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Gauss Law 2D

So I went through the Khan Academy tutorial on divergence and flux calculations for an area C encircled by a parametric function $s(t)$.

Flux

Here $C$ will be a circle with a radius $r$, centered by a point charge $+q$. $F$ will be the electric field vector $E(s(t))$, and $n$ will be the normal vector for the point $s(t)$. As electric field lines permeate outwards linearly, they are parallel with the normal vector. Hence $|E|\times|n|\times cos(\theta) = |E|\times1\times1 = |E| = \frac{q}{4\pi\epsilon_0r^2}$

For sake of calculating $ds$, we will still need a definition of $s(t) = [rcos(t), rsin(t)]$.

$$s'(t) = [-rsin(t), rcos(t)]$$ $$ds = |s'(t)|dt = r*dt$$

So let's bring them all together,

$$\int \frac{q}{4\pi\epsilon_0r^2}\times r*dt =$$ $$\int \frac{q}{4\pi\epsilon_0r}\times dt =$$ $$\frac{q}{4\pi\epsilon_0r} \int dt$$

We have to take the integral for the time interval $(0, 2\pi)$, a full period of $s(t)$. This will give us the following end result:

$$\frac{q}{4\pi\epsilon_0r} \times 2\pi = \frac{q}{2\epsilon_0r}$$

It is okay, except that annoying $r$ at below. What mistake I did here? I can't see. Please help. Also is experimenting with stuff like good for a student? Or should I just stick to my curriculum?

Qmechanic
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İbrahim İpek
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    Hint: How does the electric field fall off in 2D? Possible duplicates: https://physics.stackexchange.com/q/582104/2451 , https://physics.stackexchange.com/q/44515/2451 and links therein. – Qmechanic Mar 09 '23 at 07:55
  • The original gauss law would be $\frac{q}{2\epsilon_0}$ but we have an extra $r$ here – İbrahim İpek Mar 09 '23 at 07:55
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    Try n-dimensional space, with potential $1/r^m$ (and sphere surface $~r^{n-1}$. It won't work for just any $m$. – Roger V. Mar 09 '23 at 07:59
  • Hmm... So in 2D, $E = q/r \ne kq/r^2$, and this is the mistake? This is weird. I thought same formulas apply – İbrahim İpek Mar 09 '23 at 08:08
  • You need to distinguish two-dimensional space and a three-dimensional space with system homogeneous along one of the dimensions (which confusingly is also referred to sometimes as "two-dimensional"). – Roger V. Mar 09 '23 at 08:23

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