generally confusingly use the language that number of field lines is $4\pi$. But they are not. The number of field lines is what you want them to be.

At last let us see what is the exact statement of Gauss Law.

The net electric flux through any closed surface is equal to $\frac{1}{\varepsilon_o}$ times the net electric charge within that closed surface.

The definition of electric flux is, $\Phi_E = \iint_S \vec{E}\cdot d\vec{A} \propto$ number of field lines crossing a particular surface S but NOT equal to.

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  • $\begingroup$ So,it is not about number of "electric field line",q/epsilon is the "value" of the "Electric flux"(but not number of field line) through a surface from a charge " q" and the number of electric field line (which is roughly infinite) is proportional to the value "q/epsilon" and thus gauss law nothing to do with "number" of electric field line,it deals with only just a "value " of "flux of electric field line".Is my this concept correct? If it correct, now the question is if it deals with flux of electric field line but why not it deal with "Number" of field line (flux=number of E line pass/A)? $\endgroup$
    – Abu sayed
    Commented Aug 16, 2016 at 14:58
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    $\begingroup$ the field lines are not real. they are simple aid for visualization of electric field. once you realize what we mean by flux of electric field you should just drop the idea of this lines and deal with more quantitative definition of flux. On side note, the flux of any vector field $\vec{V}$ through a surface $S$ is given by, $\oint_S \vec{V}\cdot d\vec{A}$. This is exactly where the $A_1v_1 = A_2v_2$ comes from. Also note that this idea of field lines matches with idea of streamlines. $\endgroup$
    – The Imp
    Commented Aug 16, 2016 at 15:52
  • $\begingroup$ But what is the intuition behind ∮v.da cos(theta)?And does electric flux ="number" of electric field line passing through unit area really? $\endgroup$
    – Abu sayed
    Commented Aug 16, 2016 at 17:36
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Gauss's Law doesn't state that there are a finite number of field lines. Gauss's Law doesn't really tell anything about (at least directly) the number of field lines rather it tells about the flux of the electric field. To make a concrete meaning out of this apparent terminological web, let's define and/or state everything step-by-step.

Electric Field: It's the vector field whose value at a given point is given by the force acting on a static point-charge of unit charge by virtue of its charge.

Flux: $\phi = \displaystyle\int_{S}\vec{E}.d\vec{A}$

Gauss's Law: $\displaystyle\int_{S}\vec{E}.d\vec{A} = \dfrac{Q_{enc}}{\epsilon_0}$

Flux is not really defined as the number of field lines crossing a particular patch of the surface rather it is just the surface integration of the scalar product of electric field and elemental surface area. And Gauss's law simply talks about this flux.

However, when we try to form a picture of the electric field in the terms of things that we are primitively familiar with, we come up with the picture in which some finite number of infinitely long arrows are coming out of a static charge, each pointing radially outward. This model has an inherent deviation from the actual field because the arrows do not occupy the entire space while the field actually does. But we can still go a bit far with this model keeping in mind that it is not really in the best agreement with experiments and should not be given precedence over the mathematical laws derived via experiments while drawing the conclusions.

Let's denote a quantity defined as the number of arrows crossing a particular patch of the surface by $\phi_p$. Now it is interesting that $\dfrac{\phi_pQ}{N\epsilon_0} = \phi$ where $Q$ is the charge of the static point charge under consideration, $N$ is the number of infinite arrows imagined to be coming out of the considered point charge. So the number of arrows crossing a patch of a surface is in direct proportionality with the electric field flux for a given number of imaginary arrows. But as one can clearly see, this analogy doesn't really tell anything about the number of electric field lines, which if defined under any sensible considerations, would always be either zero or infinite through a finite patch of a surface.

Therefore, the textbookish claims of the number of field lines coming out of a point charge of charge $q$ being $\dfrac{q}{\epsilon_0}$ are false.$\dfrac{q}{\epsilon_0}$ is just the flux.

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  • $\begingroup$ Thank u.But you said that the "electric " flux is just the surface integration of the scalar product of electric field and elemental surface area.But what it measures actually? What is the intuition behind it?Does it not that electric flux ="number" of field line passing through unit area of a surface really?Does flux somehow compare itself with velocity of a fluid (like we measure the velocity of a fluid but can't easily measure the number of particle around a point) $\endgroup$
    – Abu sayed
    Commented Aug 16, 2016 at 15:55
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Where did you read this? Field lines are just a qualitative measure, you may draw them as much as you like and they may be inifnite in number.

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  • $\begingroup$ But gauss law states that if we have a charge 'Q' in free space,the total number of electric field line emitting from the charge is Q/epsilon zero.Am I not correct?It stated in my higher secondary book of physics. $\endgroup$
    – Abu sayed
    Commented Aug 15, 2016 at 7:10
  • $\begingroup$ your book is speaking qualitatively. not quantitatively. $\endgroup$
    – The Imp
    Commented Aug 15, 2016 at 7:35
  • $\begingroup$ Gauss law states a mathematical conseft. Field lines are not a mathematical conseft, and are not physical object either. They only helps us in visualizing the electric field. $\endgroup$
    – proton
    Commented Aug 15, 2016 at 10:35

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