Electrical breakdown is not always associated with high voltage.
It is often assumed that an electrical breakdown, be it in a gas, liquid,
or solid dielectric, is caused by a high voltage. It is possible to
have a very high voltage without having a breakdown, and to create a
breakdown at rather modest voltage levels; furthermore, the most common
discharges (i.e., those involving insulators) cannot be associated with
a voltage at all.
A breakdown is the result of a substantial increase in the concentration of
mobile charge carriers in the medium considered. This increase is usually
caused by the effect of an electric field on the already existing carriers.
In atmospheric air, these charge carriers are atmospheric ions and a
few free electrons from natural radioactive decay.
Air ions are molecular clusters consisting mostly of water molecules around
a singly charged oxygen or nitrogen molecule. In positive air ions,
the charged molecule may be either oxygen or nitrogen, and there may
be 1215 attached water molecules. In negative air ions, the charged
molecule is oxygen, and there may be 812 attached water molecules.1
Both ions and free electrons participate in the random thermal movement
of the molecules. It should be stressed that the thermal energy of both
ions and electrons is far too low for a thermal collision to result
in an electron being knocked off an air molecule (i.e., ionization).
However, if an electric field exists in the gas, the charged particles
would be accelerated in the field and gain extra kinetic energy. If
a particle with charge q is moved a distance Dx
by a field with the strength E, the particle gains an increase
in its kinetic energy equal to
provided that the particle does not collide with other particles over
the distance Dx.
Because an electron and a negative air ion both carry an elementary
charge e, they will gain the same increase in energy if they travel
the same distance. However, because of the difference in size, the electron
is able to travel much farther between collisions with other particles
than is the ion. The mean distance traveled between collisions is called
the mean free path.
Breakdown Field Strength and Breakdown Voltage
Figure 1 shows an electron with the mean free path le and
a negative ion with the mean free path li in a homogeneous
electric field E.
|Figure 1. Mean free paths (not to scale) of
an electron and a negative air ion.
The maximum energy DWmax,e that
can be reached by an electron is thus
= e • E • le. (2)
The corresponding energy for an ion is
= e • E • li. (3)
It takes an energy Wion to ionize an air molecule
(i.e., to knock off an electron, eventually creating a pair of positive
and negative air ions). Because the mean free paths (in atmospheric
air and at atmospheric pressure) of an electron and an ion are le
≈ 105 m and li
≈107 m, respectively, an electron is able
to reach this energy at a field strength (the breakdown field strength
Eb) that is approximately 100 times lower than would
be the case for an ion. Eb is thus given by
Because Wion ≈ 5 • 1018 J
(or ≈ 30 eV) for atmospheric air, we find from Equation 4
The result of Equation 5 is valid only for a homogeneous field, such
as may be established between two parallel electrodes (see Figure 2).
The field strength E in the space between the electrodes is
where V is the voltage difference and d the distance between
|Figure 2. Homogeneous field between plane electrodes.
The breakdown voltage is defined by
the voltage difference it takes to establish the breakdown field strength
between the electrodes and cause a discharge, in this case, a spark.
Assuming that d = 3 cm = 3 • 102
m, then the breakdown voltage is
Vb = Eb • d = 3 •
106 • 3 • 102 = 90,000 V.
If the field is not homogeneous, the conditions become more complicated. Figure
3 shows a situation similar to the one in Figure 2, except that one
electrode is a small sphere, instead of a plane.
|Figure 3. Inhomogeneous field from nonplanar
The different geometry results in higher breakdown field strength
and lower breakdown voltage. The figure shows that the field strength
has its maximum at the front of the small electrode; consequently, this
is where a discharge may start.
Let us assume that a field strength of 3 • 106 V
• m1 has been established at the tip of the electrode.
This field strength would start a discharge in a homogeneous field (see
Figure 2) because an electron would gain the critical energy Wion
over its mean free path (see Equation 4).
However, the field in Figure 3 decreases rapidly as you move away
from the electrode, even over the short distance of the mean free path,
and an electron cannot gain enough energy to ionize.
Therefore, we get a breakdown field strength Eb,sphere
> 3 • 106 V • m1. The actual
value of Eb depends on the radius of curvature and
shape of the electrode, but not in a simple way.
Although the breakdown field strength is higher in an inhomogeneous
field, the actual maximum field strength caused by a given voltage is
If the linear dimensions of the grounded conductor in Figure 3 are
much greater than the distance d and the radius r of the
sphere, then the maximum field strength at the front of the sphere,
Emax, can be approximated by
If we let Emax be the breakdown field strength Eb,
the breakdown voltage can be written as
Equation 9 suggests that the breakdown voltage for a spherical electrode
approaches a maximum value of
for increasing distances d, and that 95% of this value is reached
at a distance d = 10 • r.
Figure 4 shows the breakdown voltage Vb for parallel
electrodes (calculated from Equation 7) and for a 1-mm sphere (calculated
from Equation 9). In the case of the sphere, the voltage is normalized
relative to the maximum value of r • Eb.
|Figure 4. Breakdown voltage Vb for
spherical and parallel-plane electrodes.
In developing Equation 8, the influence on the field of the connections
to a voltage supply has not been included. This necessary simplification
can also be deduced from Equation 10, which many readers will recognize
as the relationship between the potential and the surface field strength
of a conductive sphere located far away from other conductors.
The question of absolute values for the breakdown field strength as
a function of electrode dimensions has received little attention in
recent years. The latest reference I have been able to find is
in Handbuch der Physik, which gives
as the breakdown field strength for a sphere with radius r measured
in meters.2 As r approaches infinity, Eb
approaches 3 • 106 V • m1, the
breakdown field strength for plane electrodes. There is a corresponding
formula for wires with radius r, in which the numerator 18 is
replaced by 9.
Figure 5 shows the breakdown field strength Eb calculated
from Equation 11, and the corresponding breakdown voltage Vb
calculated from Equation 10.
|Figure 5. Breakdown field strength and voltage
for spherical electrodes.
To investigate the validity of Equations 10 and 11, the following
experiment was performed. A pointed electrode with a well-defined radius
of curvature was placed above a metallic plane connected to ground through
a suitable current meter, at a distance of 1020 times the electrode's
radius. The voltage of the electrode was raised until a corona current
to the plane could be measured. The voltage at which this happens is
the breakdown voltage for the radius in question. This measurement was
performed with electrodes with radii from 0.25 to 5 mm, and the results
are shown in Figure 5 as the experimental points.
For a 0.5-mm electrode, the breakdown voltage appears to be about
5.5 kV at a distance of 1 cm, as compared with the 30-kV breakdown voltage
for plane electrodes at the same distance. The much lower breakdown
voltage at a spherical electrode, as compared with the parallel electrodes,
is not the only difference between the two situations.
When a discharge happens between parallel electrodes, the breakdown
field strength is, in principle, exceeded at all points. The discharge
may therefore start anywhere the odd electron happens to appear, and
the ionization takes place along a path connecting the two electrodes.
This is what constitutes a spark.
In the case of a spherical (or otherwise pointed) electrode, the breakdown
field strength is exceeded only in a small volume in front of the electrode,
and ions are formed only in this region. If the electrode is positive,
the positive ions outside the ionization region will move away from
the positive electrode without multiplying, and the negative ions will
move toward the electrode, where they will be neutralized. This process
constitutes a corona discharge.
According to Equation 7, we would expect to get a spark discharge
between plane electrodes at a distance d, as long as the voltage
difference V fulfills the condition V ≥ Eb
• d, where d is the distance between the electrodes,
and Eb is the breakdown field strength for plane electrodes.
In atmospheric air at 1 atm, Eb = 3 • 106
V • m1. Therefore, at a distance of 0.1 mm =
104 m, breakdown should happen at V = 3 •
106 • 104 = 300 V. However, this is
not so (see Figure 6).
|Figure 6. Paschen's curve.
Figure 6, known as Paschen's curve, shows the breakdown voltage for
plane electrodes in atmospheric air as a function of the product of
the atmospheric pressure and the electrode distance. The product is
given in units of mmHg • cm. Given a pressure of 1 atm (760 mmHg)
and an electrode distance of 0.1 cm, we find at an abscissa of 76 mmHg
• cm a breakdown voltage of 3000 V, corresponding to the breakdown
field strength of 3 • 106 V • m1
for plane electrodes.
The curve also shows that there is a minimum breakdown voltage of
about 330340 V, corresponding at 1 atm to a distance of 56
µm and a field strength of approximately
6 • 107 V • m1. It means that
it is not possible to get a discharge between parallel electrodes at
a voltage less than approximately 300 V.
What happens if the electrodes are not planar? Is it possible to get
a corona discharge at, say, 300 V?
Combining Equations 10 and 11 to solve for r, we find that
a breakdown voltage of 300 V would correspond approximately to an electrode
radius of 3 µm. However, the relationship
in Equation 10 is, at best, semiempirical and was definitely never tested
at such small dimensions. It is therefore still unknown whether a version
of Paschen's law would apply to nonparallel electrodes.
In most, at least unwanted, discharges, insulative materials are involved.
And certainly, breakdown may happen in fields from charged insulators.
However, most of the rules and concepts outlined previously for conductors
are not applicable to insulators.
Although discharges in the field between oppositely charged insulators
are possible, there is no simple way in which such events may be predicted.
The most common discharge in which insulators are involved happens in
the field between a charged insulator and a grounded conductor.
Figure 7 shows a uniformly charged free insulative disc. A grounded
conductor is placed parallel to the insulator at a distance d.
If the charge density on the disc is s, the
field strength is
where eo is the permittivity of air (8.85 • 1012
F • m1).
|Figure 7. Homogeneous field from charged insulator.
Assuming that s = 2 • 105
C • m2, then the field strength would be approximately
2.3 • 106 V • m1, that is, not
very far from the breakdown field strength for a homogeneous field.
Therefore, there is a risk of a discharge of some kindnot a spark,
but probably a brush discharge, which will quench itself while the charge
on the insulator is being (partly) neutralized.
The possibility or risk of getting a discharge can be evaluated by
measuring the field strength at the site of the grounded conductor.
At this point, some readers might say, "Why not just measure the surface
potential of the insulator?" Well, let us analyze the situation. If
the distance d = 1 cm = 102 m, then the surface
potential of the insulator is Vs = 102
• 2.3 • 106 = 23,000 V.
If this sounds like a dangerously high voltage, just decrease the
distance to 1 mm, and the voltage goes down to 2300 V. The risk for
a discharge has not changed, because the field strength is still the
The surface potential provides information about the risk of discharge
only if it is used to calculate the field strength. For that same reason,
the lower value of 300 V of Paschen's curve cannot be applied to insulators.
Figure 8 shows a common situation in which discharges are caused by
a charged insulator. A grounded conductor, in this case a small sphere,
is brought into the proximity of a charged insulator that may or may
not be uniformly charged. In all cases, the field will be strongly inhomogeneous.
The situation looks at first sight very similar to the one in Figure
3, showing a sphere at a positive voltage near a grounded plane conductor.
However, the conditions are very different between the two cases.
|Figure 8. Inhomogeneous field from a charged
In the case of Figure 3, with two conductors facing each other, the
distribution of the field is given when the geometry is given, and it
may be possible to define a breakdown field strength and a breakdown
voltage that can be compared with the voltage of the electrode.
In the situation in Figure 8, it may be possible to calculate the
breakdown field strength (using Equation 11, in case of a sphere) at
the face of the electrode and compare it with the field measured at
the site of the grounded electrode. The magnitude of this field, however,
can be achieved by an infinity of charge distributions on the insulator,
and it cannot be related to or deduced from any surface potential measured
in front of the insulator.
Electrical breakdown in air is caused by electrons being accelerated
in a sufficiently strong field. For electrodes with given dimensions,
it is possible to define the critical field strength, the so-called
breakdown field strength, and, in the case of two conductors, a corresponding
In the case of discharges caused by charged insulators to a grounded
conductor, the condition for a discharge is determined by the field
strength at the face of the conductor, and the same breakdown field
strength applies here as in the case of two conductors. The risk of
a discharge from an insulator cannot be evaluated by the measurement
of a surface potential.
1. Niels Jonassen, "Ions" in Mr. Static, Compliance Engineering
16, no. 3 (1999): 2428.
2. Handbuch der Physik 14 (Berlin: Springer Verlag, 1927):
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.