# Field lines and flux

To my knowledge, flux of a vector field through a given surface is the integral of the dot product of vector field and a unit vector given to a surface element over the entire surface. Field line is a curve such that vector at a point on it is tangent to the curve. But only a few field lines are drawn to help visualise the field and they are deliberately drawn such that the field strength is indicated by degree of closeness of lines.(right?) And I also believe that these principles are applicable in any vector field ( not just electric or magnetic field)

Now my question is that how come the number of field lines passing through a surface is proportional to the flux ? I want to clarify that my question is not in the sense that in reality there should be infinite field lines as in the case of some previously answered questions I have seen. My question is that how, from the definitions known to me as stated above, the following fact can be concluded.

Thanks

P.S. Could somebody also tell if the facts known to me are correct?

• You might find Wolf, Van Hook, & Weeks, "Electric field line diagrams don't work" (Am. J. Phys 64, 714–724, 1996) to be of interest. Section II starts by explaining why we except the electric field to be proportional to the density of field lines. The rest of the article then explains why taking that picture at face value, particularly for 2D pictures, can get you into trouble. – Michael Seifert Feb 4 '19 at 20:34

"The number of field lines passing through a surface" is a meaningless concept because it is always either infinite or zero. The only sense in which this "number" is proportional to the flux is that when people draw pictures of field lines, they try to draw a finite number of them that is in proportion to the flux, to give some visual intuition for flux.

"Field" and "flux" are the important concepts. "Field lines" can be a useful visualization. "The number of field lines" is meaningless.

I think some of your ideas are to resume. For example, the field $$\overrightarrow{V}=x\overrightarrow{{{e}_{x}}}$$ has perfectly parallel field lines and the field component increases with $$x$$.

The field increases when field lines narrow only for conservative flux fields, with $$\overrightarrow{\nabla }\overrightarrow{V}=0$$. In this case, the flux through a section of the same small field tube is constant and therefore, if the section decreases, the field must increase.

I am not sure one can define a "density of field lines" and probably it is not a useful concept.

Sorry for my poor english.

First we need to realize that electric field lines are used to visualize and analyze electric fields and therefore should be considered as a pictorial tool as opposed to a physical entity. For example, one cannot think of the space between the field lines in the picture, which, for example, get farther and farther apart the further away we go from the point charge producing the field, to be “devoid” of an electric field.

Instead of thinking of the strength of in terms of the individual, discrete field lines, think of the electric field as a continuum or smear that gets fainter and fainter the farther away from the source. The benefit of the "lines" picture vs the continuum picture is the former also allows one to see the direction of the field.

Hope this helps.