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Is it possible to have a capacitance in a system of two point charges? Since there is a potential energy between them and they both have charges then we can divide the charge by the potential and get capacitance.

However, capacitance is supposed to depend only on geometry so should therefore be zero. How does one resolve this paradox?

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How does one resolve this paradox?

As a general prelude to this answer, I would like to mention that it is well known that classical point charges lead to some unresolvable paradoxes in classical EM. Personally, I do not consider this an indication of an inconsistency in classical EM, but an indication that classical point particles themselves are inconsistent. So what remains here is to determine if this specific case is an instance of an inconsistency.

However, capacitance is supposed to depend only on geometry so should therefore be zero.

It is true that the capacitance depends only on the geometry, but that does not immediately imply that it should be zero. A pair of point charges does have some geometry, specifically the distance, $s$, between them. So all we can say from this is that the capacitance should be some function of the distance between them $C=C(s)$. While we could indeed have $C(s)=0$, that is by no means guaranteed.

Since there is a potential energy between them and they both have charges then we can divide the charge by the potential and get capacitance.

This is actually a little bit incorrect. The potential energy between two point charges is undefined. You can extract an infinite amount of energy from a system of two point charges simply by letting them get sufficiently close together. This is one of the major problems of classical point charges and this fact leads to many of the genuine paradoxes.

If we naively plug in infinity then we get $$C=\frac{Q}{V}=\frac{Q}{\infty}=0$$

Of course, since $\infty$ is not a real number, this method is more than a little suspect. But the voltage at the surface of a spherical charge $Q$ of radius $R$ is $$V=\frac{Q}{4\pi \epsilon_0 R}$$ so $$\lim_{R\rightarrow 0}\frac{Q}{V}=0$$ This result then gives a bit more confidence in the $C=0$ result.

So, although this particular aspect of classical point charges does reach close to the root of many paradoxes, it does seem that $C=0$ is not itself paradoxical.

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    $\begingroup$ How is the potential energy between two point charges undefined? Isn't it like q^2 / r ? $\endgroup$
    – DarthAster
    Dec 6 '21 at 15:46
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    $\begingroup$ @DarthAster The closer two point charges (of opposite charge, i.e. attracting) get, the larger the force between them and thus higher the gradient of potential energy. I'm not sure where your formula comes from and it does not seem to have units of energy, but even in that as r approaches 0, the result tends to infinity. $\endgroup$
    – jpa
    Dec 6 '21 at 16:56
  • $\begingroup$ @DarthAster as jpa said, if you take either charge as the reference then the other charge has infinite voltage. $\endgroup$
    – Dale
    Dec 6 '21 at 18:05
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In most common cases, the concept of capacitance was used to describe the system where two sets of conductors are separated by vacuum or some dielectric medium.

For these cases, the charge is assumed to distribute (or be trapped) on the surfaces of the conductors. If we assume that the charge is continuously distributed (therefore the charge distribution could be solved within classical mechanics and only depends on the geometry) and the medium is linearly polarized, then we could well define the capacitance of the system since the potential is just proportional to the charge.

However, for the case of point charges, the charges are no longer continuously distributed. Thus if there are some geometries, the distribution problem of the charges could be quite complicated (we may need quantum mechanics and introduce some advanced many-body approaches to solve it).

Nevertheless, solving the charge distribution should not be our business here. I would like to point that the definition of capacitance relies on the energy difference for adding/removing a quantity of charges. If we ignore the "self energy" that comes from the interaction of the same charges, and somehow can fix the charge distributions, we can still have a capacitance for the system.

However, it seems that these conditions can not be easily fulfilled by the point charge case. Therefore I think the capacitance is ill-defined for this case, and it does not make sense to say the capacitance "only depends on geometry".

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Last page in this link:

Capacitance of an Isolated Sphere

We obtain the capacitance of a single conducting sphere by taking our result for a spherical capacitor and moving the outer spherical conductor infinitely far away

Result:

$C=4 π ε_0 R$

where R is the radius of the sphere.

As you are talking of point particles, $R=0$ .

One could extend the logic to two point particles, two spheres, and again $R=0$ for each will give zero capacitance.

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    $\begingroup$ Your link is broken but it appears you are calculating the capacitance between a point and a spherical shell an infinite distance away. This is not what was asked. $\endgroup$
    – Matt
    Dec 5 '21 at 13:30
  • $\begingroup$ in the link, I hope I find it , it calculates the capacitence of a single charged sphere by using the model of two spheres in a shell and sending the outer cell to infinity.. I hope I find the llink and correct it. $\endgroup$
    – anna v
    Dec 5 '21 at 13:52
  • $\begingroup$ Here is the google search google.com/… it is a purdue u, lecture $\endgroup$
    – anna v
    Dec 5 '21 at 13:59
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If we talk about capacitors that can be charged electron by electron, then these are an everyday reality in modern nanostructure physics (for the past few decades already): see Coulomb blockade.

Remark: there is some ambiguity in the question, since it attributes capacitance to charge itself, rather than a structure/conductor containing charges.

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