…One that does not use the traditional route of formally introducing electric flux first.
Recall that the density of an object is defined as the mass per unit volume. You can stretch or bend or readjust the orientation of the object all you want. Its density might change, but the mass contained within the volume of the object is conserved; no mass is created nor destroyed. And a similar density concept and conservation law will be used to derive Gauss’ Law.
We start by imagining the electric field as a very large body that is cut up into many infinitely thin [simply-connected] shells (that do not fold back on themselves or do weird things like that). Like mass, electric field lines emanating from a source (let’s say at the center of the large body) cannot be destroyed, nor cannot they spontaneously appear out of thin air. So for each thin shell, the amount of field lines does not change. We call that amount N. Intuitively, one can think of stretching a spherical surface containing N back and forth. The density of field lines (which we will define very soon) will change but the amount of field lines cutting through the surface remains the same. Just like the earlier mass example.
We will cut an arbitrary shell into many many small rectangles of area dA that “contain” a small differential amount of field lines dN. Define the field line density for the small rectangle as:
In analogue to the mass density we recalled earlier.
Multiplying both sides by dA and integrating across the shell/surface yields:
Acknowledging that any arbitrary closed shell will contain N field lines.
We’re almost done now! We just have to recall two fundamental concepts in electrostatics:
a) the amount of electric field lines – N – is proportional to the strength of the source – Q; and
b) the closeness (“density”) of field lines – – is proportional to the strength of the field itself – E.
Making appropriate substitutions,
The constant prefactor on the RHS can be experimentally determined, and it happens to be . So finally we get
Or Gauss’ Law.
(Inspired by Professor Bloxham‘s first Physics 7C lecture, which reviewed E&M.)