Calculable Capacitor

A calculable capacitor is an electrical instrument that allows the precise realization of electrical capacitance based on one mechanical dimension. In the 2019 revision of the SI, this means electrical capacitance is directly linked to the mechanical dimension, through the fine structure constant and other defined constants.

Principle of Operation

The calculable capacitor is based on the Thompson-Lampard theorem^[1]^[2], which states that the value of the cross capacitance satisfies a simple relation for a system with uniform cross section. This implies that the capacitance per unit length only depends on permittivity of free space. In practice, this is realized with 4 cylindrical rods whose center is placed on a square. 2 central grounded rods are placed within the space within the 4 rods, which are significantly pointed inside the extend of the rods. The quantity of interest is the change in capacitance when the central rods are displaced by a known amount, measured by laser interferometry.

This fact is first proved by Thompson and Lampard for a configuration with mirror symmetry. Later on this is generalized to arbitrary cross section by Leo J. van der Pauw in 1958.^[3]

The cross section has 4 electrically isolated segments labeled 1,2,3,4. In SI units, the cross capacitor satisfies relationship $\exp {\bigg (}-\pi C_{13}/\epsilon _{0}{\bigg )}+\exp {\bigg (}-\pi C_{24}/\epsilon _{0}{\bigg )}=1$ where $C_{13},C_{24}$ are the cross-capacitance per unit length.

In a symmetric system where $C_{13}=C_{24}=C$ , the cross-capacitance is $C=\epsilon _{0}{\frac {\ln 2}{\pi }}=1.953549043...\times 10^{-12}Fm^{-1}$

In a practical realization, the average of the 2 cross capacitance is equal to the above within second order of the asymmetry. I.e. if the difference is within $10^{-4}$ , then the average is within $10^{-8}$ with the above value.

History

Realization

In practical realization, the 4 sections are usually cylindrical surface. Early implementations used gauge rods. Later realization uses specifically machines cylindrical rods with high symmetry.

References

^ Thompson, A. M.; Lampard, D. G. (12 May 1956). "A New Theorem in Electrostatics and its Application to Calculable Standards of Capacitance". Nature. 177 (4515): 888–888. doi:10.1038/177888a0. ISSN 1476-4687.
^ Lampard, D.G. (1957). "A new theorem in electrostatics with applications to calculable standards of capacitance". Proceedings of the IEE Part C: Monographs. 104 (6): 271. doi:10.1049/pi-c.1957.0032. ISSN 0369-8904.
^ Van der Pauw, L.J. (1958). "A method of measuring specific resistivity and Hall effect of discs of arbitrary shape" (PDF). Philips Research Reports. 13: 1–9.)

[1] Thompson, A. M.; Lampard, D. G. (12 May 1956). "A New Theorem in Electrostatics and its Application to Calculable Standards of Capacitance". Nature. 177 (4515): 888–888. doi:10.1038/177888a0. ISSN 1476-4687.

[2] Lampard, D.G. (1957). "A new theorem in electrostatics with applications to calculable standards of capacitance". Proceedings of the IEE Part C: Monographs. 104 (6): 271. doi:10.1049/pi-c.1957.0032. ISSN 0369-8904.

[3] Van der Pauw, L.J. (1958). "A method of measuring specific resistivity and Hall effect of discs of arbitrary shape" (PDF). Philips Research Reports. 13: 1–9.)

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