Proof for why flux is proportional to number of field lines

Question

What is the proof for this (assuming that we draw infinite field lines).

I understand why flux through some area is proportional to the number of field lines through that area only in the case of an isolated singular point charge. However, I dont understand why this would be true in other cases, like two unequal charges.

To make it clear, when an area has twice the amount of field lines going through it than another area, why is the flux through it exactly twice the other area. Why not thrice or half?

Also, why do field lines that originate isotropically from one charge always end isotropically at another charge.

Answer 1

1. Field lines are just a way to visualize the force and its direction, they do not "exist"! It is arbitrary how many field lines you draw per cm^2, so only the relative density of them tell you something of the strength of the field. Your latter statement is not true, only a isolated charge has isotropic field lines. Since the describe the force between the two charges they have to start and end between them

Answer 2

For many field line (lines of force) are a useful visual aid and it is not necessary to count all the field lines when using the concept, for example, when discussing the electric field (strength) due to a point charge.

Suppose that the strength of an electric field is measure in terms of field lines passing through unit area.

Please note that my graphic skills were not up to drawing areas $A$ and $B$ as curved surfaces.

Assuming spherical symmetry and the radial nature of the $N$ field lines starting out from charge $Q$ and passing through area $A$, the same $N$ field lines must pass through area $B$ which has four times the area of area $A$.

Thus the electric field strength at a distance $2a$ from from the charge is a quarter that of the electric field strength at distance $A$.
This is the inverse square law.

So when you work out the magnitude field at two positions as $2 \,\rm N/C$ and $10 \,\rm N/C$ you could imagine that five times as many field lines pass through an elemental area at one positions as compared with the other position.

As has been pointed out one must be wary when drawing a 2-dimentional diagram which shows field lines and then inferring a numerical value of the field strength by counting the field lines as we live in a 3-dimensional world.

Note that in a 2-dimensional world the electric field from a point charge would not follow the inverse square law.

Answer 3

I would argue that you have the logic backwards. Field lines are a convenient pictorial representation of vector fields; there is no a priori reason for the number of field lines to be proportional to the flux. You could, for example, arbitrarily decide to draw twice as many field lines on the "left side" of a point charge as on the "right side."

In order to (a) remove some ambiguity in how to draw field lines and (b) produce a meaningful picture, we adopt a convention when drawing field lines that the density of field lines at a point $x$ is proportional to the value of the field at $x$. This is a convention -- not something intrinsic about field lines that you can prove.

Once we adopt this convention, then it is the case that the number of field lines poking through a small surface with oriented area $d\vec{A}$ is proportional to the flux through $d\vec{A}$. Loosely speaking, the reason is that the flux is $\vec{E} \cdot d\vec{A}$, and by our convention above the number of field lines at the location of the area element will be proportional to $E$ at the area element. (Since the number of field lines will be the density of the field lines times the infinitesimal area).