Voltage and Field Strength: Part II, Conductors
Screening noncontacting meters will often reduce the field distortion
caused by the presence of meters.
Anyone who has worked with static or dynamic electricity is familiar
with the concept of voltage. After all, Ohm's law states that V =
R ∙ I, voltage (difference) equals resistance times current.
But this well-known relationship does not say anything about voltage;
rather, it defines resistance, and it cannot be applied to ESD problems
because there is no current. Then there is the definition of the voltage
difference between points A and B as the work done per unit charge when
a charge is brought from A to B. But here there is a metrological problem
because there is no way to measure the work that is done on a charge.
So, we have to go back to the basics and realize that voltage is not
a fundamental quantity, but rather a property of an electric field.
Figure 1 shows a section of a homogeneous field with field strength E,
where the voltage difference between points A and B is defined by
However, in most cases, fields are not homogeneous. Figure 2
shows the field from a positively charged insulator with a grounded
conductor placed in front of the insulator. In this case, the voltage
difference between A and B is defined by
Equations 1 and 2 only define voltage differences. The voltage
of a point P in a field is defined as the integral of the field from
P to infinity or to any grounded object, that is,
Figure 3 shows an insulated conductor A with a charge q. The charge
will automatically distribute itself on the surface of the conductor in
such a way that (a) the field in the interior of the conductor is zero,
(b) the field is perpendicular to the surface, and (c) the integral of
the field strength from any point P in or on the conductor to a ground
point G is constant:
V is the voltage or potential of the conductor. The voltage V and the
charge q are proportional, and this is usually written as
C is the capacitance of the insulated conductor and is determined by
the size and shape of the conductor and its placement relative to other
conductors and ground.
The charged system stores an electrostatic energy given by
which can be dissipated in a single discharge or current pulse.
Measurement of Conductor Voltage
Direct-Contact Voltmeters. The voltage of an insulated conductor may
be measured directly by connecting the conductor to an electrometer or
static voltmeter (see Figure 4). The voltmeter measures the common voltage
of the conductor and the voltmeter. If the capacitance C of the conductor
is much larger than the capacitance Ci of the voltmeter,
the voltage read on the voltmeter is, with good approximation, equal to
the voltage of the conductor without the meter being attached.
||Figure 1. A homogeneous field with
field strength E.
However, the measuring range of most static voltmeters is in the order
of tens or, at best, hundreds of volts. On the other hand, static voltages
will often be in the kilovolt range.
This problem can be circumvented by the use of a capacitive voltage divider.
In Figure 5, a capacitor with capacitance Cy is inserted
in the connection between the conductor and the static voltmeter.
If the voltage read on the voltmeter is Vi, then the
voltage V of the conductor is given by
As an example, let us assume that the maximum voltage to be read
on the meter is Vi, max = 10 V, Ci
= 10 nF = 108 F, and Cy = 10 pF =
1011 F, then Equation 7 will give a maximum voltage
The necessary high capacitance in this application of the meter is
usually obtained by running the meter in the charge-measuring mode.
It appears that using a capacitive voltage divider expanded the measuring
range of the voltmeter by a factor of 1000.
Noncontacting Measurements. Electrostatic noncontacting measurements
are always based on the effects of the fields of charges, whether they
are located on conductors or insulators. There are basically two types
of instruments: field meters, which measure the charge induced on a probe
and convert it to the field strength in front of the probe, and noncontacting
voltmeters, which raise the voltage of the probe until the field in front
of the probe is zero. The noncontacting voltmeter then takes this voltage
as the voltage of the object that it is pointing toward.
|Figure 2. The field between a positively charged
insulator and a grounded conductor.
Noncontacting voltmeters may have greater sensitivity (but not necessarily
greater accuracy) than do field meters. However, both types of instruments
may distort the original field considerably unless the meters are suitable
|Figure 3. An insulated conductor A
with a charge q, placed over a ground.
Figure 6 shows a charged insulated conductor. In the figure, the noncontacting
voltmeter reads the voltage V of the conductor and estimates the mean
field strength E = V/d between the conductor and the meter, whereas
the field meter reads the field strength E in front of the meter
and estimates the voltage V = E ∙ d of the conductor. However,
it should be emphasized that the quantities read and calculated refer
to the conditions that exist when the instruments are in place.
Conductor at Fixed Voltage
The experiment shown in Figure 7 was conducted to investigate the influence
of the meters on the field from and the voltage of the charged conductor.
A 35 ∙ 35-cm metal plate was connected to a voltage supply kept at
a constant voltage of 3 kV. A field meter was placed perpendicular to
the plate, pointing at the center of the plate, and the field strength
E was measured as a function of the distance d between the
plate and the field meter. For each distance d, the product E
∙ d was calculated.
Figure 4. The direct measurement of voltage.
The results of the measurements are shown in Figure 8. It appears that
the field strength E decreased with increasing distance d,
as expected. However, if the voltage V of the plate is calculated
from Equation 1 as V = E ∙ d, the result would be a very poor
approximation of the true value (3 kV) of the plate voltage.
The reason for this is that Equation 1 assumes the field to be homogeneous,
as shown in Figure 1. But the setup in Figure 7 resembles much more closely
the situation in Figure 2 because the housing of the field meter (or for
that matter, the housing of a noncontacting voltmeter) is essentially
at ground potential. The field strength read on the field meter (or compensated
for in a noncontacting voltmeter) is therefore higher than the mean field
strength between the meter and the target, and the E ∙ d approximation
of the voltage will therefore be too high. Figure 8 shows that in the
range of distances from 4 to 30 cm, the estimated voltage E ∙
d varies between 4.5 and 6.2 kV, rather than the true value of 3 kV.
|Figure 5. A capacitive voltage divider.
The problem of the instruments distorting the field can be corrected
partly by surrounding the meter with a grounded screen placed parallel
to the face of the target, as shown in Figure 9. The experimental setup
had a 25 ∙ 25-cm screen and a 35 ∙
35-cm metal plate as the target.
Figure 10 shows the field strength E and the apparent voltage
E ∙ d as a function of the distance d. The results
demonstrate that, with the screen attached, the voltage V of the
metal plate is adequately determined by the product E ∙ d
out to a distance of approximately 15 cm between the plate and the field
meter. In this range, the field is homogeneous and inversely proportional
to the distance to the field meter, that is, the E-field curve is a hyperbola.
At larger distances, the field again becomes inhomogeneous, and at this
range, the field meter underestimates the voltage.
|Figure 6. Noncontacting measurements.
The distance to which the voltage can be determined with reasonable accuracy
also depends on the target size. If the measurements in Figure 10 were
repeated with a 15 ∙ 15-cm plate, the readings
would yield reliable results only out to a distance of approximately 67
Conductor with Constant Charge
In the previously discussed cases, the target conductor was locked to
a voltage supply. The voltage of the conductor would therefore be kept
constant, independent of field meter placement. The charge, on the other
hand, might vary with the intercapacitance of the conductor and the field
meter, that is, with the distance d.
The previous cases do not represent the ordinary, everyday situation
in which a conductor has been charged and the voltage is measured by pointing
a meter at the conductor. In this more common case, the charge stays constant
while the voltage may change because of the coupling with the meter capacitance.
Figure 11 shows an experimental setup for investigating this situation.
In the experiment, a 35 ∙ 35-cm metal plate was charged to an initial
voltage of 3 kV (in the absence of the field meter), and then the connection
to the voltage supply was broken. Next, the field meter was placed at
various distances d from the metal plate, and the field strength
E was measured.
Figure 7. An unscreened field meter.
Figure 12 shows the product E ∙ d (the apparent voltage)
as a function of d for plate capacitances C @ 20 pF (the plate
alone) and C @ 220 pF (the plate and an additional external capacitor).
The greater plate capacitance of 220 pF provided a curve that is very
similar to the one plotted in Figure 10, where the metal plate was locked
at 3 kV. This means that the presence of the field meter does not significantly
change the total capacitance and, hence, the plate voltage for a given
distance. The lesser plate capacitance of 20 pF resulted in a calculated
voltage that is lower at all distances than that found with the plate
of greater capacitance. This is due to the added value of the meter capacitance.
At the very short measuring distance, the presence of the meter increases
the original value of the capacitance from 20 to about 45 pF, resulting
in the voltage dropping from 3 to about 1.3 kV.
Figure 8. Measurement results
from an unscreened field meter.
The measurements reported in Figure 12 were repeated with an unshielded
field meter. The general trend was the same as demonstrated in Figure
8. At all distances (and with both capacitances tested), the unshielded
field meters overestimated the true values of the plate voltage by up
Probably the most common way to do a fast static survey is to point
a handheld meter at the suspicious item and pronounce a voltage. Often,
this is the only measurement done. And very often, this is not enough.
These handheld meters are known as, and often even called, static locators.
And that is exactly what they are instruments to locate a static
electric field. As long as that is the only thing they are being used
for, everything is fine. But often, their use is being extrapolated into
Figure 9. A screened field
Figure 13 is an illustration of a static locator. The meter may have
two ranges, both of them scaled in volts. However, the meter is not a
voltmeter, meaning that it does not react to a voltage, but rather to
an electric field. Often, it is a regular field meter, for instance, a
field mill, or it may essentially just contain an operational amplifier
that reacts to the charge induced on a sensor plate at the front of the
The meter also has a stipulated measuring distance. In the case shown,
it is 0.25 in. This means that the meter was calibrated by placing it
0.25 in. from, and parallel to, a metal plate, which was then raised to
a range of voltages, and a corresponding scale was drawn.
Figure 10. Measurement results from a screened
So the question is, after this calibration, what can the meter be used
for? The answer is very simple: The meter can be used to measure the voltage
at a distance of 0.25 in. from a metal plate with the same dimensions
and the same capacitance as the one used for the factory calibration.
Figure 11. Measuring a conductor with constant
The problem is that manufacturers seem very reluctant to mention this,
or to just describe what the calibration conditions were and what happens
if the instrument is used under other, and maybe even more-everyday, conditions.
It is very rare, if it ever happens at all, for the dimensions of the
calibration plate, not to mention its capacitance, to be given in the
manual. Nor is there any warning that if the meter were to be pointed
toward an insulator, the reading in volts would never refer to the insulator
as a whole. As was mentioned in Part I of this article, an insulator does
not have a voltage. If the user is lucky, a kind of surface voltage may
It is something of a puzzle why static locators are always calibrated
in volts. After all, they are just ordinary field meters pretending
to be voltmeters, without really being so. All they can do is measure
the voltage of a certain metal plate at a certain distance. If these
meters were calibrated in units of field strength, that is, V ∙ m1,
they could be used much better to evaluate the static conditions of
insulators as well as conductors.
Figure 13. A static locator.
But could the explanation simply be that most people understand voltage
better than they do field strength? No, that does not seem possible. Just
look at Equations 1, 2, and 3 of this article. A voltage is always defined
by a field strength (and a distance), so if someone does not understand
one, that person would not understand the other.
This article has analyzed the problems connected with measuring the
voltage of a charged insulated conductor. The emphasis was placed on
noncontacting measurements, that is, measurements based on the effect
of the field from the charge. It has been demonstrated that the instruments
used will often distort the fields and hence change the properties to
be measured. It was also shown that, by screening the meters, it is
often possible to reduce the field distortions considerably.
1. Niels Jonassen, "Surface Voltage and Field Strength: Part I, Insulators"
in Mr. Static, Compliance Engineering 18, no. 7 (2001): 2633.
Niels Jonassen, MSc, DSc, worked for 40 years at
the Technical University of Denmark, where he conducted classes in electromagnetism,
static and atmospheric electricity, airborne radioactivity, and indoor
climate. After retiring, he divided his time among the
laboratory, his home, and Thailand, writing on static
electricity topics and pursuing cooking classes. He passed away in 2006.