Electric flux -- Which of these three definitions is correct?

[sunil36]

I have read three definitions of electric flux in textbook which is confusing me..
1. Electric flux is the number of electric lines passing through any area of a surface.
2. Electric flux is the number of electric lines passing through unit area per second held perpendicularly.
3. Electric flux is the number of electric lines passing through any area perpendicularly.

Whuch one is correct?

 

[Doc Al]

The first one is correct. (Note that "electric field lines" are just imaginary lines to help visualize the electric field.)

What book are you using?

 

[vanhees71]

It's a bit misleading to formulate it in this way. For a given surface $S$ with surface normal vectors aribtrarily chosen in one of the two possible directions (making the surface an oriented surface) the flux of the electric field is
$$\Phi_{\vec{E}}=\int_{S} \mathrm{d}^2 \vec{f} \cdot \vec{E}.$$
The name originates, of course, from fluid dynamics, where the flux of particles can be taken literally: It's the number of particles streaming through a surface per unit time. If $n(t,\vec{x})$ is the number density (momentary number of particles per unit volume around the position $\vec{x}$ at time $t$) the flux is
$$\Phi_{N}=\int_{S} \mathrm{d}^2 \vec{f} \cdot n(t,\vec{x}) \vec{v}(t,\vec{x}).$$

 

[Ibix]

I'd be inclined to say that none is correct, although 3 is closest. The flux through an arbitrary surface $S$ is, as @vanhees71 says, $\int_S\vec E\cdot d\vec S$. If you consider a uniform field passing through a flat surface that simplifies to $\vec E\cdot\vec A$, where $\vec A$ is the vector area of the surface, a vector perpendicular to the surface with magnitude equal to its area. So the flux is the component of the electric field perpendicular to the surface times the area (and the full integral formulation is just the same thing evaluated for each tiny bit of surface and summed over the surface).

Notice that I haven't mentioned field lines. Field lines are a way of visualising a field. Formally, they are integral curves of the field, made by picking a point, taking a small step in the direction of the field vector there, and repeating (this Insight on magnetic field lines has more detail). There are always infinitely many of them passing through any surface, which is why I said none of the three definitions in the OP is correct. However, we usually choose to draw a fairly regular set of field lines emnating from a simple surface, and how many of them pass through another surface is a reasonable approximation to the flux as long as they are pretty much perpendicular to the surface. When the lines aren't perpendicular to the surface then the approximation isn't really useful.

So I would take the third definition and modify it a bit: 3. Electric flux is the number of electric lines passing through any area as long as they all pass through all surfaces of interest perpendicularly. You could apply similar caveats to definition 1, though, which is what I would guess that @Doc Al has in mind.

 

[Doc Al]

I'd be inclined to say that none is correct, although 3 is closest.

I still say that 1 is closest. Note that the electric field "lines" do not have to be perpendicular to the surface of interest, they just have to pass through. And since this is just a means of visualization, you cannot use an "infinite" number of lines: You have to space them out so that the number of lines per area is proportional to the field strength.

That said, I don't think defining flux using electric field lines is particularly useful in most cases. I prefer using the field and the normal to the surface, much like @vanhees71 was doing. A simpler version of that can be found here (for the OP's benefit): Electric Flux

 

[vanhees71]

As Einstein said: You should explain physics as simple as possible, not simpler. Field lines are just a way to visualize vector fields, i.e., you draw a line through a given point which are everywhere tangential to the vector field. An even better visualization is to draw little arrows in addition so that you see also the direction.

A flux is better thought about in the analogy with a literal flux of a fluid, which tells you how much of a quantity transported with the fluid is going through a given surface. Prominent examples for such quantities are mass, electric charge, the number of atoms/molecules making up the fluid, etc.

 

[sunil36]

I'd be inclined to say that none is correct, although 3 is closest. The flux through an arbitrary surface $S$ is, as @vanhees71 says, $\int_S\vec E\cdot d\vec S$. If you consider a uniform field passing through a flat surface that simplifies to $\vec E\cdot\vec A$, where $\vec A$ is the vector area of the surface, a vector perpendicular to the surface with magnitude equal to its area. So the flux is the component of the electric field perpendicular to the surface times the area (and the full integral formulation is just the same thing evaluated for each tiny bit of surface and summed over the surface).

Notice that I haven't mentioned field lines. Field lines are a way of visualising a field. Formally, they are integral curves of the field, made by picking a point, taking a small step in the direction of the field vector there, and repeating (this Insight on magnetic field lines has more detail). There are always infinitely many of them passing through any surface, which is why I said none of the three definitions in the OP is correct. However, we usually choose to draw a fairly regular set of field lines emnating from a simple surface, and how many of them pass through another surface is a reasonable approximation to the flux as long as they are pretty much perpendicular to the surface. When the lines aren't perpendicular to the surface then the approximation isn't really useful.

So I would take the third definition and modify it a bit: 3. Electric flux is the number of electric lines passing through any area as long as they all pass through all surfaces of interest perpendicularly. You could apply similar caveats to definition 1, though, which is what I would guess that @Doc Al has in mind.

I believe that the word 'perpendicularly' is contradicting because the electric flux would still be there(more or less) even if the electric field lines are not perpendicular to the surface. So, Is it necessary to use that word here?

 

[vanhees71]

A flux is a functional of the field and the surface it refers to. I never understood, why the integral form of the Maxwell equations is considered to be didactically favorable to the much more simple local laws, which after all are the fundamental laws of all physics, as is known since the discovery of (special) relativity.

 

[Ibix]

I still say that 1 is closest. Note that the electric field "lines" do not have to be perpendicular to the surface of interest, they just have to pass through.

Yes, but the flux is the component perpendicular to the surface (or the integral thereof), which is why I said 3 was closer. It's true that, as long as there's a consistent angular relationship between a field and a sequence of surfaces, you can just count field lines.

You have to space them out so that the number of lines per area is proportional to the field strength.

Sure. The obvious approach is to pick a surface of constant field strength and start your field lines from a regular set of points. But the point I was trying to elucidate is that this is a finite selection from an infinite number of field lines - hence the counting process is an approximation, and does rely on a reasonable definition of "regular" in your starting points.

That said, I don't think defining flux using electric field lines is particularly useful in most cases.

Agreed.

 

[Ibix]

I believe that the word 'perpendicularly' is contradicting because the electric flux would still be there(more or less) even if the electric field lines are not perpendicular to the surface. So, Is it necessary to use that word here?

This post has only just appeared for me - it wasn't there when I replied earlier, so apologies for apparently ignoring you.

The point is that the flux depends on the component of the electric field perpendicular to the surface (that's what the dot product with the vector area does). So there's a variable relationship between the number of field lines (assuming the caveats Doc Al and I were discussing about uniform spacing), unless the field lines are always perpendicular to the surface. So, in short, I wouldn't use the field lines model unless they were always perpendicular to all surfaces of interest (or at least, all lines always make the same angle with the surfaces).