# Electric Flux - What is the point?

Electric flux is a defined quantity that is proportional to the no. of field lines passing through a given area element for a given electric field. It is not proportional to the relative density of field lines, which would supply information regarding the strength of the field at that point. Electric flux, it seems to me, does not supply us with any practical information. It seems to me that electric flux is a quantity defined and modeled specifically for Gauss' law, to introduce some kind of mathematical elegance to it and to introduce an additional visual aspect to the concept of electric fields. Perhaps this is why for symmetrical situations especially, Gauss' Law can be used to easily determine the electric field due to the given charge distribution. Am I wrong here? Is there a physical significance to electric flux that I do not understand?

• Why is "amount of electric field that passes through a surface" not physically significant for you? (You can eliminate almost all defined quantities out of equations by simply substituting them with their definition, so "I can just substitute the integral for it in Gauss' law" is not a good reason) Feb 19 '15 at 16:54
• @ACuriousMind. Yes you are correct, any defined quantity can be substituted with its constituents, unless it is a fundamental quantity like length or time. If we look at the concept of electric field, it is not just a mathematical construct. If we switched an electromagnetic source on briefly, there would be an EM field propagating through space transporting energy. But to me, electric flux seems to be a concept with no physical counterpart, and what other application does it have other than to determine the electric field easily? Feb 19 '15 at 18:49

You're not particularly spot on with your definition of electric flux. Most fundamentally, the electric flux $\Phi$ through a given surface $S$ is defined to be $$\Phi=\int_S \mathbf E\cdot\mathrm d \mathbf S.$$ If you introduce a well-defined model in terms of field lines, then this does end up describing the number of field lines that cross $S$, to within the limits of the model. When the model is accurate, both share the same features: saying that $\Phi$ depends on the number of field lines instead of their density is the same as saying that $\Phi$ stays constant if we increase the decrease the line density and proportionally increase the surface area, and indeed the same is true if you decrease the electric field strength and proportionally increase the surface area.

The reason we define the electric flux is because it is useful. It is precisely the correct quantity to relate the electric field to the existing charges, and this is done via Gauss's law, $$\oint_S \mathbf E\cdot\mathrm d \mathbf S=\frac{1}{\epsilon_0}Q_\text{enc}.$$ This is the fundamental law of electrostatics, really, and it all flows from here (and also the superposition principle). What else do you need for it to be physically meaningful?

Another misconception is to say that the electric flux

is not proportional to the relative density of field lines, which would supply information regarding the strength of the field at that point.

It cannot tell you anything about what happens at any given point because it's not a function of any point, it talks about what happens on a given surface as a whole. And, if you're given a space-dependent vector field and a surface, there's not really many invariant ways to combine them other than through the flux.

• It is implicit that we are dealing with an accurate model of a charge configuration with an arbitrary no. of field lines. In such a model, no. of field lines passing through a given area does not tell us anything about the electric field in that region. If I increased or decreased the area element, the flux would also change. Only when we inspect the relative density of said field lines do we get an idea of the strength of the electric field in that region (not at that point, as you said). Therefore electric flux on its own, is physically insignificant right? Correct me if I am wrong.Thank you Feb 19 '15 at 18:56

Yes electric flux has a physical significance. It measures the charge enclosed inside the surface where we are measuring the electric flux.

We already know electric flux is the number of field lines passing through any arbitary area or surface. So we can say electric flux is directly proportional to the electric field lines. The more electric field lines passing through a area or surface the more the flux is.

Now we can say the electric field lines depend on the total charges.As,if $q$ produces 5 field lines $2q$ must produce 10.

So we can obviously say electric flux has a physical significance.It measures the total charge enclosed inside any arbitrary surface.The more the electric flux is the more charge must be there inside the surface to produce the needed number of electric field lines.And from here Gauss created his theorem I guess.

What practical info. Does electric flux provide us ?

What is the physical significance of electric field ? And according to you, the only reason why this quantity is introduced is to explain Gauss' law !

Well, we must know the physical significance and meaning of "flux" before understanding electric flux. Before I tell you anything you must be able to picture field of physical quantities properly and not just take them as a function of position. I mean talking about apple you must be able to see red round apple in your mind and not just take it as a spell, a collection of letters "a,p,p,l,e"

Field of any physical quantity is an environment where the physical quantity persists with discrete values for different points in the environment. It is the environment influenced by the concerned physical quantity where the physical quantity persists, prevails and can be felt.

Therefore, an electric field is an electrical environment (which has its own speed) where every point has an electric field intensity And a velocity field is a speedy environment where every point has different velocities as in case of flowing water, it would have a velocity field where the velocity of every water particle is defined.

The situations I would describe now would need imagination...so close your eyes and picture them properly.

Consider two pipes through which water flows, the two pipes have same cross section area. In the first pipe, water flows slowly. In second pipe, water flows faster. Tell me, in which pipe the strength of water flow is greater?

Obviously, the second pipe where "the same quantity of water flows faster" Now consider another pipe (third pipe) which is actually a sever pipeline (cross section area is many times greater) where water flows with the same speed as in case of second pipe. Think, in which case the strength of water flow is more ? Well, the third pipe has greater strength of water flow where "with same speed greater quantity of water flows." The strength of water flow in other words mean the strength of velocity field of the flowing water. And we conclude the strength of velocity field is directly proportional to the velocity of water particles and the amount of water particles.

Notice, The strength is defined on the basis of concerned cross section areas i.e. the strength of field is defined for a surface (in this case cross section area).

If we draw the field line diagram of the velocity field of flowing water we notice the relative closeness of field lines (no. of field lines passing perpendicular to unit area) tells us about the velocity of particles. and the no. of unit areas tells us about the amt. of water particles. (in more area more water particles would reside)
(o.t.s would mean 'over the surface' t.t.s means 'through the surface' d.p.t means 'directly proportional to' for the following)
So, we reach the conclusion:
Strength of vel.field o.t.s d.p.t velocity of water particles passing t.t.s
Therefore, strength o.t.s d.p.t no. of field lines per unit area...(1) And
Strength of vel.field o.t.s d.p.t amt. of water particles passing t.t.s
therefore, strength o.t.s d.p.t no. of unit areas o.t.s...........(2)

Using (1) and (2) we get, Strength of vel.field o.t.s is d.p.t no. of field lines passing t.t.s The strength of a field over a surface is nothing but the flux of the field over the surface And we get Flux of a field over a surface d.p.t no. of field lines passing through the surface Now, imagine electric field as flowing water in case of electric field the velocity of water particles would be replaced by intensity of electric field at those points and the amt. of water be replaced by amt. of electric field.

So, Electric Flux provide us the info. About the strength of field over the surface concerned as a whole. The physical significance of flux of any field is in providing the info. About the strength of the field over the surface

The physical significance of electric flux really appears in the Ampere-Maxwell law involving displacement current. Electric flux is necessary to explain the continuity of electric current in capacitive circuits.