"How Fast Does a Charge Decay?" the neutralization of
charges by conduction through the bulk or along the surface
of a material was discussed.1 Although bulk
and surface conduction are important processes in static
neutralization, in many situations, the materials involved
cannot be made sufficiently conductive in an acceptable
way. In these cases, the only solution is neutralization
by air ions. The physical properties of air ions were
discussed in "Ions."2 However, their use in
charge neutralization was touched on only briefly.
ions or atmospheric ions differ from electrolytic ions in
several ways. Air ions are small molecular clusters consisting
mostly of water molecules grouped around a nitrogen or oxygen
molecule that has lost or gained an electron. In the case
of positive ions, the number of water molecules may range
from 10 to 15, and in the case of negative ions, from 8
to 12 (see Figure 1). In the latter case, the (negatively)
charged molecule is always oxygen.
1. Positive and negative air ion.
ions have only a limited lifetime. In undisturbed air,
that lifetime is in the order of minutes. They disappear
in a number of ways. Positive and negative ions can recombine,
air ions can combine with airborne particles, and air
ions can plate out on surfaces, either by diffusion or
with the aid of an electric field. It is the latter process
that is utilized in charge neutralization.
will move in an electric field with a velocity v
proportional to the field strength E given by the
E . (1)
factor k is the mobility of the ion, and the unit of measurement
For positive ions, k+ ≈ 1.31.4
for negative ions, k ≈ 1.8
atmosphere with concentrations n+ and
n of positive and negative ions,
respectively, may be characterized by polar conductivities
l+ and l,
which are given by
= n+ke (2)
= nke , (3)
e is the (numerically) common charge of the ions.
it is more convenient to use the corresponding polar resistivities
2 shows a positively charged conductor A surrounded by an
atmosphere containing positive and negative ions. The field
created by A moves the surrounding ions. Positive ions are
repelled from the body and move away. Negative ions are
attracted, and eventually, they plate out on the charged
body. The field from the plated-out ions is superimposed
on the original field from the positive charge, thus making
it appear as if the charge on A has been reduced.
2. Charge neutralization.
us consider a section of the surface of A, where the charge
density is s+(C •
The charge will create a field perpendicular to the surface
with the strength
eo is the permittivity of air and of vacuum (8.85
to Ohm's law, this field causes a current directed toward
A, with the current density j(A
m2) given by
current density is the rate at which the charge density
is being neutralized (the rate at which it decays), that
the initial charge density and
the time constant for positive charges.
the charge density is integrated over the whole surface
of A, the total charge would be
the initial total positive charge. Similarly, for a negative
charge, we get
11 and 12 state that a charge on a conductor decreases exponentially
according to the time constant eor,
where r is the relevant (opposite)
resistivity of the air.
11 and 12 are valid only if the field from the charge
extends exclusively through air with a given resistivity.
However, in many (probably most) practical cases, the
electric flux from the charge is spread over several regions
of different permittivities and resistivities.
3 shows a relatively simple example of this situation. A
metal disk with an area A is placed on a slab of
insulative material (a dielectric) with a thickness d
and permittivity e, and resting
on a grounded plane. The disk has a positive charge q,
which creates a field Ed in the dielectric
and a field Ea in the ambient air. If
we assume the dielectric to be perfectly insulative, only
the field Ea will contribute to the neutralization
of q. The relationship between Ed,
Ea, and q can be estimated by considering
the capacitance of the disk with respect to ground, Cd,
and with respect to the surrounding air, Ca.
3. Flux distribution from a charged disk.
we assume that the thickness d of the dielectric
is much smaller than the linear dimensions of the metal
disk, the capacitance Cd can be written
Ca depends on the shape of the disk and
the distance (above the disk) to grounded surroundings.
An appropriate approximation of Ca seems
to be that of a semispherical capacitor (see Figure 4) with
r is the equivalent radius of the disk,
R is the average distance to grounded surroundings.
4. Semispherical capacitor.
we can usually assume that r << R, Equation
15 can be written as
total capacitance of the disk is
= Ca + Cd . (17)
charge q will be distributed on the disk, with
the topside, and
field in the surrounding air from the charge qa
causes the neutralizing current I
to flow to the disk. Using Equations 6, 7, and 8, we find
I is also the total charge's decay
(or neutralization) rate, that is,
21 has the solution
the time constant t+
is given by
using Equations 14 and 16,
A = pr2
and e = ereo,
where er is the relative
permittivity of the insulative slab, Equation 24 can be
Equations 5 and 10, Equation 25 may also be written as
a given ion environment, Equations 25, 26, and 27 give
the rate of neutralization by air ions of a positive charge
as a function of the geometrical and dielectric location
of the charge. Similar symmetric equations hold true for
the neutralization of negative charges.
25, 26, and 27 were developed for a charge located on
a conductor, that is, in a situation in which the concept
of capacitance can be applied. However, with some approximations,
the same formulas are also valid in the case of charged
5 shows the distribution of the electric flux from a charged
insulator disk. It is the same situation as the one shown
in Figure 3, except that in Figure 5, the metal disk has
been removed and the charge is uniformly distributed on
an area A of the insulator surface.
5. Flux distribution from a charged insulator.
fields Ea and Ed are
therefore the same in the two situations; consequently,
the neutralization processes are identical. Therefore, Equations
25, 26, and 27 still express the rate of charge neutralization.
some readers may ask, "When everything seems to be the same
for the conductor and the insulator, why not use the same
concepts, especially the concept of capacitance, in the
two cases?" The short version of the answer is the following.
If a conductor with a capacitance C and a charge
q is grounded, an energy
be released in a discharge (most likely a spark). In the
situation of Figure 3, the energy WC would
this energy can be released by approaching the surface
of the metal disk with a grounded wire. In the situation
of Figure 5, exactly the same energy is stored in the
field, as the flux distribution is the same in the two
cases. However, touching the surface of the charged insulator
with a grounded wire may provoke a brush discharge, releasing
a fraction of the total energy stored in the field. As
capacitance is intimately related not only to the energy
stored in the field but also to the energy that may be
dissipated in a discharge from the charged item, capacitance
has no place in the description of charged insulators.
has been shown that it is possible theoretically to relate
the neutralization rate of a charged item with given dimensions
and dielectric properties to quantities that are, at least
in principle, measurable. In Part II of this article,
the theory will be applied in practice.
Jonassen, "How Fast Does a Charge Decay?" in Mr. Static,
Compliance Engineering 17, no. 2 (2000): 2633.
Jonassen, "Ions" in Mr. Static, Compliance Engineering
16, no. 3 (1999): 2428.
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. HAfter 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.