An absolute electrometer for the physics laboratory

F EATURES
www.iop.org/journals/physed
An absolute electrometer for the
physics laboratory
S Straulino and A Cartacci
Dipartimento di Fisica dell’Università di Firenze, Via G Sansone, 1 50019 Sesto Fiorentino,
Italy
E-mail: straulino@fi.infn.it
Abstract
A low-cost, easy-to-use absolute electrometer is presented: two thin metallic
plates and an electronic balance, usually available in a laboratory, are used.
We report on the very good performance of the device that allows precise
measurements of the force acting between two charged plates.
Introduction
The electrometer is a device which is used to
measure electric potentials; it is absolute when
the potentials are evaluated through the direct
measurement of a force and consequently no
previous calibration, using a known voltage,
is necessary. The absolute electrometer was
first proposed by Lord Kelvin (an interesting
description of this instrument was given by
Maxwell [1]). We realized an up-to-date version
of the absolute electrometer with an electronic
precision balance and two circular metallic plates.
The classical absolute electrometer
Lord Kelvin’s instrument is schematically sketched
in figure 1: it can be imagined as a two-arm balance
connected to a parallel-plate capacitor. The
lower plate of the capacitor is fixed and attached to
the table by means of an insulating rod. The upper
plate, hung up by a wire and connected to the balance’s
arm, is equilibrated using some weights on
the opposite pan of the balance. The lower plate
is taken to an unknown voltage. Then opposite
charges appear on the facing plate because of electrostatic
induction. Positive and negative charges
generate a force that produces a small rotation of
the balance’s arm. The previous equilibrium can
GR P1
Figure 1. The classical scheme for the absolute
electrometer. P1 and P2 are the metallic plates and
GR is the guard ring.
be restored by adding some known weights that
give us the amount of the electric force mutually
exerted by the charged plates 1 . An electrometer
can work at either constant charge (when plates are
isolated) or constant potential: we used this second
configuration as the plates were connected to a
voltage generator. Sometimes the upper plate may
1 The measurement however is not trivial: since the
electrostatic force depends on the distance between the plates
while the gravitational force is constant, the equilibrium of the
balance is unstable.
0031-9120/09/030301+05$30.00 © 2009 IOP Publishing Ltd P HYSICS E DUCATION 44 (3) 301
P2

S Straulino and A Cartacci
Figure 2. A photograph of the electrometer. The upper plate is sustained by two microscope slides, glued on the
disc and laid on a pair of adjustable laboratory jacks. The other disc is laid on the pan of the balance through an
insulating support. The high-voltage generator is not shown in the picture.
be surrounded by a guard ring, kept at the same potential
as the internal plate: it can be used to reduce
the fringing effects. Different aspects of the electrometer
have already been discussed within educational
articles [2–4].
For a calculation of the force acting between
the plates, let us review some basic relationships of
electromagnetism. The capacitance of a charged
conductor body, far away others, is defined as the
ratio between its charge Q and its potential V :
C = Q
V .
In the case of a parallel-plate, infinitely large
capacitor in vacuum, Q is the absolute value
of the charge present on both plates and V the
potential difference between them. In this case C
can also be written by means of the geometrical
characteristics of the capacitor in the following
form:
S
C = ε0 , (1)
d
S being the surface of each plate and d the distance
between them.
The energy stored in the capacitor is given by
W = 1
2 CV2 = 1
2 ε0
S
d
V 2
and the force is obtained by calculating the
derivative of the previous quantity with respect to
d:
F =−
ε0SV 2
2d 2
. (2)
The potential difference V can be written as
�
2 |F|
V = d
ε0S .
When an insulating material is filling the
space between the plates, the force F depends on
εr, the dielectric constant of the material. The
force measured in the air will be, to an excellent
approximation, the same as that measured in
vacuum because εr = 1.0006 for the atmosphere
in standard conditions.
Construction and setup in our laboratory
We commissioned from a mechanical workshop
several circular aluminium plates with identical
diameters (300 mm) and various thicknesses (3,
4, 5 mm). A couple of discs were selected for
the device on the basis of the best planarity 2 .
A thin copper wire was attached to each plate
for the electrical connections. The first disc was
laid on a plastic support, laid in turn on the pan
of the balance: it was electrically connected to
ground. We set the plate horizontal by adjusting
the screws on the balance’s base. The other disc
was suspended by two insulating supports exactly
above the first one, at a small distance d (figure 2).
2 At first we used two stainless steel discs, 1 mm thick. We
realized after a few measurements that they were not planar,
because they were too thin (a good planarity—within 0.1 mm—
is not easily achievable for thin metal plates). Then we used a
light metal such as aluminium to fall within the range of the
balance (1500 g) with a greater thickness of the disc. Besides,
the use of a non-ferrous material helps students by removing
any source of misunderstanding about a supposed magnetic
origin of the attraction between the plates.
302 P HYSICS E DUCATION May 2009

Figure 3. The height gauge used to determine the
positions of the discs.
A potential difference produced by a highvoltage
(up to 2000 V) DC generator was applied
to the plates. The sensitivity of the balance
(0.01 g) was enough for us to see visible effects
when voltages were larger than about 100 V. An
electronic precision balance is suitable for this
purpose as the pan is practically motionless and
consequently the distance between the plates is
always the same, allowing a direct determination
of the force without any instability or oscillation 3 .
A Plexiglass box covered the electrometer: it
was intended not only for safety reasons (the
high voltage is obviously dangerous 4 )butalsoto
avoid unexpected movements of the surrounding
air possibly influencing the measurements. The
distance between the plates was measured using
a height gauge (figure 3), with a sensitivity of
0.02 mm, at four equidistant points A, B, C, D on
the disc (figure 4), to obtain a mean distance with
a small level of uncertainty.
3 The weight of both the lower disc and the plastic support
can be automatically included as a tare. Readings are negative
numbers, the electrostatic force being attractive.
4 As a further caution, the current output of the generator
was limited to 100 μA. This does not represent a restrictive
condition for the use of the electrometer, since in stationary
conditions the nominal current flow is zero.
An absolute electrometer for the physics laboratory
D
A
Figure 4. Labels on the disc to measure the distance
from the other plate.
Table 1. Three different distances from the upper and
lower discs, as measured by the height gauge.
Label Distance (mm) Mean (mm)
No 1 A 18.30 18.4 ± 0.1
B 18.40
C 18.54
D 18.30
No 2 A 12.16 12.1 ± 0.2
B 12.10
C 11.90
D 12.22
No 3 A 6.14 6.1 ± 0.1
B 6.04
C 6.12
D 6.10
Results
In this section we report some measurements taken
with the device. The electrometer works very
well, showing results that are always reproducible.
We report three sets of measurements at different
distances for voltages ranging from 0 to 2000 V. In
table 1 the distances between the upper and lower
discs are reported: from the measurements taken
at different points we calculated the mean and the
maximum difference from the mean. In table 2 the
weight 5 measured by the balance for every voltage
can be found.
In figure 5 the weight is reported versus the
squared voltage and a law of direct proportionality
can be verified. A linear interpolation of the
5 The force read on the balance was in gram-force while, using
the international system of units, the force of equation (2) is
expressed in newtons.
May 2009 P HYSICS E DUCATION 303
C
B

S Straulino and A Cartacci
weight (g)
Table 2. Readings of the balance at various voltages for three distances between the plates.
Voltage (V) 0 200 400 600 800 1000 1200 1400 1600 1800 2000
No 1 Weight (g) 0 0 −0.01 −0.03 −0.06 −0.09 −0.13 −0.18 −0.24 −0.31 −0.38
No 2 Weight (g) 0 −0.01 −0.04 −0.08 −0.15 −0.23 −0.33 −0.45 −0.58 −0.74 −0.91
No 3 Weight (g) 0 −0.03 −0.14 −0.32 −0.57 −0.90 −1.30 −1.77 −2.31 −2.93 −3.61
0
–0.5
–1.0
–1.5
–2.0
–2.5
–3.0
d = 18.4 mm (fit)
d = 11.8 mm (fit)
–3.5
d = 5.9 mm (fit)
–4.0
0 1 2 3 4
squared voltage (kV 2 )
Figure 5. The measured weight, reported as a
function of the squared voltage for three different
distances, and the linear fits. The experimental error
bars are smaller than the point size. The dotted curves
represent the theoretical behaviour if the measured
distances are used. For the greatest distance, the two
lines are superimposed.
points allows an a posteriori determination of
the distance of the plates that is written in the
graph. The agreement with the corresponding
measured distances is quite good. In the figure
the theoretical straight line corresponding to the
value of the measured distances is also shown.
The small discrepancy between measured and
fitted distances can be attributed to the fringing
effects, which were not included in our analysis,
and to the imperfect alignment of the plates. In
figure 6 the difference of each measured weight
from the interpolating straight line is reported.
The instrument allows also a measurement of the
applied voltage through the value of the force
(absolute electrometer). The discrepancy is of the
order of 2% for voltages greater than 1000 V (as
measured weight – fitted weight (g)
0.02
0
–0.02
0.02
0
–0.02
0.02
0
–0.02
0 0.5 1.0 1.5 2.0
0 0.5 1.0 1.5 2.0
0 0.5 1.0 1.5 2.0
voltage (kV)
Figure 6. Differences between the measured weight
and linear fit, reported as a function of the voltage,
for the same measurements as in figure 5. An
uncertainty of 0.005 g (half the sensitivity of the
balance) is assumed for each point.
can be deduced from table 2) and depends on the
uncertainty of the distance and the sensitivity of
the balance. A better precision could be obtained
by means of a more expensive balance with a
sensitivity of 1 mg. In any case, the instrument
is not designed for measuring an unknown voltage
in our laboratory but as an experiment which can
easily indicate the existence of the force between
the plates and verify its dependence on the applied
voltage.
Note on the fringing effects
The formulae describing the parallel-plate capacitor
that we have used, assume a constant electric
field inside the capacitor, regardless of fringing
effects. Many articles can be found in the literature
(for example [1, 5, 6]) where the fringing effects
are discussed and often a guard ring is introduced
304 P HYSICS E DUCATION May 2009

in experimental devices to preserve the uniformity
of the field near the edges also.
Therefore we expected that the deviation
from the ideal case might become significant for
increasing distances of the plates. However, in our
measurements we find only a small discrepancy
between the parallel-plate description and the
results. This matter has been discussed by
Hale [4, 7]: he showed that the corrections due to
the fringing effects are always quite small when
measuring the force between facing plates. In
fact the difference between the real capacitance
and the ideal one (equation (1)) is approximately
independent of the plate separation for a limited
range of distances and barely affects the attractive
force (equation (2)) connected with the derivative
of the capacitance.
Educational remarks
The instrument described was included in the
physics laboratory of the OpenLab project of
the Florence University, partially supported by
the Italian Instruction Ministry in the context of
the ‘Lauree Scientifiche’ project. Every year
many exercises in the laboratory are organized.
The high-school students who attend the activity,
divided into small groups, assemble and use
the apparatus under the supervision of university
researchers. The absolute electrometer has been
presented to some classes visiting this laboratory:
it has proved to be of great educational interest in
the high school. Students immediately perceive
the presence of a force between the plates of the
capacitor and they can appreciate the simplicity
and the accuracy of the measurements. Usually
in high-school laboratories qualitative experiments
on the electric charge are performed: here, in
contrast, quantitative measurements can be easily
taken. Feedback on the activity is usually obtained
An absolute electrometer for the physics laboratory
from the teachers who are present in our laboratory
together with their classes.
Acknowledgment
We wish to thank Professor Oscar Adriani and Dr
Cecilia Gambi for fruitful discussions concerning
the subject of this work.
Received 8 January 2009, in final form 13 February 2009
doi:10.1088/0031-9120/44/3/011
References
[1] Maxwell J K 1873 A Treatise on Electricity and
Magnetism (Oxford: Clarendon)
[2] Cross J D and Smalley C 1969 Electrical
measurements with an electronic microforce
balance J. Phys. E: Sci. Instrum. 2 633
[3] Metz R N 1973 The absolute electrometer revisited
Am.J.Phys.41 1170–5
[4] Hale D P 1978 A measurement of electrostatic
force Phys. Educ. 13 344–7
[5] Wintle H J 1986 Capacitor edge corrections IEEE
Trans. Electr. Insul. EI-21 361–3
[6] Carlson G T and Illman B L 1994 The circular disk
parallel plate capacitor Am. J. Phys.
62 1099–105
[7] Hale D P 1978 Fringe capacitance of a
parallel-plate capacitor Phys. Educ. 13 446–8
Samuele Straulino graduated at Florence
University in 1999. In 2003 he was
awarded a PhD in physics from Bologna
University; his thesis was on cosmic-ray
physics. Since 2004, thanks to a
post-doctoral fellowship, he has been
working on educational physics at
Florence University.
Annamaria Cartacci was Associate
Professor at Florence University until
2002, where she taught elementary
particle physics. She is the author of
numerous papers in international
journals. Now retired, she remains
involved in physics popularization and is
responsible for laboratory courses for
teachers working in high schools.
May 2009 P HYSICS E DUCATION 305