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Is there is any difference between Electric Flux ($\Psi$) and Total number of Electric Field lines ($E\times$Area)?

$\psi = \Sigma Q$, and $\phi$ = Electric field intensity $\times$ Area

where Area = Total surface integral over Gaussian surface

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I would say that "number of electric field lines" is an inexact way of phrasing things. It's not as though the electric field lines are real physical objects that you can count. But it works well to convey the intuition of what flux is. –  MatrixFrog Dec 27 '12 at 7:09
    
@Crazy Buddy - Is the expression "Area = Total surface integral over Gaussian surface" correct? –  Inquisitive Dec 29 '12 at 13:43
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@Inquisitive: Hi Inquisitive. That you might wanna ask the author. I'm just the editor who reproduced his writing. BTW, Yes, Its right that $A=\oint ds$ where $ds$ is the closed surface (taken as the Gaussian surface). He forgot to mention that in formulas :-) –  Waffle's Crazy Peanut Dec 29 '12 at 13:57

3 Answers 3

"Number of electric field lines" is not a well-defined notion. In a picture, you can always choose to draw twice as many field lines out of every positive charge so there's clearly no meaning to that phrase.

Flux is an attempt to rectify the problem I just stated. That is, the electric flux through a surface is quantity that is (1) well-defined and (2) tends to be proportional to the number of lines you would draw in a picture going through a surface.

Flux is defined as follows: Let $\bf{E}$ be the electric vector field and let $S$ be a surface (more precisely, an oriented 2-dimensional submanifold of $\bf{R}^3$). Let $\hat{n}$ be a unit vector field on $S$ that points orthogonal to the surface everywhere. Then, the flux is $\int_S \bf{E} \cdot \hat{n} dA$ where $dA$ is the area element on $S$. This is the mathematical quantity which best cooresponds to the idea of the number of lines through $S$

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The electric flux is defined as the number of electric field lines passing through a surface.

I'm going to assume by "total number of electric field lines" you mean the total number of electric field lines passing through the surface of a sphere surrounding the charge.

The flux of the electric field is $$\phi = \oint\bf{E}.d\bf{S}$$ integrated over the whole surface of the sphere. In this case, the flux of the field will go down exactly as the number of field lines per unit area of the sphere, i.e., as the radius of the sphere increases, the flux of the field and the number of field lines per unit area will decrease by the same amount.

More generally, the flux through a surface is always proportional to the number of lines of force passing through that surface.

Also, the number of electric field lines is not given by $\bf E \times $ Area, because if the area you are integrating over increases, that implies that the total number of field lines increases, which in turn implies that the flux through all surfaces regardless of how far away it is from the source would either be increasing or constant. The total number of field lines are constant for any given charge.

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Flux is the surface integral of the electric flux density $\boldsymbol{D}$, which is related to the electric field $\boldsymbol{E}$ by the dielectric constant $\epsilon$: $$\boldsymbol{D}= \epsilon \boldsymbol{E}$$

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