Gauss Law in 2D

Picture the electric field as something streaming out of a charge. Now wrap any closed loop around it and count how much of that stream crosses the loop on the way out: that total is the flux. Gauss's law says something remarkable, the flux depends only on the charge enclosed, not on the size or the shape of the loop, and not on where the charge sits inside it. Stretch the loop, dent it into a blob, slide the charge around inside, and the flux does not budge. Move the charge outside and the flux drops to exactly zero, because every field line that enters the loop on one side leaves on the other. The scene streams the field through a Gaussian loop you can drag and deform, with the outflow and inflow marked along it; the diagnostic is the flux contribution all the way around, whose shaded area is the total flux, landing on the enclosed charge every time.

Figure 1. Gauss's law in 2D. Top: the field of point charges streaming through a Gaussian loop (draggable, resizable, deformable); red marks outflow and blue inflow along the loop. Bottom: the flux contribution E.n around the loop versus arc parameter; its signed area is the total flux, equal to the enclosed charge over epsilon-zero. Method: Simpson integration of the closed line integral.

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WHAT TO TRY

Drag the loop off the charge: the flux drops to zero, every line that enters also leaves.
Change the shape to ellipse or blob and resize it: while the charge stays inside, the flux does not change at all.
Pick the + and - pair and enclose both: the net charge is zero, so the flux is zero even though the field is strong.
Drag a charge across the loop boundary and watch the flux jump by one full unit as it crosses.