How Fast Does a Charge Decay?
There's a phrase that has been bothering me for years: "How do
you remove static electricity?" At one level the question makes
sense. Everybody involved in electrostatics understands what the
inquirer is trying to ask. But at a physics level, as well as a
linguistic one, the phrasing is more dubious. A better expression
of the question would be "How do you neutralize the field from static
Why is this phraseology better? Well, first of all, the
field from a charge (distribution) is a well-defined concept, which
static electricity is not. And secondly, when you do neutralize
a field (or "remove static electricity"), you very rarely remove
anything from the charged body. (When you ground a negatively charged
conductor with a metallic wire and avoid all kinds of discharges,
you lead away the excess electrons. But that is the only case where
charges can be removed.)
In order for neutralization to happen, the charged area
has to be in contact with a medium containing charge carriers of
the opposite polarity. A force from the field then acts upon these
charge carriers, and, if they have some ability to move, they'll
eventually plate out on the charged area. The field from the plated-out
carriers will superimpose the original field, resulting in a steadily
decreasing "total" field. In other words the static charge is decaying.
So let's change the question from how to remove static electricity
to how fast does a charge decay.
Bulk and Surface Decay
It is easier to describe the decay if we consider separately bulk
decay, where charges move through the interior of the material,
and surface decay, where the movement of charges takes place primarily
in a surface layer.
Bulk Resistivity. The rate at which neutralization takes place
in a given field depends upon the conductivity of
the medium. A field E will release a current with the density (current
per unit area) j given by
Equation (1) is often written
is the bulk resistivity of the medium. Equations
(1) and (2) are both versions of Ohm's law (in differential form).
The field from a given charge will always be proportional to the
charge, but the factor of proportionality will depend upon the geometry
and dielectric properties of the charged body and its surroundings.
|Figure 1. Bulk decay of charge, situation 1.
Let's look at a simple example. Figure 1 (situation
1) shows a piece of a material, with the resistivity
and the permittivity ,
resting on a grounded plane. A charge q is evenly distributed on
the surface of the material. We assume that the distance to other
grounded objects is much larger than the dimensions of the charged
sample. If the charge density is (C
m2), then the field strength in the material
According to equation (2) this field will produce
a (negative) current
But the current density j is also the rate at which
the surface density decreases, that is
The solution to equation (5) is
is the initial value of the charge density. Thus it appears that
the charge is being neutralized exponentially with the time constant
Equation (7) is generally valid when the field from
the charge to be neutralized extends exclusively through the medium
with the resistivity
and the permittivity .
Consider a sample of Plexiglas with
3 1011 F m1 (r
3.4). A charge on it will decay with a time constant of about 300
seconds. It should be appreciated that the rate of decay is determined
not only by the resistivity, but also by the permittivity. So if
we have a sample with the same resistivity as the Plexiglas, but
with twice the permittivity, the rate of decay will be half that
of the Plexiglas.
The situation is more complicated, however,
if the field from the charge, or rather the flux, extends through
several dielectrics with different resistivities and permittivities.
Thus, a brief digression to discuss electrical flux is useful here.
The electrical flux or E-flux E
through a surface S is defined as
If the surface S is a closed surface surrounding
a charge q, then, assuming you have the same permittivity all over
the surface S, the previous equation becomes
This is simply Gauss' theorem, which enables calculation
of the field from various charge distributions. Flux being a rather
abstract concept, it can be helpful to envisage the situation as
a charge "emitting" a certain number of field lines. The number
of those field lines through a unit area (perpendicular to the field
strength) is equal to the field strength. So the flux through a
given area is, roughly speaking, the number of field lines through
Now back to the more-complex situation. Figure
2 (situation 2) shows a sample with the thickness d, permittivity
and resistivity ,
resting on a grounded plane, like that shown in Figure 1. But in
this case another grounded plane is placed parallel to the sample
at a distance x. Let us assume that the sample is Plexiglas,
and that the space above the sample is vacuum (or air) with
The field (flux) from the charge is now shared between the Plexiglas
and the air in such a way that the surface potential of a point
on the charged surface is the same whether it's calculated as the
field strength in air multiplied by x or as the field strength
in the dielectric multiplied by d. Thus the charge is expected to
decay exponentially again, but now with a time constant given
For instance if we choose d = 0.01 m and x
= 0.003 m (r
= 3.4, the relative permittivity of Plexiglas), we find that
= 594 seconds. In other words, it takes about twice as long for
the charge to decay, even from the same sample of material, simply
because of the proximity of another grounded conductor.
The example shown in Figure 2 is a very simplified
case, and often it will not be possible to predict the relevant
value of the time constant for a given sample in a given geometrical
|Figure 2. Bulk decay of charge, situation 2.
Surface Resistivity. Special cases are
the ones in which neutralization takes place in a shallow layer
on the surface of the material. This could be a material treated
with an antistatic agent or an insulative substrate onto which a
conductive layer is evaporated. If such a layer is highly conductive
as compared with the contacting materials, the neutralizing current
will run only in this layer. However, part of the flux from the
charge will run in the adjoining layers, and the "driving field,"
that is, the field in the conductive layer, will depend upon the
permittive properties of the adjoining insulators. Thus the rate
of decay (and the time constant) will depend not only on the properties
of the region where the decay takes place, but also on properties
outside the region of decay. This is, in principle, the same problem
shown in Figure 2. Usually the processes in thin layers are characterized
by defining a surface resistivity s
(in a way similar to the definition of bulk resistivity) by the
This version of Ohm's law states that a field Es
along a surface with the surface resistivity s
will cause a current with the linear current density js
(current per unit length, A m1) in the layer
given by equation (9).
Although in the matter of bulk resistivity
it is possible in certain simplified cases (Figures 1 and 2) to
derive a connection between the resistivity and the rate at which
a charge is being neutralized, it is not nearly as simple in the
matter of surface resistivity.
Figure 3 (situation 1) shows a piece of material
A. At one end of A is a spot of negative charge and, at the other
end, a grounded electrode B in direct contact with A. Between B
and A is a field. Only that fraction of the flux that runs through
the conductive layer will cause a current to neutralize the charge.
There is no doubt that if the charge is, say, doubled, then the
field strength will be doubled in every point, but the field distribution
will be the same. And if the surface resistivity is doubled, the
decay rate will be halved. With this geometry, it seems likely that
we will have a time constant proportional to the surface resistivity.
But, in contrast to simple situation 1 for bulk decay (Figure 1),
we cannot theoretically predict--even if we measure the surface
resistivity and know the permittivity of the conductive layer--the
time constant for surface decay. This is because we don't know how
the flux is distributed between the conductive layer and the environment.
|Figure 3. Surface decay of charge, situation
Figure 4 (situation 2) shows a state similar
to situation 2 for bulk decay (see Figure 2). Another grounded conductor
C is in the neighborhood of the charged sample, but not in direct
contact with it, so no neutralizing current will flow to C. And
since the flux to B is now lower, so is the neutralizing current,
and the time constant will have increased, even if the sample, the
charge, and the grounding electrode arrangement is the same.
|Figure 4. Surface decay of charge, situation
This discussion has tacitly assumed that there
is only one value for the resistivity (be it bulk or surface) independent
of the field applied. Yet it is often found that the resistivity
increases with decreasing field strength. Nevertheless, resistivities
are usually determined at only one field strength (one voltage difference
between a set of electrodes on the sample), and we have no way of
knowing if this particular field strength is typical for the physical
conditions during a decay process.
Measurement of Decay Time
The previous considerations illustrate that
only under very ideal conditions is it possible to calculate reasonably
accurately from material parameters (resistivity and permittivity)
how fast a charge on an insulator will decay. This is because of
two main reasons:
- The resistivity depends on the field strength from the
decaying charge (and we rarely know this relationship), and even
- The driving field from a given charge depends on the
permittive properties of the environment in a usually incalculable
So the obvious question is why not measure the decay
time directly? If we are dealing with a highly resistive item, it
is certainly possible to charge the material and measure how fast
the field from the charge decays when the item is placed in a relevant
environment. Usually we are interested in semiinsulative materials
where the charges are neutralized in seconds or less. And the procedures
of measurement have to allow for this.
Over the years several procedures have been
developed, and, to be kind, none of them were very successful. A
general shortcoming of all these methods is that they do not measure
in situ. That is, the measurements are performed not on the material
as it normally appears when it gets charged, but rather on sheet
samples suspended in such a way as to facilitate charging as well
as field measurement.
Probably the most commonly used method is Federal
Test Method (FTM) Standard 101C, method 4046.1, where a sample is
clamped between two electrodes (see Figure 5). A field meter is
mounted pointing at the center of the sample midway between the
electrodes. The sample is allegedly charged by the electrodes when
they are connected to a voltage supply, and the charge decay is
taken as the reading of the field meter after the electrodes are
grounded. It seems difficult (at least to this author) to be sure
that a reading of the field meter is a sign of an excess charge
on the material, unless the material is truly conductive. Polarization
may certainly show itself, at least with some materials, as an external
field, and the rate of relaxation of polarization is not necessarily
the same as that of a true excess charge.
|Figure 5. FTM Standard 101C decay of field
Several other questions could be raised concerning
this method. The most important one is that a decay time obtained
by method 4046.1 for a sheet of a material of a given small size
does not reveal much about how fast the field from a charge will
be neutralized on a larger sample or item in another location.
In another method, the sample (again a suspended
sheet) is charged by a corona discharge. The charger is then removed
and replaced by a field meter. (Incidentally, we developed this
method, which has the merit of placing a real charge on the surface
of the material under investigation, at our laboratory as early
as 1977, but ultimately abandoned it since our instrumentation was
not fast enough.) Although it has been argued that the corona charging
with air ions may be irregular, one could also argue that the charging
experienced in everyday life is irregular too. So this should rather
be deemed a virtue of the method. Still the main argument is that
one does not measure the charge neutralization (decay) rate under
circumstances that resemble normal use of the materials.
It should also be mentioned that it is not
possible to distinguish between bulk and surface decay using either
of these methods, or probably any other method for that matter.
It may even be argued that the distinction does not make sense at
all. Another objection to any principle, suggested or applied, for
determination of charge decay time is that any method capable of
detecting the presence and time variation of a charge on an object
will occupy a certain fraction of the electrical flux from the charge,
a fraction which, without the presence of the measuring equipment,
might contribute to the rate of neutralization or decay of the charge.
Thus the measured rate of decay will normally be different from
(often larger than) the "natural," undisturbed rate.
The considerations presented in this paper
may make it seem as if we know nothing about the laws of decay of
charges on insulators. This is not the case.
Although we can accurately predict the current
I through a resistor with the resistance R from a voltage supply
with output voltage V, we have to accept that static electricity
(ESD, if you insist) is a little more complicated (and interesting).
We also have to accept the fact that there's no way you can predict
the decay behavior of a manufactured item placed in an arbitrary
environment by doing some laboratory measurements on a sample of
the material of said item.
So what do we do when we have to choose between
different materials? Well, we know that if we have two materials
with different resistivities, bulk or surface, under similar circumstances
the one with the lowest resistivity will mean the fastest decay
time, although not in an unambiguous way. So the obvious advice
is to choose the material with the lowest resistivity.
Niels Jonassen, MS, DSc, is retired from the Technical
University of Denmark, where he has conducted classes on static
electricity, ions, 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.
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