I've been looking on the net for electric point dipoles but couldn't find something useful (at least to me). The strange thing about an electric point dipole (the dipole moment pointing to the right on the x-axis, which could just as well point in any direction, but this the convention) is that in the dipole point the charge is zero, of which you expect no electrical field lines to emerge. Yet, it radiates electric field lines emerge outward from the point and entering the point symmetrical on the other side of the vertical axis. I know an electric dipole at a great distance can be approximated by a point dipole. But the "point" is, why constructing a point dipole? Does this make calculations easier, while keeping in mind a point dipole doesn't really exist?
Now is this a mathematical construct, with no connection to reality (at least, I can see none, except for making calculations easier so this point dipole has no realization in the real world)? If we bring two equal charges infinitely close together, the charges will add up to zero when being on top of each other. And above that their distance becomes zero so how can we speak of a moment (you can, of course, let the charges grow to infinity, but these are fixed)? So how can a neutral point object have field lines coming out and getting into it?
Is this picture the depiction of the passing of the real electric dipole moment to a (supposed) non-existent electric point dipole? I can't understand how an electrically neutral point creates electric field lines, wich is the case as the blue and red charge coincide. And what's the use (apart from maybe simplifying calculations, in which case it's certainly non-existent)?
Answer
There are no observed point-like electric dipoles (though at least one type of particle is predicted to have one: the $W$-boson). The smallest permanent dipoles I, personally, know have been seen are molecular in size (e.g. water, carbon monoxide, and phosphine). You can even build up ferroelectric materials that have a macroscopic permanent dipole moment like magnets have a magnetic one.
That said, the dipole fields are a useful component in calculations that even pop up in practical applications. When I say, "dipole fields" that isn't a typo, there are two types of dipole fields. The first is the type discussed in the question, the "outside" dipole field that has a potential that falls like $r^{-2}$ and electric field that falls like $r^{-3}$. That field is useful when calculating the fields outside of an imaginary sphere centered on the origin that contains all of the charges.
The "inside" dipole field is something you've already encountered in your studies. A constant electric field is, in fact, the field produced inside of a dipole. With the point charges in the original question, you just take the limit the other direction in distance, but also with increasing magnitude of charge, such that the electric field halfway between them is constant. "Inside" multipole fields are used when you're working inside of an imaginary sphere sitting at the origin that contains no charges.
The mixed case is difficult to describe with words, but straightforward algebraically. The basic of it is to separate the charges into the two above cases, then add up the results.
In terms of building a practical dipole, if you place a conducting sphere in a constant electric field the charges you induce on its surface produce a dipole field that is, in the ideal situation of uniform external field and perfectly spherical conductor, also perfect. In more detail, they produce an "outside" dipole field outside of the conductor, and the exact "inside" one needed to cancel the external field inside the conductor.
Notice how I said earlier that a constant electric field is a dipole field, and it induces a dipole in a spherical conductor? That's not an accident. The math behind dipoles, and other multipoles, has a deep connection to rotations (group theory), linear algebra, etc that are at the center of more advanced physics.
The other thing to keep in mind is that even though we haven't seen any physical point-like electric dipoles, point-like magnetic dipoles are everywhere: electrons, muons, all of the quarks, etc.
The structure of the dipole field is universal, so studying the ideal electric dipoles helps you understand the very real magnetic ones. Compare the electric field of an electric dipole $$\mathbf{E}(\mathbf{r}) = \left(\frac{1}{4\pi\epsilon_0}\right)\frac{3(\mathbf{p}\cdot\hat{r})\hat{r} - \mathbf{p}}{r^3}$$ with the magnetic field produced by a magnetic dipole $$\mathbf{B}(\mathbf{r}) = \left(\frac{\mu_0}{4\pi}\right)\frac{3(\mathbf{m}\cdot\hat{r})\hat{r} - \mathbf{m}}{r^3}.$$
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