Talk:Capacitance

Wrong citation for examples?

In section https://en.wikipedia.org/wiki/Capacitance#Capacitance_of_conductors_with_simple_shapes , the "Pair of parallel wires" has the reference of Jackson's book, but the indicated page does not say that. Problem 1.7 of 3rd edition of his book gives an approximation of the formula though. — Preceding unsigned comment added by 179.154.141.69 (talk) 00:01, 25 February 2019 (UTC)

I agree. The AcrCosh form is substantially different from Jackson and the logarithm form does not appear in Jackson but it does converge to converge Jackson when d >> a. Constant314 (talk) 00:50, 25 February 2019 (UTC)

Rewriting the Wiki page

Hello to all concerned,

I would like to help improve the capacitance page. I have a strong background in electromagnetic theory and equally strong working background with bulk capacitors and strays. I am an analog designer that formally worked for Hewlett Packard Company / Agilent designing precision front end signal conditioning for high end digital multimeters. If the parties that monitor and oversee this page would be interested in my participation, please contact me via EMAIL or skype:

   [email protected]                   name:  George Hnatiuk

   skype account:   georgeandkhan

Thank you July 11, 2019 — Preceding unsigned comment added by 140.186.255.232 (talk) 05:00, 12 July 2019 (UTC)

George. We do not usually correspond by email because it compromises anonymity. If you create an account, it will be easy to communicate with you.Constant314 (talk) 02:23, 13 July 2019 (UTC)

The expression for an oblate spheriod is wrong

The correct value could be found in http://www.coe.ufrj.br/~acmq/tesla/capcalc.pdf. The given reference is very dubious (non-technical). Furthermore, an oblate spheroid rarely occurs, it is suggested to remove the entry.radical_in_all_things (talk) 12:37, 10 August 2019 (UTC)

The formula should reduce to the formula for a sphere when a=b, but it does not. So I agree, it probably is wrong. Also, the source does not specify the page number. I agree with removing the entry. Constant314 (talk) 19:31, 10 August 2019 (UTC)

A conductor is generally not necessary for an object to have capacitance.

I notice that the opening sentence of this article states

"Capacitance is the ratio of the amount of electric charge stored on a conductor. . ."

This is not a general definition. Conductors are not a prerequisite for a capacitor.

In general, capacitance is the ability of an object to store charges. That's as general a statement as one can make! It includes both concepts of self-capacitance and mutual capacitance as described later in the article.

An object may be a box of air, a silver metal plate, a brick of silicon semiconductor, a window pane (typically a silicon oxide "insulator"), or even in more current electronic technologies biomolecules and other organic structures. In fact, the most sophisticated computer (according to many) is the human brain that has absolutely no metallic interconnects whatsoever -- no "conductors". In the nanoscale, single silver or gold atoms do not have conducting properties like their bulk forms, but each atom has a capacitance. These form point contacts in quantum nanoelectronics. A single quantum dot may have no conducting properties whatsoever, yet each has a particular capacitance -- the ability to store charges.

Though it is true that in most conventional capacitors, charges are found in metallic structures, it is generally not true that all capacitors have metallic structures.

Nor is capacitance an exclusive property of electronic capacitors. Hence, "electric charge" is also not part of a general definition of capacitance. It's a conventional concept that electrons are the exclusive domain of capacitive devices, but we historically have capacitive devices in which ions and protons can be stored that have capacitance as well.

Even the singular term "charge" isn't general enough as it doesn't reflect the physical nature of discrete electrons, protons, nor ions. Thus, "charges" (plural) is part of a general definition.

In the least, I would rather this article not include "conductor" in its definition of capacitance.

Conventionally, in accordance with far too many textbooks, however, we can disagree that "electric charge" is appropriate, but it doesn't help beginning scientists today who more frequently use, study and develop technologies that involve capacitances that have nothing whatsoever to do with "electric charge".

Just my two cents' worth for the penny you may have offered for my thoughts. Knowledge is profit. TJ LaFave (talk) 17:19, 19 October 2022 (UTC)

I completely agree. In fact, I would argue that "charge" or "charges" is not even general enough because capacitance is really an abstract notion like that of a an empty bowl. It is a geometric potential to hold generalized charge or hypercharge. It need not be electric charge at all. Although to connect it back to electrostatics or electrodynamics one would like to think about distributing classical Maxwellian electric charge over the geometric region. Please go ahead and make the changes you suggest, especially to the opening paragraph. As you point out, it is completely wrong. It is somewhat even worse that it makes a link to the page on conductors, which is neither necessary nor sufficient (an open ended idealized conductor where charge "slides" out of it is not a capacitor at all). MMmpds (talk) 19:45, 20 October 2022 (UTC)

I disagree that it is completely wrong. I would characterize it as incomplete. Remember, the lede should be written at about a ninth-grade reading level. We cannot cover it all in the lede; that is what the body of the article is for. What is there, is a good starting point. Also, given enough time, just about all objects, even rocks, are conductors. I suppose we could talk about the charge on an equi-potential surface without discussing how the surface became equipotential, but saying conductor pretty much implies an equi-potential surface without getting bogged down in technical terms. Constant314 (talk) 01:42, 21 October 2022 (UTC)

It is true that as with most concepts, the notion of capacitance can be subjected to various generalizations. However, for the reader wanting to understand electrostatics, such abstractions are unhelpful. It is surely easier to consider the capacitance of conducting bodies before considering the capacitance of equipotential surfaces in general. Useful generalizations of a concept can be covered in a Generalizations section near the end of the article. catslash (talk) 19:07, 21 October 2022 (UTC)

@Constant314 A 9th grade level varies greatly across a given country and the planet. In addition, one ought to distinguish between 9th grade level and fictional or mythical level. Simplification does not require misrepresentation of the truth. This includes lies by omission. One might think that omission is unavoidable given the goal of simplicity and brevity (such as an article lead) but my experience teaching and explaining very difficult things to humans of all ages, stages of study/professional status, and a wide variety of natural and prepared skill level has proven to me that it is disingenuous to talk down to an audience/readership of any type. The real issue is one of creativity on the part of the writer to know when to rely on metaphor for simplicity, in order to draw on common human experiences and normative understanding.

The point made about the rock is an excellent point because it shows that capacitance in its full generality (but not so abstract as to be confounding) is nothing more than a metaphoric receptacle to hold generalized charge (which itself is tricky to define). Charge is really no more fancy than points or objects, which attract or repel (including things like fixed points or configurations of stability/instability).

There is also no need for the mentioning of equipotentials in the lead. Actually, equipotential need not even appear in any article/passage on capacitance because the equipotential nature is relative to the structure (composition and geometry) of the capacitor. This is automatically encapsulated in the notion of a metaphoric receptacle. That is to say at some point, the charges must be held fixed in a given region. In short, it can be taken as a given.

Incomplete is one of many definitions of wrong. Think about the consequences of incomplete data: an incomplete phone number, an incomplete homework or test assignment, an incomplete job or legal application/form, an incomplete testimony, an incomplete transaction, etc. Each one of these are wrong on more than one level. I very much think User:TJ Lafave should make changes consistent with the observations pointed out. MMmpds (talk) 19:09, 21 October 2022 (UTC)

(Please excuse my increment of your indentation there). What definition do you propose for the capacitance of a metaphoric receptacle - and can you cite sources for it? catslash (talk) 19:17, 21 October 2022 (UTC)

No worries on the indentation, however, I am using the "reply" button so I am assuming the indentation is as intended. The question asked will push me to re-writing the introduction, which I do not have the intention of doing. But to be fair since the question is a reasonable one, I can perhaps make a brief sketch. I think starting out with the first historically understood notion is perfectly fine and then moving on to the more modern generalized version. The rest of the article can then be filled with examples and exposition that show how the specific is tied to the general. As for a source, "Introduction to Electrodynamics" 2nd ed by D. Griffiths initially p.106 defines capacitance as a very general constant of proportionality and then goes on to say that it is "determined by sizes, shapes and separation...". Admittedly, it then mentions conductors but as has been discussed, this is not necessary to introduce (in the part of the lead devoted to generality). MMmpds (talk) 19:36, 21 October 2022 (UTC)

Replying to myself, personally, I would have added the word "composition" to the defining list D. Griffiths gives. But again, since in that section the topic there is in the context of idealized conductors, it is understandable why the word was not added. But it does not do the reader justice to define important and quite general concepts in physics and really in any science with only narrow examples in mind. It might be a nice starting tool for beginners but it cannot hurt to add the mention of the generality. In an elementary context one can simply skip elaboration on the generality. However in a robust encyclopedic article, it should fairly represent the full breadth of the current human understanding of the concept to date. MMmpds (talk) 20:06, 21 October 2022 (UTC)

While you have brought up Griffiths' text and his practical introduction to the entire subject of electrodynamics, I agree in his introduction to the capacitance subsequently followed by how capacitance may be determined. I hesitate to accept his introduction of capacitance as a general constant of proportionality, however, as capacitance is a very real, tangible, physical quantity.

We can define π as the ratio of two physical quantities, but this leaves π as a dimensionless number. Capacitance isn't dimensionless. It is a physical quantity with units of Farads because mathematically, in electrostatics it is the ratio of charge to potential difference -- two different physical quantities. Thus, this opening statement to define capacitance is disingenuous about the physical nature of capacitance. With π it's clear there is some number relating two measurable distances. It's merely a unitless number. The ratio of one charge to another charge is a unitless mathematical number. Capacitance isn't a number, and this opening statement about a physical quantity requires additional explanation -- especially if we want to constrain this encyclopedia to an arbitrarily defined 9th grade reading level.

Opening with a mathematical statement about two different physical quantities and then muddying the definition with yet another specific material property (conductor) all in the same sentence/thought isn't something any introductory science student would digest readily. Why not just give a tangible explanation like one 9th grade friend to another would do? An object you can hold in your hand. Any object. It can store charges in it. How many charges? Well, that's the capacitance of that object. Now, let's talk about what those charges are, how they're stored, how different objects store charges, and so on.

The idea that a conductor is a necessary prerequisite in the definition of capacitance because it represents an equipotential surface puts the cart before the horse. It's therefore a bad definition.

Also, to address the idea of how people in "the trades" might define capacitance, as an example, a company that manufactures tools that measure capacitance don't mention conductors either: "Capacitance is the ability of a component or circuit to collect and store energy in the form of an electrical charge.(Fluke)"

Finally, w.r.t. Griffiths' text, it's great. It's practical, but on the subject of storing energy in a dielectric with free charges -- since you're an educator -- consider his "spring energy" term -- and think about why it's completely wrong to include it. But it's a great way to get the result to agree with Jackson's expression, so compare it with Jackson's derivation that includes holding external charges fixed (unphysical and not a general case, but it's a good assumption if we assume there are metallic plates nearby providing the electric field that polarizes the dielectric. If there is no movement of these external charges, how is this term zero in general?) Jackson admits there's an energy term missing in the general case of stored energy. Griffiths goes the extra mile to get the same non-general solution, but presents it quietly as though it were a general solution. ....it's another reason why use of "conductor" here is very shortsighted and destructive to an introductory science student/reader. TJ LaFave (talk) 16:42, 27 October 2022 (UTC)

I admire that you are a teacher, but when teaching in person you can get immediate feedback and adjust the level. Wikipedia articles must be a wide selection of readers. Yes, the lede must be written down to a lower level. Of course, it depends on the subject. An article about a calculus topic could have calculus in the lede. However, capacitance is a trade school topic and perhaps a high school topic. Wikipedia articles, especially the lede, should not be written at the level of people who already understand the subject.

But, aside from that, any formula for capacitance that uses the total charge on a surface tacitly implies that the surface is an equipotential surface, because otherwise you would have to consider the charge distribution. If you do not have the charge on an equipotential surface, then there is no unique capacitance for a given total charge. Of course, simply saying that the surface has a voltage also implies an equipotential surface. Specifying a conductor is a simple way to tacitly imply an equipotential surface without a lot of ancillary explanation. Constant314 (talk) 01:29, 22 October 2022 (UTC)

Recall that an encyclopedia is not a book on an introduction to topics. It is meant to be the sum total of all that is known on various topics. However, since the inception of the idea of the encyclopedia, human knowledge has grown exponentially and one had begun to see the rise in encyclopedias of specialized topics such as the encyclopedia of mathematics or life sciences etc. With Wikipedia, one is not constrained by the number of pages or volumes in a set. Ergo, Wiki articles can and do contain vast amounts of information on highly specialized topics. There are even special articles devoted to the introduction to certain more advanced topics. The name of the article here is "Capacitance" and not "Introduction to (electrostatic) capacitance". Compare General relativity to Introduction to general relativity. Therefore, there is no issue with including the more abstract and more general in the intro of this article on capacitance. MMmpds (talk) 18:20, 23 October 2022 (UTC)

Yes, put all you want down in the body of the article. Keep the intro simple. Constant314 (talk) 19:30, 23 October 2022 (UTC)

The current introduction is NOT simple.

The first sentence says something like "capacitance is a mathematical ratio of two different physical quantities, and by the way, there's a conductor involved." It sounds like the opening line of a bad sci-fi movie where the main character flaps their hands in the air and then says "ugh, nevermind all that. I'll explain when we get there."

Simple would be something like "Here's an object in hand. How many charges it can hold is its capacitance. . . Now let's talk about how it stores charges. . ."

Also, as I've noted above, a company that makes tools that measure capacitance define it without mentioning conductors at all: "Capacitance is the ability of a component or circuit to collect and store energy in the form of an electrical charge.(Fluke)" That's a very substantial way to introduce a tangible concept to a novice tradesperson. That statement literally sells products. The buyer has MANY applications for that product. They have many kinds of components and many kinds of circuits to measure.

As well, intentionally talking down to introductory science students does not help them. It puts a glass ceiling over their heads. The best success I've had with high school students (through outreach programs -- as I work at the university level) is to speak to students as human beings working right alongside me just as any tradesmen would do on a job site. TJ LaFave (talk) 17:02, 27 October 2022 (UTC)

@Catslash If I understood the argument correctly, it is more of a philosophical one as to whether an individual prefers top down or bottom up thinking. There is no single correct way to do things. However, since the only other redirect is to physiology, I think it is important to impress upon a reader learning for the first time that capacitance exists outside of electrostatics despite being in the section on articles about electromagnetism. It is merely an historical coincidence (due to the structure of matter and its distribution throughout the universe and specficailly here on earth) that capacitance first arose in electrostatic theory. There is absolutely no reason why a reasonable introduction cannot do both and still remain a size in character count appropriate for an article lead. MMmpds (talk) 19:24, 21 October 2022 (UTC)

It is possible that I was missing the point. If there is something called capacitance in (say) economics, then the proper course is to have an article called Capacitance (economics), an article called Capacitance (electrostatics), and a disambiguation page to direct the reader to the one relevant to their interest. catslash (talk) 19:35, 21 October 2022 (UTC)

There is a generalized mathematical concept but it is named differently here on Wikipedia as Capacity of a set. Nonetheless, the mathematical abstraction is based upon the physical concept. MMmpds (talk) 19:40, 21 October 2022 (UTC)

This type of capacitance is new to me. It seems to be a straight-forward generalization to n-dimensional Euclidean space. The article says "...with the boundary conditions u(x) = 1 on Σ and u(x) = 0 on S...", which I take to be a statement that Σ and S are equipotential hypersurfaces. catslash (talk) 20:02, 21 October 2022 (UTC)

Yes precisely under those special choices The real beauty is that it allows one to generalize the purely geometric and composite extent of capacitance to abstract mathematical objects like sets, surfaces, topologies etc. MMmpds (talk) 20:10, 21 October 2022 (UTC)

• A point that is being missed from this discussion is that, on Wikipedia, the lead is meant to be a summary of the article (WP:LEAD). If the body of the article has no discussion of more esoteric or generalised definitions of capacitance then that certainly should not be in the lead either. If there is another article covering this, then a hatnote or see also entry would be helpful. SpinningSpark 11:25, 22 October 2022 (UTC)

  I have yet to review the entire discussion above, but I have glanced over it coarsely.

  Your point is poignant. This article does involve more esoteric and generalised definitions and descriptions of capacitance. It's largely why I've questioned the integrity of the statement. TJ LaFave (talk) 16:02, 27 October 2022 (UTC)

  But no, it doesn't, or at least not until one gets down to the "Nanoscale systems" section. For the great bulk of the article conductors are assumed. Nanoscale issues might deserve a sentence in the lead, but it doesn't justify a massive restructuring of it. A massive restructuring of the article is not justified because something funny is going on at nanoscales. For the vast majority of situations, the article is fit for purpose already. SpinningSpark 13:18, 28 October 2022 (UTC)

  So, yes the article does include discussion about generalized definitions and descriptions of capacitance, but no it doesn't in its beginning. This is akin to describing a page on automobiles starting with a Maserati as part of the definition.

  A massive restructuring is not necessary. The opening sentence is simply too specialized and requires special assumptions of equipotential that the reader isn't exposed to until a more comprehensive read. As I've noted in other parts of the discussion above, the opening sentence essentially says the following:

  -- capacitance is the mathematical ratio of two different physical quantities and oh by the way it somehow involves something called a conductor. --

  This is not a definition as it requires explanation of a well-defined material equipotential surface for justification. It's unjustified because many capacitors and other objects that exhibit the physical property of capacitance have no such well-defined material equipotential surfaces -- emphasis on material.

  Also, there isn't anything "funny" happening at the nanoscale. In fact, the expressions for capacitance in the nanoscale sections are general, and if one applies them to macroscopic capacitors, one recovers the macroscopic expressions -- but only in the presence of well-defined equipotential surfaces (without the necessity of a material conductor, incidentally).

  A great macroscopic example of capacitance that frankly is at your fingertips is the capacitive touchscreen. A common conducting material, indium tin oxide, which isn't generally a conductor as it can be used as a dielectric and so forth, is used in many capacitive touchscreens. Another great example are biological systems like nerves and ion pumps -- neither of which have any material conductors in them.

  But, your original point is correct. "The lead is meant to be a summary of the article." The article includes a lengthy discussion of nanoscale systems that may or may not depend on conductors. So, the lead should either be, as you suggested may be the case, massive[ly] restruct[ed] or introduce the article with a practical definition of capacitance suitable as a summary of the article.

  In short, my argument is that the current first sentence misleads the reader with the false assumption of a conducting material (a conductor) as a prerequisite for a capacitor to have capacitance. It's telling a lie and then later walking it back. TJ LaFave (talk) 15:23, 28 October 2022 (UTC)

  @Tjlafave I think you have hit the nail on the head. Well done. I particularly like the Maserati metaphor. MMmpds (talk) 23:12, 29 October 2022 (UTC)

  I offered the Maserati metaphor because of its elite position in the field of all automobiles.

  The use of conductors to describe capacitance is precisely that -- an elite example. This is because conductors were a technologically common way to access capacitive properties in electronics and microelectronics for decades while today's students -- scientists and engineers -- are being more and more exposed to capacitances in systems that have none.

  Conductors make the math easier and are conventionally part of electronic circuitry. Today we can set up circuitry with dielectric waveguides, point contacts that ultimately depend on the electric properties of a handful of "metal" atoms, and even biological action potentials in nerves.

  I don't understand why when I come to the page on capacitance I'm told that a Maserati is the prototype for the concept of an automobile when any horseless driven box on wheels will suffice. Even Mies van der Rohe didn't expose beginning architecture students to the secrets of his work. He showed them how to make good boxes and then how to make buildings functional things. (esoteric example? maybe) TJ LaFave (talk) 19:17, 2 November 2022 (UTC)

I would like to thank editor @Kbrose for implementing the astutue observations of editor @Tjlafave. MMmpds (talk) 04:48, 29 October 2022 (UTC)

Isn't the ability of a material to store charge called capacitivity rather than capacitance? Constant314 (talk) 18:50, 29 October 2022 (UTC)

So I can say with certainty the modern answer in the 21$^{\text{st}}$century is no. The only correct technical term is "capacitance". I would have added in scientific standard American English, except that what makes the answer so definitive is Gustave Choquet's mathematical abstraction into general Euclidean space and then abstract metric spaces in the absence of classical electrostatics including static charge. That is to say, capacitance is inherently geometric. Now however, that is not to say the term has never been used or that it was not in usage at some point in the past 200 years. I have not traced the literature back to make any determination historically. But I can confidently say that all other abutting terms and synonyms are deprecated today, when it comes to a precise scientific delineation of function and consequence, as well as dynamics.

There may be sources that use the terminology just as Electrical conductance and Electric conductivity are used, as well as Electrical resistance and Electrical resistivity. The issue being that the disjoint antecedent pairs (when numbered from left to right, the odd pair and the even pair) are mutual multiplicative inverses of each other. There is no analogous mathematical relationship for capacitance, though in the (mathematics) literature some authors have tried to coin such a term. However, no practical real world use is known for such an analog. Also, electrical conductance is the element component application of the more general concept of electrical conductivity. Said another way, one would speak about nano- and pico-conductivity in general contexts, while nano-and pico-conductance would only apply to a particular static nano/pico-electric configuration or device/design. At the pico-scale, quantum magnetic inductance becomes nontrivial, and so the very essence of our overly simplistic language breaks down. Note that there is no proper notion of inductivity, much in the same way there is no notion (at least today) of capacitivity. I would also add that all the other (non-hyphenated) scientific terms used were not marked as incorrectly spelled until I typed "inductivity" and "capacitivity", which were both marked as misspellings. Though, this is not a definitive test, as science often changes faster than language. But it does give a flavor for the why of it being not correct. An historic review of the literature and more careful etymological tracing of usage throughout the decades would yield more definitive results, especially with actual citations/references. MMmpds (talk) 00:37, 30 October 2022 (UTC)

The Maserati-automobile relationship is not analogous to the conductor-capacitance relationship. A Maserati is merely an example of an automobile, and it is possible to define automobile without reference to Maseratis.

A better analogue is the circle-diameter relationship; the primary meaning of diameter is only applicable to something circular. The concept of diameter may of course be generalized to refer to all manner of objects, including squares, but such generalizations are esoteric and are not uniquely determined. In any case, if one were to proceed from some generalized definition of diameter, them one must supply that definition - and we do not currently have any definition for the capacitance of a perfectly non-conducting body. catslash (talk) 00:42, 30 October 2022 (UTC)

@Catslash I disagree because just like with the diameter, one changes the name to acknowledge the rather important differences. The opening second sentence is exactly what is being addressed. A Maserati is merely an example of an automobile, which is true and well said. However likewise, a conductor is merely an example of a capacitor and it is completely possible to define a capacitor with no reference to conductors. There is an exact definition of capacitance in the absence of any notion of conduction. As I have mentioned several times, please see Capacity for a broader notion that includes the classical electrostatic notion. MMmpds (talk) 19:51, 31 October 2022 (UTC)

@MMmpds The Capacity page says: "The harmonic capacity is a mathematically abstract version of the electrostatic capacity of the conductor K..." (my italics), and in the previous section it defines the boundary conditions to be equipotentials. If it is completely possible to define capacitance (rather than a capacitor) without any reference to conductors/equipotentials, then please do so.catslash (talk) 22:10, 31 October 2022 (UTC)

No problem. Unfortunately, the Wiki article is not very comprehensive nor is it definitive. For a thorough discussion see "Analysis 2nd ed" by Lieb and Loss published by the AMS, starting on page 291. There it is shown that an equipotential approach does not give the correct answer insofar as a proper generalization includes the specific already knowns as subcases. In fact, neither does an energy minimizer with specific constraints. Lieb and Loss's 4th definition is quite general and encapsulates all of what is expected from classical electrostatics but does not lead to contradictions when generalized. It can then be seen that equipotential is completely superfluous, rather it is correct by accidental degeneracy of overlaps in logic. MMmpds (talk) 22:20, 1 November 2022 (UTC)

"If it is completely possible to define capacitance (rather than a capacitor) without any reference to conductors/equipotentials, then please do so."@Catslash

Here's the answer to your question. Take any few-electron system. Any system with N electrons. It has capacitance regardless of whether there's a conductor or an equipotential surface. Why? Because it is a system that stores charge, and as a result, stores energy. You can use it in a computer design, for example, perhaps to store a 1 or 0 bit, or even in a quantum computer as some flavor of multi-valued qubit if you like. Choose any such system and you can use it as a capacitor. The trick of course is learning how to use it, but it can be done. That's what engineering is all about. It's also what natural biology is all about -- the human brain, for example, has no rigidly defined equipotential surfaces nor conducting materials, yet it's an excellent computer.

The problem is that with a small number of electrons present in the object/device, the equipotential surfaces in its three-dimensional volume are defined by the locations of the electrons in their electrostatic configuration. Equipotential surfaces are only needed in order that we can exactly define mathematical expressions for multiple capacitances (in series, parallel, etc), but they're not required for a general system of charges to have capacitance. In fact, it is physically impossible in general to define a three-dimensional equipotential surface for a given N-electron system (except for special few-electron cases like N=1,2,3,4). Only for large-N systems and/or systems with metallic objects containing (many) electrons is it possible to approximate equipotential surfaces!

The interaction of every electron in a system with the electric potential at its location ultimately determines the capacitance. After all, these quantities, q (=|e|) and V, are the fundamental material quantities of interest when calculating capacitance. The complication arises when we see it's physically impossible for every electron in any given system to be subjected to an exactly equal electric potential. In fact, only for N=1, 2, 3 and 4 electron systems, is it possible that all electrons may be subjected to an identical equipotential surface without potential superposition from multiple charge sources. For N=3 and 4 this occurs when the electrons reside at the vertices of a equilateral triangle and a regular tetrahedron. There are no three-dimensional configurations for N>4 in which all electrons are equidistant from each other (this physical impossibility is on-par with the physical impossibility of matter exceeding light speed), but in some special cases, all electrons can experience an identically equal electric potential (by superposition). Therefore, in general, the charges in an N-electron system do not experience the same electric potential unless there is either a very large collection of electrons or the electrons reside in the presence of metallic structures.

So, the definition of capacitance of an arbitrary N-electron system requires the definition of an average electrostatic potential experienced by all electrons. In the limit of large-N, this tends toward the textbook expression for capacitance of macroscopic capacitors. However, for a single electron system the textbook expression is in error by a factor of 1/2 (50%). This work is consistent with the use of a chemical potential used by Iafrate, et al., as shown on this article's sections on nanoscale capacitance in which they defined differential capacitance in terms of macroscopic chemical potential, whose sum becomes the general expression for capacitance. So the use of an average electric potential is symmetric with their use of a chemical potential.

No conductor is needed in the general concept of electric capacitance. It is entirely dependent on the number of charges stored within a capacitor. No equipotential surface is needed (and generally impossible!) except for cases when we want an exact mathematical expression for multiple capacitors -- an equivalent capacitance -- in series and parallel. In short, the notion of an equipotential surface is primarily related to capacitance because it simplifies the math. In fact, when I turned in my definition of capacitance based on my electrostatics interactions framework to my doctoral committee (electrical engineering) they exclaimed "why would you make things so complicated for us?! We prefer a simple constant capacitance." In general, capacitance is also not a constant value specifically because there is no global equipotential surface in any system! TJ LaFave (talk) 18:54, 2 November 2022 (UTC)

It is useful that this was explained in real life physical terms. One can also add the following. Equipotential literally means constant or fixed potential contours in 3 Euclidean dimensions. This is not a fundamental concept because it is the charge that is conserved and constant and not the potential. This is why Lieb and Loss reject the constant potential approach because it fails to generalize to more robust logical circumstances, or as explained above, at the microscopic scale (by which one means nano to pico scales) the quantum nature of matter matters a lot and cannot be overlooked. As was also pointed out, any fixed configuration of a finite number of charges possesses capacitance. In short, Laplace's equation is an idealization of a state of nature, which does not really exist. The quantum vacuum is anything but a vacuum. It is a highly dynamic and exotic space of constantly fluctuating states. The existence of the Casimir Force and its nontrivial dependence upon temperature illustrates how rather poignantly reality departs from idealized novice textbooks. I strongly believe that a good introduction is concise but also precise and accurate, while still being clear enough and appealing to the widest range of readership possible. Just because it is difficult to attain this balance does not mean it is impossible or that one ought to give up entirely. One tool is to utuilize copious amounts of (relevant) internal links so that less familiar readers can take a quick detour to get their bearings. MMmpds (talk) 19:34, 3 November 2022 (UTC)

Surely the property corresponding to capacitivity is permittivity. The term is a coining of Oliver Heaviside derived from his name for capacitance, permittance. There is indeed a multiplicate inverse of capacitance. That is elastance, also coined by Heaviside and still occassionally found in analysis (I know Wilhelm Cauer used it for instance). The term is particularly liked by systems engineers who have to deal with multiple energy domains and need common terminology – see this book on thermodynamics for instance. SpinningSpark 09:51, 30 October 2022 (UTC)

Yes elastance is exactly the term I was referring to and Cauer and others do make uses of the term. However, the system engineers I know do not find much use for it because they are all working on the pico-scale now where quantum magnetic inductance is no longer negligible and classical thermodynamic intuition breaks down. One must use quantum statistical mechanics and pieces of condensed matter field theory to get good control (theory-wise) over such systems. The reason thermodynamics fails is because it has no notion of sudden local spontaneous ordering, which is the basis of the Third law of thermodynamics, but not the explanation. In particular, the first stated definition, in the Wiki article, is the most correct.

Unfortunately while it is true that the entropy constant does not depend upon characteristic parameters like pressure or magnetic field, it does depend upon the path of all such parameters. In the simplest of cases, the path of all other characteristic parameters is null because the paths are constant (points not tracing out paths) and one need only track a single path of a single parameter. The reason for this complexity is easy to understand because it does not require a reversible isothermal process. Hence, the most general form need not approach zero entropy. The careful reconciliation between the Third and Second laws of thermodynamics is only resolved by a careful accounting of the microstates, as well as the global to local paths taken. These are only able to be accounted for through quantum considerations and hence quantum statistical mechanics and condensed matter field theory are mandatory and indispensable. As a specific example, the ground state of bosonic systems need not be unique, while the ground state of fermionic systems often is unique (But, it still may not be zero; degeneracy, i.e. non-uniqueness is not the only cause of nonzero entropy at extremely low temperatures. This is a specific consequence of the mass gap problem in field theory.). Let alone the spontaneous formation of bosons by fermions through Cooper-esque like pairing. For those wonder what "path" I am talking about, it is the path taken by the system variables in quantum multi-particle phase space. If only thermodynamics were necessary, it would be the path in classical multi-particle phase space.

The suggestion of permittivity as the terminological substitute for a would-be "capacitivity" is not unreasonable. The justification is based upon analysis of units. The general relationship of resistivity to resistance is to take resistance and multiply it by a length scale. Since conductance and conductivity are multiplicative inverses of resistance and resistivity respectively, this would immediately imply that conductivity is obtained from conductance by dividing by a length scale, and it is easy to see this through the most basic mathematical relationships. The units of capacitance are Coulombs^2 / Joules. The units of permittivity are Coulombs^2 / (Joule * meters). Thus indeed, permittivity is obtained from capacitance by dividing by a length scale just like conductivity is obtained from conductance by dividing by a length scale. Ergo, there is already a term in existence and no need for a new (or old as the case may be) term,"capacitivity", which is why it does not exist. MMmpds (talk) 21:02, 31 October 2022 (UTC)

I understand how an object made out of material could have a capacitance. I understand how there could be a capacitance between two electrodes buried in a material. But I don't understand how a material could have a capacitance. Constant314 (talk) 17:56, 30 October 2022 (UTC)

Well, you are right in that material is just an abstract term that does not by itself constitute an object. It is just a name or description of matter, I think. Material is usually something in raw form that has an intended characteristic, purpose, a raw material to manufacture something from. One can say that an object of a certain material with a certain geometry has self-capacitance, but perhaps not the material. I am not sure that this needs to be pointed out in the article though, but perhaps it is best to switch material with object. kbrose (talk) 00:11, 31 October 2022 (UTC)

I would be fine with switching material with object. Constant314 (talk) 00:29, 31 October 2022 (UTC)

Yes I tend to agree because of the anthropic perception of material being some sort of malleable substance. In the absence of a particular geometry, defining a capacitance is not possible. The benefit of object is, again from a pure language and human perspective, that it tends to invoke notions of a fixed configuration (of substance). Thus geometry is assumed implicitly, whether consciously acknowledged or not. Unfortunately, due to charge and baryon number conservation in materials stable, say at standard temperature and pressure, one can write an integral with the constants adjusted for each material and make the following conclusion. Given a specific material, one can calculate the capacitance (mutual or not) of any geometry one settles upon given said material. This is a very general and very powerful statement. It is too bad that our minds impose meaning on words that need not be there, as other definitions of the same word would suffice and ring just as as true. That is to say, a single word has multiple definitions and the most popular one need not be the one intended by the writer. MMmpds (talk) 21:16, 31 October 2022 (UTC)