- #1

coquelicot

- 299

- 67

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/2ε

(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/2ε

So, the total field E at M is the sum of four fields generated by the four faces above:

E = ±σ1_ext/2ε

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/ε

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/2ε

ES = (q1 - q2) / 2ε

ΔV = (q1-q2)d/2Sε

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

C = d/Sε

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 ?

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/2ε

_{0}**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/2ε

_{0}**n**_{S}.So, the total field E at M is the sum of four fields generated by the four faces above:

E = ±σ1_ext/2ε

_{0}± σ1_int/2ε_{0}± σ2_int/2ε_{0}± σ2_ext/2ε_{0}, 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/ε

_{0}+ σ2_int dS/ε_{0}= 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/2ε

_{0}+ σ1_int/2ε_{0}- σ2_int/2ε_{0}- σ2_ext/2ε_{0}, hence, integrating along a face between the plates and parallel to the plates, we have:ES = (q1 - q2) / 2ε

_{0}, henceΔV = (q1-q2)d/2Sε

_{0}.This shows that the notion of capacitance can be defined for this type of capacitor, with

C = d/Sε

_{0}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 ?

Last edited: