# Is there a static capacitance between the metals in coaxial cables and other metals in the environment (or at infinity)?

In a coaxial cable, we are familiar with the capacitance per unit length between the two metal conductors, which is given by C=(2pi epsilon)/log(r/R). However, in my circuit design, the static mutual capacitance between the metal in the coaxial cable and the surrounding environment increases the power transmission losses. Therefore, I want to understand the magnitude of this stray capacitance.

According to electrostatics theory, an ideal infinitely long coaxial cable (infinite cylinder) should not produce any electric field outside the conductor. Does this mean that after adding other electrically neutral metals in space, the electric field in the space still remains zero everywhere? (Uniqueness theorem? I'm not sure.) Can we deduce from this that there is no mutual capacitance between the metal in the coaxial cable and any electrically neutral metal in the surrounding space? And can this be extended to practical non-ideal coaxial cables, where the mutual capacitance with environmental metals is expected to be very small in magnitude?

I apologize for the many questions here. I greatly appreciate any replies, guidance, and discussions. Thank you.

Actually, my circuit at this moment functions similarly to a switched capacitor DC-DC converter, but there is one capacitor that needs to be connected to the circuit via a cable (about 30 cm), as shown in the figure below. In this case, the cable is only used for the electrical connection between the capacitor and the switching circuit. The stray capacitance of the cable results in voltage changes during the switching process, which implies losses.

• If you have losses due to the coax cable's proximity to external metals then please explain how you are driving the coax cable. An external electric field is not produced when an ideal coax is driven correctly. Any amount of capacitance between the coax shield and external metals does not necessarily imply losses. So, it boils down to how you drive and terminate the cable and you need to explain that. Commented Aug 1 at 13:50

The classic result of electrostatics that the electric field outside an infinite coaxial capacitor is 0 relies upon the rotational symmetry of electric field.

Gauss's Law states, in essence, that the total electric flux passing through a closed surface is equal to the net electric charge within that surface. In the case of a set of conductors with zero net charge, that means that the total flux passing through a closed surface surrounding those conductors will be zero.

But to make the move from the total flux through the surface being 0, to the flux at each point on the surface being 0, one needs the assumption that the magnitude of flux passing through each point of the surface is the same.

If one adds to the system of conductors, a third conductor, which lacks rotational symmetry around the axis of the coaxial conductors, which conductor is at a different potential from that of the outer coaxial conductor, one (in most cases) breaks the symmetry of the electric field. Thus, the classic result about the electric field being 0 no longer holds. There IS mutual capacitance between the additional third conductor, and the outer coaxial conductor.

Note that I am answering the question asked, about mutual capacitance between coaxial conductors and a third conductor in an electrostatic situation. It is possible that the real intent of the question is whether coaxial cables ensure signal integrity. That is largely a different question because

1. signals imply that you are no longer working in an electrostatic regime
2. the mutual capacitance between the outer coaxial conductor and a third, non-coaxial conductor does not negate the Faraday cage effects of the outer conductor in shielding the inner conductor
3. static, irrotational, electric fields typically have no real affect on electronic circuits that are galvanically isolated from the source of the electric field. On a physics level, such an E field might alter the distribution of charges in the circuit, but on a practical level, the circuit (usually) works the same.

Regarding the electrostatics part, perhaps a more logical explanation is: since an infinitely long coaxial cable does not produce an electric field in space, even if a third electrically neutral metal is added, this metal will not become polarized (because there is no electric field), and the electric field in space will not be altered by the metal. I am not sure where this reasoning goes wrong.

If the outer coaxial conductor and the third conductor (which has no net charge) are at the same electrical potential, then indeed there will be no electric field between them. That is why I wrote:

If one adds to the system of conductors, a third conductor, which lacks rotational symmetry around the axis of the coaxial conductors, which conductor is at a different potential from that of the outer coaxial conductor,...

But the fact that there is no electric field when there is no potential difference between two conductors does not entail that there is no mutual capacitance between those conductors. In general, the electric field between the plates of a capacitor only arises when the plates are at different electric potentials.

Your reasoning goes wrong in assuming that if there is no electric field when the two conductors are at the same potential that there is therefore no mutual capacitance.

What I truly want to know is indeed just the issue of mutual capacitance. Here is my understanding: my circuit is essentially similar to a switched capacitor DC-DC converter, but there is one capacitor that needs to be connected to the PCB via a cable. If the mutual capacitance between the cable conductor and the third conductor is too large, the energy loss during the switching process will be significant. Therefore, my initial question is actually: does a shielded cable like a coaxial cable have the characteristic of not having mutual capacitance with the environment?

To answer the question of capacitance once again, yes, the outer conductor of a coaxial cable has mutual capacitance with conductors in the environment.

To answer the question of whether that mutual capacitance will affect your circuit, one must consider whether the electric potentials of any of the conductors are changing. Currents flow into or out of capacitors only when the electric potential difference between the plates is changing.

The inner coaxial conductor may be "shielded" from third conductors in the environment by the outer conductor if the outer conductor is not "floating" (for example if it is grounded).

You have added an image to your answer that shows wires labeled "coax cable" connected to a capacitor, and a box labeled "switch capacitor". It is not clear to me whether the two wires connected to the capacitor are meant to be the inner and outer conductors of a single coaxial cable, or whether each plate of the capacitor is connected to the "switch capacitor" box by the inner conductor of two separate coax cables. I am guessing the former, but I may be mistaken.

Significantly, what is NOT clear from your diagram is whether the outer conductor of the "coax cable" is grounded. (Or, if there are two coaxial cables, whether the outer conductors are grounded.)

If the capacitor is connected via the inner and outer conductors of a coaxial cable, and the outer conductor is grounded, or at least held at an exactly constant voltage with respect to ground, then any capacitance between that outer coaxial conductor and a third conductor, also held at an exactly constant voltage, will have no effect on your circuit, because the electric potential difference between the two will not be changing, and hence there will be no current resulting from that capacitance.

If in your circuit, the outer conductor of the coax cable or coax cables is not already grounded, be aware that connecting it to ground will shield the inner conductor from third conductors, but will add capacitative effects between the inner and outer coax conductor. Depending upon your circuit, this may make matters worse rather than better. Or not. It depends on the circuit.

[As an aside, the model for Stack Exchange is not a free flowing back and forth, but simply a question and answer. In general, one should avoid making significant additions or edits to a question after an answer has been posted, as this may prompt significant additions or edits to an answer, or may render an existing answer incomplete or worse.]

• Thank you very much for your response. Regarding the electrostatics part, perhaps a more logical explanation is: since an infinitely long coaxial cable does not produce an electric field in space, even if a third electrically neutral metal is added, this metal will not become polarized (because there is no electric field), and the electric field in space will not be altered by the metal. I am not sure where this reasoning goes wrong. Commented Aug 2 at 2:23
• What I truly want to know is indeed just the issue of mutual capacitance. Here is my understanding: my circuit is essentially similar to a switched capacitor DC-DC converter, but there is one capacitor that needs to be connected to the PCB via a cable. If the mutual capacitance between the cable conductor and the third conductor is too large, the energy loss during the switching process will be significant. Therefore, my initial question is actually: does a shielded cable like a coaxial cable have the characteristic of not having mutual capacitance with the environment? Commented Aug 2 at 2:23