# Does the limit from a finite to a point dipole violate the superposition principle?

In the course of electrostatics I have come across a basic condition to hold the superposition principle for electric fields, and that is the electric field of one charge should not affect the field of others, that is the first criterion, may be the charges should be placed a minimum distance apart so that one's field can't affect the other field ,and principal of superposition holds perfectly.

Now suppose I have two equal and opposite charges, $+q$ and $-q$, and I place them at distance $d$ from each other. Now, say $d$ is large now so that superposition principle holds, and now I decrease $d$: watching the field vs time, I should see that at first the superposition principle holds, but by the time that $d$ decreases by a significant amount, then it becomes dipole. Now the electric field of a dipole can't be written as sum of two electric fields.

Now my question is how just that happened? All of a sudden the superposition principle does not hold? So what is the deal with the distance between them?

And if the distance should be small for it to be a dipole, then small with respect to what?

If it should be small with respect to field point then what would happen if I place those charges far away from each other but my field point (where I am checking the electric field) is at even further away from them, will then they form dipole?

If the small distance is with respect to something then dipole field is not absolute it is relative! But it makes me crazy and confused. Too much.

You are misunderstanding what the superposition principle says and doesn't say. The superposition principle states that

• for any two valid electrostatic fields $\mathbf E_1$ and $\mathbf E_2$, the superposition $\mathbf E_1+\mathbf E_2$ is a valid electrostatic field.

It does not state that any valid electrostatic field can be written as a superposition of point-charge fields (or any other kind of field). As such, the limiting procedure $d\to0$ does not at all affect the superposition principle - what it does is produce a genuinely new, valid electrostatic field, as the limit of a family of valid electrostatic fields.

Regarding your second question: yes, the precise statement is that a finite-distance electric dipole with two charges separated by a distance $d$ can be approximated well as a point-dipole field when it is interrogated at field points a distance $R\gg d$ from the center of the dipole.

Thus, if your two charges are "far" from each other (whatever that means), but your field observation point is an even longer distance away, then the field can be approximated as a point dipole at that observation point.

And yes, when we manufacture a point dipole out of finite dipoles, the distance is always relative - and that's OK. It doesn't stop the dipole field from being an excellent model in many situations, and it is an integral part of rigorous asymptotic expansions for the electrostatic field produced by a spatially-bounded charge distribution.

For many situations, we just use the point-dipole field as an (excellent) approximation, and there are further corrections (quadrupole, octupole, and so on) available if they are required. However, there are simple charge configurations for which the field is exactly given by the point-dipole field (outside of a certain region), the simplest example being a surface charge on a sphere of the form $\sigma=\sigma_0\cos(\theta)$. (For fancier examples of this in action, see this thread.)

• And I forgot to ask that if I taking the charges far away from each other and field observation point is not at longer distance , but extra what I do is I increase the charge q itself then there field will be significant at field point and so field of both q and -q can affect each other at field point and produce a dipole , so it seems that the minimum distance between two charges to become a dipole is also depended on charge q itself. Is it true? – user101134 May 20 '17 at 3:12
• That's completely unclear, but no, nothing in the limiting process depends on the magnitude of the charge. There also isn't a discrete distance at which a finite dipole "becomes" a point dipole: so long as $d$ is finite, the point-dipole field remains an approximation, and there are always quadrupole corrections of order $d/R$. The smaller that is, the better the approximation, but it is never exact. – Emilio Pisanty May 20 '17 at 11:42