ABSOLUTE CAPILLARY VOLTMETER

If two flat vertical capacitor plates, separated by a small gap, are partially immersed in an insulating liquid (figure), then the level to which the liquid rises between the plates will be determined not only by capillary forces, but also by the ponderomotive forces of the electric field, i.e., by the voltage applied to the capacitor. It is not difficult to see that, from the condition of minimum potential energy, it follows that:

\[ \Delta h=\frac{\varepsilon-1}{8\pi g\rho}E^2=\frac{\varepsilon-1}{8\pi g\rho d^2}V^2, \tag{1} \]

where \(\Delta h\) is the change in the level of the meniscus, \(\varepsilon\) is the dielectric constant of the liquid, \(\rho\) is the density of the liquid, \(d\) is the distance between the plates, and \(g\)—

acceleration of gravity. Relation (1) is completely exact (for an infinite condenser) and is equally applicable both to a constant voltage and to a rapidly varying voltage \(V\).

This method was used by H. Greinacher*) to measure the dielectric constants of a number of liquids. The measurements showed that, under real conditions, over wide ranges of variation of \(\varepsilon\), \(\rho\), and \(V\) (both constant and alternating), relation (1) is strictly satisfied. On this basis, Greinacher proposes using the device described as an absolute voltmeter.

Diagram labels: ebonite; insulating liquid; glass cuvette.

The voltmeter he constructed had plates 4 mm wide, placed at distances \(d = 1.04\) mm and \(d = 0.84\) mm. Machine oil and nitrobenzene were used as the insulating liquid. Observation of the meniscus was carried out with a horizontal microscope at a magnification of \(N = 70\). The displacement was measured on an ocular scale.

It follows from (1) that if \(x\) is the reading on the ocular scale, then

\[ V = k\sqrt{x}, \quad \text{where} \quad k = \sqrt{\frac{8\pi g\rho d^{2}}{(\varepsilon - 1)N}} . \]

If, for any reason, calculation of \(k\) is difficult, then to determine it, i.e., to calibrate the voltmeter, it is sufficient to determine \(x\) for only one value of \(V\) (whether constant or alternating). The absolute accuracy of the measurements of \(V\) is, obviously, determined by the value of \(k\), i.e., by the choice of liquid, the distance between the plates \(d\), and the magnification \(N\). Apparently, the smallest technically easily attainable error is \(\pm 1\) volt. Relative accuracy is determined by the dimensions of the microscope’s field of view and is unlikely to exceed 1%. The accessible range of measurements extends from several hundred to several thousand volts.

Among the merits of the voltmeter one should note, first of all, the simplicity of its design. The absence of easily movable mechanical systems with inevitably thin filaments or movable plates considerably reduces the voltmeter’s sensitivity to external mechanical influences, while at the same time ensuring high zero stability. The small capacitance (of the order of several centimeters) makes the voltmeter very sensitive to electric charge, and also permits its use in circuits with weak currents, including high-frequency ones. (The internal resistance of the voltmeter, measured by the author, is of the order of \(5 \cdot 10^{13}\) ohms.) Its low inertia, rapidity, and convenience of measurement make it suitable for observing not excessively rapid changes in voltage. Finally, the simplicity of manufacture—the voltmeter can be made even in a moderately equipped school laboratory—makes it widely accessible.

In this connection it should be noted that the capillary voltmeter is undoubtedly an excellent task for a physics practicum.

It may be assumed that, despite its comparatively low accuracy, the absolute capillary voltmeter will find application in various areas of physics.

G. Rosenberg

*) Helvetica Physica Acta 21, 261, 273 (1948).