By definition, insulators do not have a voltage.

Niels Jonassen

This article, the first of a two-part series on measuring voltage and field strength, examines the controversial topic of an insulator's surface voltage and field strength. The discussion will include both theory and actual measurements, and will begin with a review of the most important features for a charged conductor and how these features differ for a charged insulator.

Charged Conductors

 

Figure 1. Charged conductor.

Figure 1 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 the field in the interior of the conductor will be zero, the field will be perpendicular to the surface, and the integral of the field strength E from any point P in or on the conductor to a ground point G is constant and given by

where V is the voltage or potential of the conductor.

The voltage V and the charge q are proportional, and q is usually written as

where C is the capacitance of the insulated conductor and is determined by the conductor's size and shape, and its placement relative to other conductors and ground. The charged system stores an electrostatic energy of

which can be dissipated in a single discharge or current pulse.

Charged Insulators

 

Figure 2. Charged insulator.

Figure 2 shows a charged insulator. The field conditions here are very different from those at a charged conductor: The polarity of the charge may be different from point to point, the field in the interior may be different from zero, the field is not necessarily perpendicular to the surface, and the integral of the field strength from a point on or in the insulator to ground is usually different from point to point.

In Figure 2, the integrals of the field strength for P₁ and P₂ are

respectively. V_P1 and V_P2 are the surface voltages (or surface potentials) of the two points. In general, the surface voltage of an insulator will vary from point to point, as will the voltage of any point in the interior. It is therefore not possible to characterize a charged insulator with a single voltage figure. In other words, an insulator does not have a voltage.

 

Figure 3. Uniformly surface-charged, spherical insulator.

Many people do not like to accept this simple fact, so specifics need to be discussed. There are cases in which the surface of an insulator has a constant surface voltage. But apart from such instances, there is only one situation in which all points in and on an insulator can be ascribed a well-defined (but unmeasurable) voltage. If a spherical insulator with radius R and uniform charge q (see Figure 3) is placed infinitely far (a distance much greater than R) from any conductors, the sphere would have a voltage of

However, this very theoretical situation is the only case in which it makes sense to talk about the voltage of an insulator.

Similarly, the concept of an insulator's capacitance is meaningless. Although it is possible to get a discharge from a charged insulator, the discharge will always be partial, and the energy dissipated can neither be related to the total charge nor be related to any kind of voltage. In other words, voltage and capacitance are quantities of a conductor, not an insulator.

So a natural question arises: what measurements can be taken from a charged insulator? The simple answer is that the effect of the field from the charge, and sometimes the total charge, can be measured. This article will concentrate on the direct effect of the field. As with conductors, the instruments used for measurement are field meters and noncontacting voltmeters. Both types of instruments will distort the fields to be measured unless properly screened. Uniformly charged free insulative sheets and uniformly charged insulative sheets backed by a grounded conductor are the only two cases in which it is possible to make quantitatively reliable measurements of charged insulators.

Uniformly Charged Sheets

 

Figure 4. Static measurement on free charged sheet.

Figure 4 shows a uniformly charged insulative sheet. If the field strength indicated on the meter is E, the charge density s on the part of the insulator in front of the meter should be

If a noncontacting voltmeter is placed at a distance d from the sheet, then the surface voltage V_s indicated on the meter would be given by

 

Figure 5. Field strength from and surface voltage of free plastic sheet.

Figure 5 shows the field strength E from a free plastic sheet with a total charge q @ 0.5 ∙10^–7 C. The area of the sheet is 21 x 29 cm², which gives an average charge density of

 

Figure 6. Uniformly charged insulator disk, backed by grounded conductor.

The figure shows that the field strength E is relatively constant at about 88 kV∙m^–1 to a distance of approximately 5–6 cm. According to Equation 5, this corresponds to a charge density of s = 8.85 ∙ 10^–12 ∙ 88 ∙ 10³ = 0.78 ∙ 10^–6 C∙m^–2. Considering the uncertainty of the measurements of the total charge and of the field strength, the agreement between the calculated and measured values of the charge density (s_avg = 0.82 ∙ 10^–6 C∙m^–2 versus s = 0.78 ∙ 10^–6 C∙m^–2) seems satisfactory.

It therefore appears that measurement of the field strength near a free charged sheet leads to information about the charge density and charge distribution on the surface. In the region where the field is homogeneous, the surface voltage of the sheet is proportional to the distance from the sheet and is measured, using Equation 6, by a noncontacting voltmeter. This measurement then leads to the surface charge density, given that the measuring distance can be estimated with reasonable accuracy. However, it should be stressed that a measurement of the surface voltage does not provide any more or better information about the charged state of the insulative sheet than a measurement of the near-surface field strength does.

Insulator Disk

Figure 6 shows an insulator disk with permittivity e and thickness t. The disk is resting on a grounded plane and has a positive charge with density s (C∙m^–2). If the disk is far from other conductors, the field inside the material will be given by E₁ = ^s/_e, and each point on the surface will then have a voltage of

It should be stressed that V_s is not the voltage of the insulator disk, but only of the surface. Any point inside the insulator has a different, unmeasurable voltage.

Figure 7. Uniformly charged insulator disk between grounded backing electrode and free grounded electrode.

The situation shown in Figure 6, with the disk far from conductors other than the grounded base, is of little practical interest because it excludes the presence of meters. A more common situation is shown in Figure 7, in which a grounded electrode A is parallel to the charged disk at a distance d. The field strength in the space between the charged disk and A would be given by

The grounded plane A might typically be the place where a field meter or noncontacting voltmeter is placed, with distance d being much greater than thickness t. The charged disk can be, for instance, an electret or a web. With these conditions, Equation 9 can be written as

The surface voltage, which is almost equal to the undisturbed value, can be written as

It appears that, under these conditions, it is possible to estimate the charge density by measuring either the field strength or the surface voltage from the charged disk, assuming the permittivity and thickness of the disk are known.

Sheet with Grounded Conductor

 

Figure 8. Uniformly charged insulator backed by a grounded conductor.

Figure 8 shows an experimental set-up corresponding to the conditions described in Figure 7. This could, for example, be a charged web or an electret. The charged insulator is a 1-mm plate with dimensions of 0.21 x 0.29 m². The relative permittivity (dielectric constant) of the material is e_r » 2 (e = 1.77 ∙ 10^–11 F∙m^–1). The total charge on the free surface of the insulator is q » 2.7 ∙ 10^–7 C, leading to an average surface charge density of s » 4.4 ∙ 10^–6 C∙m^–2.

In the absence of a field meter (and other grounded objects, not including the backing plate), the surface potential of each point on the surface can be calculated using Equation 7 as

When the field meter is placed in front of a charged plate, the electric flux from the charge is shared between the field meter and the backing plate. Consequently, the internal field and the surface voltage will be reduced slightly, depending on how far away the meter is placed. There will also be a field E_d in the space between the charged plate and the field meter. This field is the only quantity of the charged plate that can possibly be measured.

Figure 9 shows the field strength from and surface voltage of the disk shown in Figure 8. At 5 cm, the field strength and surface potential are measured to be E₅ » 4.6 kV∙m^–1 and V_s » 235 V, respectively. According to Equation 9, this corresponds to a charge density of

Comparing this with the calculated value of s = 4.4 ∙ 10^–6 C∙m^–2 and considering the uncertainties in the quantities involved, especially in the uniformity of the initial surface charging and the effective distance to the meter, the agreement between the calculated and measured values is surprisingly good: 4.4 ∙ 10^–6 C∙m^–2 and 4.1 ∙ 10^–6 C∙m^–2, respectively.

 

Figure 9. Field strength from and surface voltage of a uniformly charged plastic sheet backed by a grounded conductor.

As shown in Figure 9, the surface voltage, E∙d, is relatively independent of the distance to the meter, and this feature will be even more pronounced in the cases of thinner insulators such as real electrets and webs, which have thicknesses on the order of 50–100 µm.

General Comments

Free insulative sheets and insulative sheets backed by a grounded conductor are the only cases in which it is possible to extract reliable information from a noncontacting measurement of the charged state of an insulator. In both cases, the electric field from the charge is the deciding factor. With a free sheet (or just a relatively planar insulator), the electric field measured at a short distance (a few centimeters) will provide all the possible information—that is, the charge density. If a noncontacting voltmeter is used, the distance will have to be measured in order to convert the surface voltage to surface charge density. Surface voltage in itself does not provide extra information.

In the case of a sheet backed by a conductor, the surface voltage is relatively constant. If the thickness and permittivity of the material are known, then the surface voltage could be used to calculate the surface charge density. If a field meter is used, then the distance would also have to be measured. Field strength depends on the surface parameters (thickness and permittivity) in the same way surface voltage does.

Even in the well-defined situations of a free charged sheet and a backed charged sheet, a noncontacting measurement will, at best, only provide information about the charge density. Sometimes a field measurement (free charged sheet) is the most relevant, whereas at other times a direct surface-voltage measurement (backed charged sheet) is the most relevant. However, either measurement will only lead to the charge density.

 

Figure 10. Static measurement of the field strength and surface voltage on a plastic container.

But what happens if the charged insulator is not one of the well-defined objects previously described, and the meter is just pointed toward an ordinary object? The answer can be found in Figure 10, which shows a plastic container. A screened field meter very close to the container identifies a field strength E = +100,000 V∙m^–1. A noncontacting voltmeter at a distance of 2 cm (as well as the distance can be measured) identifies a surface voltage V_s = +2 kV. What can be concluded from these measurements? A prudent and safe answer is that the container is positively charged.

If the situation in Figure 10 is approximated with that of Figure 4, using Equations 5 and 6, both readings would suggest that the surface charge density in front of the meters is positive and on the order of 1 µC∙m^–2. This result, however, is very uncertain, especially when using a noncontacting voltmeter, because the reading is approximately inversely proportional to the measuring distance. If the measuring distance of 2 cm can be read with an accuracy of ±2 mm, then there is already an uncertainty of 10%, regardless of meter sensitivity. If the distance is increased, then charges other than those on the surface immediately facing the meter will influence the reading and make the interpretation even more uncertain.

Static Locators

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.

The meters so used are known as static locators. And that is exactly what they are: instruments used to locate a static-electric field. As long as that is the only thing they are used for, everything should work fine. Static locators are scaled in volts and have a stipulated measuring range. However, the meter is not a voltmeter, meaning it doesn't react to voltage, but rather to an electric field.

If a static locator is a real field meter (e.g., a field mill) and has a scale in V∙m^–1 (or kV/in.), it may be used close to charged insulators to estimate the surface charge density, as explained above. If the scale is in volts, the reading may approximate the surface voltage and can, using Equation 6, lead to the surface charge density.

With both types of measurements, the results may have a high uncertainty and even errors, especially if the meters are not screened. Even if the meters are screened, there is also the influence of charges other than the ones immediately facing the meters—for instance, the charges on the other side of the insulator. The second part of this series on voltage and field strength will discuss static locators in more detail.

Conclusion

It is easy to determine whether an insulator is charged. Just point a suitable meter at the insulator and take a reading. If the measurement is done carefully, then the reading may provide information about how much charge is located on a unit area of the facing surface (i.e., the surface charge density, C∙m^–2), as well as the polarity of the charge.

However, the problem is that no meters are calibrated for this unit of measurement. The meters with the closest unit are field meters with scales in volts per meter, V∙m^–1. Fortunately, the volts-per-meter measurement can be multiplied by e_o (8.85∙10^–12 F∙m^–1) to arrive at the charge density.

The bad news, however, is that most meters have scales in volts. In all cases, these meters have been calibrated relative to conductors, where the concept of voltage makes sense. Used in connection with insulators, the reading may at best be an approximation of the surface voltage, which characterizes only a part of the insulator's surface, not the insulator. In this case, the reading in volts, when multiplied by e_o and divided by the measuring distance, can also lead to the surface charge density. It should be stressed that the voltage of an insulator has no meaning. All that can be found by any noncontacting measurement on a charged insulator is the polarity of the charge and, if the measurement is done carefully, the surface charge density.

Niels Jonassen, MSc, DSc, worked for 40 years at the Technical University of Denmark. 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.

Niels Jonassen

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