Capacitance - does a general approach really exist?

[coquelicot]

I have spent hours on the web and in libraries to obtain a general and careful approach to the notion of capacitance, without success. Please, read carefully the question: I already know the usual blabla about capacitance. Also, I appologize in advance for the length of this question.

I consider here only the notion of mutual capacitance between two conductors. In some textbook, it is shown that if it is supposed that one conductor is loaded with a charge q, and the other with -q, then q is proportional to the potential difference between the two conductors, which defines the proportionality constant as the mutual capacitance between the two conductors. So far so good. But this NEED NOT be always the case: contrarily to what is asserted in some documents, the charge inside the first conductor need not be equal to the opposite of the charge inside the second conductor: an extreme and obvious case is whenever the two terminals of a plate capacitor are connected to the (+) terminal of a 20000V PSU: obviously, the two plates are electrified and bear the same positive charge. One may consider other cases, for instance whenever one plate is connected to +20000V, and the second one to +18000V.
I've seen very few documents that consider the general case, and then they do this very sparsely at the best.

The following facts are shown in the literature, and need not be dealt here:
At electrostatic equilibrium:
(1) inside the body of a conductor, the electric field vanishes,
(2) the electric potential is constant inside the body and at the surface of a conductor
(3) the electric charge is located at the surface of the conductor
(4) the electric field generated by an infinite plane is σ/2ε₀ n_S. This is also the field at a point very close to the surface of the conductor.
(5) finally, it is postulated that Coulomb law, hence also Gauss law, remains true even if the electric field passes through a conductor or a dielectric: this is only the response of the free charges inside conductors (or the dipoles inside dielectrics) that causes the electric field to differ to what it would be in vacuum, but the CONTRIBUTION of a given charge to the electric field at a point continue to follow Coulomb law.

To give more motivation to the discussion, I write here what I believe to be fairly general considerations about parallel plates capacitors (critics and comments will be greatly appreciated).

Assume that two thin parallel plates A and B of same dimensions are separated by a distance d.
Let q1 be the total charge born by plate A, and q2 be the total charge born by plate B.
Let σ1_int and σ1_ext be the (non necessarily uniform) charge density at the inner and outer face of plate A resp., and σ2_int and σ2_ext be the charge density at the inner and outer face of plate B resp.
Since d is small with respect to the other dimensions, the electric field generated by a face at a point M near the face can be approximated by the field generated by an infinite plane bearing the same charge density as the charge density near M, that is, σ/2ε₀ n_S.
So, the total field E at M is the sum of four fields generated by the four faces above:
E = ±σ1_ext/2ε₀ ± σ1_int/2ε₀ ± σ2_int/2ε₀ ± σ2_ext/2ε₀, where the ± sign is determined by the position of M with respect to each face .
Writing that E=0 inside the body of plate A gives a linear equation, and integrating along the face gives a linear equation in the four variables q1_int, q1_ext, q2_int, q2_ext.
Take a thin cylinder between the plates, orthogonal to the faces , and whose extreme faces dS1 and dS2 are inside the body of the conductors (dS1 = dS2 = dS), and apply Gauss law: this leads to
σ1_int dS/ε₀ + σ2_int dS/ε₀ = 0, and integrating along the faces, we have q1_int = - q2_int (one more equation).
Finally, we have obviously the equations q1 = q1_int + q1_ext and q2 = q2_int+q2_ext.
So, we can solve the system of linear equations formed by the above equations and obtain

q1_ext = q2_ext = (q1 + q2)/2, and q1_int = - q2_int = (q1-q2)/2.

As a particular case, whenever q1 = -q2, we obtain that the charges q1 and q2 are entirely distributed at the inner faces of the plates, and that no charge appears at the outer faces.

Furthermore, applying Gauss law with a cylinder between the plates leads immediately to E = Const along a line orthogonal to the plates and between the plates. Integrating along this line gives Ed = ΔV = const (see (2) above). Hence E = const between the plates.
But between the plates, E = σ1_ext/2ε₀ + σ1_int/2ε₀ - σ2_int/2ε₀ - σ2_ext/2ε₀, hence, integrating along a face between the plates and parallel to the plates, we have:
ES = (q1 - q2) / 2ε₀, hence

ΔV = (q1-q2)d/2Sε₀.

This shows that the notion of capacitance can be defined for this type of capacitor, with

C = d/Sε₀ and ΔV = C (q1 - q2)/2.

Note that this method can be applied to any number of parallel plates of equal dimensions (but small d_max), whose total charge q_i is known for each plate, and gives the charge at the faces of the plates, as well as the ΔV between the plates.

Regarding parallel plates capacitors, there is another question for which I have not seen and don't know any rigorous proof: when two such capacitors are put in series, what is the charge at the faces of each plate? I've tried to apply the tools above, and charge conservation gives another equation, but one equation is still lacking in order to solve the system. The problem is that the tools above works only thanks to the infinite plane approximation.

Regarding the general case of two arbitrary conductors, one of my friends has suggested me a way to generalize a part of the previous discussion: one can define the "inner faces" of the conductors as the faces from which the electric field lines go from one conductor to the other, the "outer face" being those faces where the field lines go to infinity.
A simple application of Gauss law with thin tubes going from one of the inner faces to the other (parallel to the field lines) leads easily to q1_int = -q2_int. It would be nice if it could be shown that the charge at the outer faces are equal to (q1+q2)/2, or at least that this charge is negligible. Then a notion of capacitance could be defined (or at least approximately defined). I'm not sure this is true though.

To sum up the questions here:
1. textbook or article dealing rigorously with the notion of capacitance in the general case?
2. is my derivation about parallel plate capacitor fairly correct?
3. parallel plates capacitors in series, what charge ?
4. generalization to two arbitrarily shaped conductors ?

 

[BvU]

So far so good. But this NEED NOT be always the case: contrarily to what is asserted in some documents, the charge inside the first conductor need not be equal to the opposite of the charge inside the second conductor: an extreme and obvious case is whenever the two terminals of a plate capacitor are connected to the (+) terminal of a 20000V PSU: obviously, the two plates are electrified and bear the same positive charge.

Sure. A single sphere has capacitance too. But a mere 111 pF for a 1 m diameter sphere calculation makes clear why the textbooks focus on capacitors with opposite charges not too far from each other.

 

[coquelicot]

Sure. A single sphere has capacitance too. But a mere 111 pF for a 1 m diameter sphere calculation makes clear why the textbooks focus on capacitors with opposite charges not too far from each other.

I know this, but you are mixing self-capacitance with mutual capacitance, and even though this is probably true, I see no theoretical proof here. Furthermore, these picofarads are not as negligible as you think, as is known by those acquainted with precision electronics, and high voltage capacitance is also interesting, e.g. to design precision electrostatic voltmeters. Finally, the theoretical aspect of capacitance is interesting by itself.

 

[BvU]

but you are mixing self-capacitance with mutual capacitance

Am I, or are you ?

obvious case is whenever the two terminals of a plate capacitor are connected to the (+) terminal of a 20000V PSU: obviously, the two plates are electrified and bear the same positive charge

Re

these picofarads are not as negligible as you think, as is known by those acquainted with precision electronics

Compared with a capacity you get from two sperical areas 0.1 mm apart and with a decent dielectric in between ?
Im not saying (or thinking, as you seem to assume) that 110 pF in itself is negligible at all.

 

[coquelicot]

BvU, first, thank you for answering me.
Second, I've not said that mixing self capacitance with mutual capacitance is a bad idea, but only that I don't see a theoretical proof here.
Also, regarding what you've quoted in my post, let me point out that the symmetry immediately allows to conclude that the electric charge born by each plate is the same: q1 = q2. It is fairly rigorous to use symmetry in physics. But we know nothing, a priori, whenever the potential at each plate is different.
Regarding your last words, you are indeed free to consider that this question is useless, and you may or may not be right. Nevertheless, this is the object of this post: is it possible to define the notion of capacitance rigorously, without assuming that q1= -q2 (which may not hold, as we have proved by symmetry).

 

[f95toli]

Of course there is a general approach. However, you usually have to use FEM calculations to get a numerical answer (although the calculations are quite efficient) and you also have to be careful about setting up he boundary conditions properly. It is the most "basic" calculation you can do in electrostatics.
Also, the full answer to any such calculation is always a matrix (which is you have two object will be 2x2); it will include both the self and mutual capacitance.

Have a look at e.g. the COMSOL website for some examples and a description of how this is done. There is also some free software which will allow you to do this (at least in 2D)

Note also that a more general definition of capacitance should include phenomena like quantum capacitance; i.e. you can't define it in terms of two charges,

 

[coquelicot]

f95toli. Thank you. Indeed, the distribution of charges can surely be computed using numeric tools for a given geometry, and given the potential difference and the total charge (say). I'll have a look at the site you've indicated, but in brief, does numerical methods provide a rigorous theoretical definition of the capacitance between two arbitrarily shaped conductors?

 

[coquelicot]

Also, the full answer to any such calculation is always a matrix (which is you have two object will be 2x2); it will include both the self and mutual capacitance.

This seems much interesting. It seems like you have some reference you could share here, or better, an explannation.

 

[sophiecentaur]

these picofarads are not as negligible as you think, as is known by those acquainted with precision electronics

Yes but would you ever use a 1m sphere as a 111pf Capacitor? Even that 111pF would actually correspond to the Mutual Capacitance between it and the Earth.
I agree that a 'fully charged' person could do a lot of damage, though.

 

[The Bill]

This seems much interesting. It seems like you have some reference you could share here, or better, an explannation.

Wikipedia has a decent, if brief, explanation in the section of the capacitance article devoted to the capacitance matrix: https://en.m.wikipedia.org/wiki/Capacitance#Capacitance_matrix

Edit: A source used for that section is a web page on exactly the topic of your original question and a bit more: http://www.av8n.com/physics/capacitance.htm

 

[Nugatory]

does numerical methods provide a rigorous theoretical definition of the capacitance between two arbitrarily shaped conductors?

No, it provides a way of calculating that capacitance. We already have a rigorous theoretical definition of the capacitance between two arbitrarily shaped conductors; we just don't have a convenient closed-form way of calculating its value in many configurations.

 

[Dale]

I've tried to apply the tools above, and charge conservation gives another equation, but one equation is still lacking in order to solve the system.

You are correct, there is a missing constraint, the net charge on the capacitor. In circuit theory that is assumed to be 0 at all times. In EM you would need to supply that constraint some other way.

 

[coquelicot]

No, it provides a way of calculating that capacitance. We already have a rigorous theoretical definition of the capacitance between two arbitrarily shaped conductors; we just don't have a convenient closed-form way of calculating its value in many configurations.

Can you explain what is the rigorous theoretical definition of the capacitance referred in this quote?

 

[coquelicot]

Wikipedia has a decent, if brief, explanation in the section of the capacitance article devoted to the capacitance matrix: https://en.m.wikipedia.org/wiki/Capacitance#Capacitance_matrix

Edit: A source used for that section is a web page on exactly the topic of your original question and a bit more: http://www.av8n.com/physics/capacitance.htm

Humm - Actually, I have contributed to this page in Wikipedia, and yes, I already knew this source. Unfortunately, the author assumes that the charge distributions must be of this form, and don't really prove why. Also, at some point, he only evokes the possibility of charges q1 ≠ -q2 in the case of two conductors, saying this can be handled by the methods in his article, but I don't see how.