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Файл:Steady electric current. Tutorial
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61
movement, similar to the movement of a ball in a viscous liquid. Let us find
the average velocity of ions under the influence of an electric field.
The movement of ions under the action of the electric field force
qEF
E
=
(4.2)
is impeded by the friction force, proportional to the velocity of their movement, and determined by the Stokes formula1
rVη6πF
fr
−=
, (4.3)
where η is the viscosity of the medium, r is the radius of the ion, V s the
velocity of its movement; The minus sign means that the friction force is
directed opposite to the ion movement. The charge of an ion q is equal to the
product of its valence Z and the elementary charge е.
The acceleration of ions in an electric field continues until these forces
become equal to each other
rV6ZeE ηπ=
, (4.4)
and the average velocity of ion movement will be established, equal to
E
r6
Ze
U =
ηπ
. (4.5)
The proportionality coefficient, numerically equal to the ion velocity
at E = 1 V/m, is called the ion mobility:
r6
Ze
b
ηπ
=
. (4.6)
Let us write the corresponding expressions for the mobilities of positive and negative ions of the electrolyte:
+
+
+
=
r6
eZ
b
ηπ
, (4.7)
−
−
=
r6
eZ
b
ηπ
, (4.8)
where Z+ and Z–, r+ and r– are the valences and radii of positive and negative
ions.
1
George Gabriel Stokes (1819–1903) was an Irish mathematician and physicist.
As a physicist, Stokes made seminal contributions to fluid mechanics, including the Navier–Stokes equations; and to physical optics, with notable works on polarisation and
fluorescence.
62
The current density created by the movement of positive and negative
ions in the electrolyte will be equal to the sum of the current densities of
positive and negative ions:
j = j+ + j–. (4.9)
Taking into account the relationship between the current density and
the average velocity of current carriers (3.5), we obtain
j = q+n+U+ + q–n–U–, (4.10)
where q+ = Z+e and q– = Z–e are the charges of positive and negative ions, n+,
n– are their concentrations, U+, U
–
are the average velocities of ordered move-
ment of ions under the influence of an electric field.
Substituting (4.5), (4.7) and (4.8) into expression (4.10) we obtain
EbnqbnqEbnqEbnqj )(
−−−+++−−−+++
+=+=
. (4.11)
Expression (4.11) is a differential form of Ohm's law for electrolytes,
it means that the current density in electrolytes is proportional to the electric
field strength.
The proportionality coefficient is the specific electrical conductivity
of the electrolyte
−−−+++
+= bnqbnqσ
. (4.12)
The conductivity of electrolytes increases with increasing ion concentrations and their mobility. With increasing temperature, the conductivity of
electrolytes increases due to an increase in mobility (since the viscosity of
the liquid decreases) and the degree of dissociation.
The mobility of ions according to (4.6) and Stokes' law (4.3) also
depends on their size: the smaller the ion radius, the greater the mobility.
For example, in an aqueous solution of hydrochloric acid, the mobility of the
hydrogen ion H+ is 32.6∙10−8 m2/V∙s, and the mobility of the Cl− ion is
6.8∙10−8 m2/V∙s (almost 5 times less). Also, the current densities created by
hydrogen and chlorine ions will differ by ~5 times.
Since the conductivity of electrolytes is ionic conductivity, the passage
of current in them is accompanied by the phenomenon of electrolysis1—the
1
Electrolysis is one of the industrial methods for producing aluminum, copper, and
hydrogen. It is used for the extraction of metals from ores (electroextraction, electrorefining), in wastewater treatment (electrocoagulation, electroextraction, electroflotation).
It is used for applying metal coatings (galvanoplasty), reproducing the shape of objects
(galvanoplasty).
63
release of components of dissolved substances on the electrodes (fig. 4.4).
Two Faraday laws (1832) are valid for the phenomenon of electrolysis.
Fig. 4.4. Electrolysis diagram, negative ions (anions) move to the anode,
positive ions (cations)—to the cathode
Faraday's first law. The mass M of the substance released on the
electrodes is directly proportional to the electric charge Q that has passed
through the electrolyte
tIkkQM ==
, (4.13)
where k is the electrochemical equivalent.
Faraday's second law. The electrochemical equivalent of a substance
k is proportional to the ratio of the molar mass μ of the ions of this substance
to their valence Z
Z
μ
F
1
k =
, (4.14)
where F = 96485.33 C/mol is the Faraday constant.
Let's consider the derivation of Faraday's laws. Let a constant current
I flow through one of the electrodes, then in time t the charge will be trans-
ferred
tIQ =
, (4.15)
equal to the number of ions N released at the electrode, multiplied by the
charge of one ion q = Ze
ZeNQ =
. (4.16)
64
The mass M of the substance released at the electrode will be equal to
A
N
μ
NmNM ==
, (4.17)
where m is the mass of the ion, μ is its molar (atomic) mass, NA is Avogadro's
number.
Substituting successively into expression (4.17) the number of ions N
from (4.16), and expressing the charge Q through the current I (4.15)
tIkkQQ
Z
μ
F
1
Q
Z
μ
eN
1
N
μ
Ze
Q
M
AA
=====
, (4.18)
we come to Faraday's laws (4.13) и (4.14).
Based on the laws of electrolysis and by measuring the Faraday number experimentally, one can determine the magnitude of the elementary
charge. The Faraday number can be easily measured; for this, it is only necessary, for a given current I*, to measure the time t* during which a mass
M equal to the atomic mass μ of a given element is released on the electrode.
As follows from (4.18), the Faraday number measured in this way is equal to
Z
*t*I
F
=
. (4.19)
From (4.18) it also follows that the Faraday number is equal to
F = e . NА (4.20)
Substituting the Faraday number and Avogadro's constant into (4.20) gives
.
4.3. E l e c t r i c c u r r e n t in g a s e s
Gases in a normal state, including metal vapors, consist of electrically
neutral atoms and molecules. Gases, such as air, are insulators because they
do not contain current carriers. Only ionized gases can be conductors of elec-
tricity. In addition to neutral atoms and molecules, they contain electrons,
positive and negative ions. Ions in gases can arise under the influence of high
temperatures, X-rays, ultraviolet rays, and cosmic rays. In all these cases,
65
electrons are ejected from the electron shell of the atom or molecule.
This process is called ionization. The released electrons can attach to neutral
atoms, then, in addition to positive ions, negative ions appear in the gas.
Ions and free electrons make the gas an electrical conductor. However, even under normal conditions, gases, such as air, have electrical conductivity, although small, which is caused by the radiation of radioactive
substances present in the earth, as well as cosmic rays. Simultaneously
with ionization, electrons, positive and negative ions combine with each
other to form neutral molecules and atoms. This process is called recombination. After the ionizer stops working, the gas conductivity disappears
due to recombination.
The process of passing an electric current through a gas medium is
called a gas discharge. Gases become electrically conductive as a result of
their ionization. If an electric discharge in a gas occurs only under an external influence that causes and maintains ionization, then it is called
a non-self-sustaining discharge. An electric discharge in a gas that continues after the external ionizer has ceased to operate is called a self-sustaining discharge.
The variety of conditions determining the initial state of the gas (composition, pressure, etc.), external effects on the gas, materials, shape and location of the electrodes, configuration of the electric field arising in the gas
leads to the fact that there are many types of gas discharges. The main types
of self-sustained discharges are glow discharge, spark discharge, corona discharge, arc discharge.
The laws of gas discharges are significantly more complex than the
laws of current flow in metals, electrolytes, and semiconductors. Electric discharge in gas obeys Ohm's law only at low voltages, so their electrical properties are described using volt-ampere characteristics. The dependence of the
current in the gas on the voltage between the electrodes (fig. 4.5) has several
characteristic sections. Let the external ionizer have a constant ionization intensity. Section OA corresponds to the applicability region of Ohm's law.
In section AB, the current changes nonlinearly with voltage. Section BC corresponds to the saturation current, with all ions and electrons created by the
external ionizer reaching the electrodes.
At point C and further on in section CD the discharge from non-selfsustaining becomes self-sustaining. This phenomenon is caused by impact
ionization and a sharp increase in the number of current carriers in the gas.
66
Fig. 4.5. Volt-ampere characteristic of gas discharge
Let us consider the basic laws inherent in a non-self-sustaining discharge. Let the gas between the electrodes (fig. 4.6) experience a constant
intensity ionizing action. Let us denote the number of ion pairs appearing
under the action of the ionizer in a unit of volume per second1 as Δni, and the
number of pairs of ions recombining in unit volume per second as Δnr.
Fig. 4.6. Non-self-sustaining gas discharge
The probability of ion recombination, i. e., the meeting of two ions of
opposite signs with each other, is proportional to the number of both positive
and negative ions. Therefore, the number of ion pairs Δnr recombining in
a unit volume per second is proportional to the square of the number of ion
pairs n in a unit volume
2
r
nrΔn =
, (4.21)
where r is the recombination coefficient.
1
The number of ion pairs of both signs formed under the action of the ionizer in a unit
of air volume per unit of time is called the power of the ionizer.
67
At equilibrium, the number of ions produced by the ionizer is equal to
the number of recombining ions
ri
ΔnΔn =
, (4.22)
therefore
2
i
nrΔn =
. (4.23)
Thus, the equilibrium concentration of ion pairs
r
Δn
n
i
=
(4.24)
is determined by the power of the ionizer and the rate of ion recombination.
Let us consider a gas discharge (see fig. 4.6) between parallel electrodes, S is
the area of each electrode, l is the distance between them, E is the electric
field strength in the discharge gap. Let us consider two extreme cases,
namely, the case of a weak field and the case of a strong field.
Under the influence of an electric field, positive ions move toward the
negative electrode, and negative ions move toward the positive electrode.
Let us denote by Δnj the number of ion pairs leaving for the electrodes from
a unit of volume per unit of time. In this case, the number of ions produced
by the ionizer, on the one hand, and the number of recombining ions together
with the ions leaving for the electrodes, on the other hand, are equal to each
other. The equilibrium condition has the form
jri
ΔnΔnΔn +=
. (4.25)
In the case of a weak field, Δ
Δ
, the current density is propor-
tional to the concentration and charge of ions n and q, as well as the velocity of
movement of positive and negative ions under the influence of the field
,
:
)(
−+−+
+=+= UUqnjjj
~~
. (4.26)
Average velocities
−+
U,U
~~
,
−+
~~
of ions, as in electrolytes, are proportional
to the field strength:
−+
U,U
~~
= b+ E;
−+
~~
= b– E, (4.27)
where
−+
bb и
is the mobility of ions in the gas.
Taking into account expression (4.24), we obtain
Ebb
r
Δn
qj
i
)(−++=
. (4.28)
68
Therefore, in the case of weak electric fields, a non-self-sustaining gas
discharge obeys Ohm's law (2.9), section OA (fig. 4.5). Specific electrical
conductivity of gas
)(
−+
+= bb
r
Δn
q
i
σ
(4.29)
increases with increasing ionizer power and ion mobility.
In the case of a strong field, Δnr much less than Δnj and all ions created
by the ionizer leave the gas discharge gap to the electrodes, without having time
to recombine noticeably. In this case, the condition is satisfied Δnj = Δni. Current density
Δ
Δ
Δ
Δ
Δ
. (4.30)
is determined by the power of the ionizer, the distance between the electrodes and does not depend on the field strength. It is called the saturation
current density.
The maximum value of the current at which all the ions formed go to
the electrodes is called the saturation current. With a further increase in the
applied voltage, the current stops growing and remains constant, section BC
(see fig. 4.5).
4.4. E l e c t r i c c u r r e n t i n a v a c u u m
Electric current in a vacuum can be obtained using electron emission
phenomena: thermionic, photoelectron, and autoelectron emission. Thermionic emission is the phenomenon of electron emission by heated metals. Photoelectron emission (external photoelectric effect) is the phenomenon of electron emission by metals under the influence of electromagnetic radiation.
Autoelectronic emission is the phenomenon of electron emission by conducting solid and liquid bodies under the influence of an external electric field E
of sufficiently high intensity (E ~ 105 V/m).
Electrons are held inside the metal. This means that near the surface
there are forces acting on the electrons and directed inside the metal.
The origin of such forces can be explained by two reasons. The first is the
inductive action of an electron leaving the metal. Such an electron induces
a charge of the opposite sign on the metal surface. Coulomb forces of attraction arise between the electron and the induced charge. The second reason is
that some electrons can escape from the metal due to thermal motion and,
69
thus forming an electron cloud above the metal surface, will prevent other
electrons from escaping.
The electron cloud together with the layer of induced positive ions of the
lattice forms a double electric layer, the field of which is similar to the field of
a flat capacitor. The thickness of this layer is equal to several interatomic distances (10
−10
−10−9 m). It does not create an electric field in the outer space, but
prevents the electrons from leaving the metal. To extract an electron from
a metal, work W must be performed, called the work function, equal to
φΔ=eW
, (4.31)
where e is the electron charge, Δφ is the surface potential difference between
the electron cloud and the metal.
At room temperature, only a tiny fraction of the electrons inside the
metal has enough kinetic energy to escape. As the temperature rises, the number of fast electrons increases, and so does the number of electrons that escape from the metal. At high enough temperatures, the metal begins to emit
electrons. This phenomenon is called thermionic emission. Thermionic emission is the basis for the design of vacuum tubes. The electric current in a vac-
uum tube is the current in a vacuum, where current carriers are formed as
a result of the emission of electrons from the surface of the cathode.
Heating of the cathode can be done in two ways: by passing electric
current (filament current) directly through the cathode or through an auxiliary heating spiral (filament). Accordingly, a distinction is made between
electron tubes with a direct or indirect heating cathode.
The simplest two-electrode electron tube (fig. 4.7) a diode has only two
electrodes (cathode and anode) placed in a vacuum glass or metal bulb. The air
pressure in the bulb is ~10−8 mm Hg—this is a high or deep vacuum.
а b c
Fig. 4.7. Schematic diagram of a two-electrode electron tube with
a cathode: a is direct; b is indirect heating; c is designation on electrical
circuits (1 is cathode; 2 is anode; 3 is glass or metal bulb; hs is filament)
70
At a constant cathode temperature, the magnitude of the anode current
depends on the anode voltage (the voltage of the anode relative to the
cathode). The graph of the dependence of the anode current on the anode
voltage (fig. 4.8) is called the anode characteristic.
Fig. 4.8. Anode characteristic of a two-electrode electron tube
This characteristic is nonlinear and, therefore, the vacuum tube is an
example of a conductor that does not obey Ohm's law. With an increase in the
anode voltage, the current increases in accordance with the Boguslavsky
1
–
Langmuir2 law (the “three–half” law):
Iа = В . U
а
3/2
, (4.32)
where B is a constant that depends on the shape, size and relative position of
the cathode and anode, as well as on the temperature of the cathode.
At a certain value of voltage Ua, the anode current takes on the maximum value possible at a given cathode temperature and is called the saturation current. The saturation current is numerically equal to the charge of all
electrons emitted by the cathode per unit of time
NeI
sat
=
, (4.33)
1
Sergei Anatolyevich Boguslavsky (1883–1923) was a Russian physicist. His works
are devoted to electrodynamics, kinetic theory of crystalline dielectrics, pyroelectricity.
He calculated the dependence of the constant B on the ratio of the radii of the outer and
inner cylinders two-electrode electron tube.
2
Irving Langmuir (1881 –1957) was an American chemist, physicist, and metalurgical
engineer. He was awarded the Nobel Prize in Chemistry in 1932 for his work in surface
chemistry.
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