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Файл:Steady electric current. Tutorial
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11
1. B A S I C C O N C E P T S A N D D E F I N I T I O N S
1.1. E l e c t r i c c ur r e n t
Electrical or electric current is the ordered directed movement of electric charges. Charges can move inside bodies, together with bodies, or by
themselves in the absence of any medium. Due to this, there are three main
types of electric current: conduction current, convection current, and current
in a vacuum.
Conduction current is the directed movement of microscopic charge
carriers inside a stationary conductor. For example, electrons in metalic conductors, electrons and “holes” in semiconductors, ions in electrolytes, electrons and ions in gases (the electric current through a gas is called a gas discharge). Lightning (fig. 1.1) is a natural phenomenon formed by electrostatic
discharges through the atmosphere between two electrically charged regions,
either both in the atmosphere or one in the atmosphere and one on the ground.
Fig. 1.1. Lightning. Current strength reaches 0.5 MA
(most often 20–40 kA), voltage up to 1 GV. Discharge duration
about 1 ms, length about 10 km, channel thickness up to 20 cm.
Discharge energy about 10 MW-hour
12
Convection or transport current is the movement of charged bodies in
space in the absence of an electric field. For example, the movement of
charges inside galvanic cells as a result of oxidation-reduction reactions, the
transfer of charges by a dielectric tape in a Van de Graaff generator, the
movement of charged clouds in the atmosphere.
Vacuum current is the movement of microscopic charge carriers in
a vacuum, such as electrons in a vacuum tube, electrons and ions in a plasma,
in the form of streams of protons, alpha particles, heavy ions and electrons
emitted from the surface of the Sun at speeds of 300 to 800 km/s as part of
the “solar wind”.
Further we will mainly consider the laws inherent in conduction current.
The conductors themselves are usually divided into conductors of the first and
second kind. Conductors with electronic (metalic) electrical conductivity are
called conductors of the first kind. Conductors of the second kind, or electrolytes, are solutions (in particular, aqueous) and melts of salts, acids, alkalis and
other substances with an ionic structure of molecules. The conductor contains
a large number of free charged particles that can move freely inside the conductor under the action of any small force, creating a conduction current.
For the long-term existence of an electric conduction current it is necessary:
a) the presence of free charge carriers – electrons, ions; such charged
particles capable of moving are called current carriers;
b) the presence of an electric field in the conductor, the energy of
which would be spent on moving charges;
c) for the long-term existence of an electric current, a current source is
required that converts some type of energy into the energy of an electric field.
1.2. E l e c t r o m o t i ve f o r c e , p o t e n t i a l
d i f f e r e n c e , v o l t a g e
The forces acting on charges in an electric circuit are divided into two
types: Coulomb forces and extraneous forces.
Coulomb forces are electrostatic forces, forces of interaction between
electric charges, provide attraction of opposite charges and repulsion of like
charges. Coulomb forces, connecting opposite charges, equalize potentials at
all points of the conductor and can provide only a short-term electric current.
13
For example, a lightning discharge is a giant spark gas current pulse that lasts
only thousandths of a second, while the current strength in the main lightning
channel can reach 0.5 MA, and the discharge energy is up to 10 MW-hour.
Extraneous forces are all other forces, except Coulomb (electric) forces,
capable of acting on electric charges. These may be chemical forces in galvanic
cells, arising as a result of chemical reactions between the electrodes of the cell
and the electrolyte. Extraneous forces arise at the contacts of dissimilar conductors (Seebeck effect). Extraneous forces can also include mechanical forces that
separate opposite charges that arise on contacting bodies due to friction (triboelectricity). Finally, the basis of industrial current generators are extraneous induction forces acting on electric charges from the vortex electric field that occurs when the magnetic field changes through the rotor windings1.
Extraneous forces, separating opposite charges, transform some other
type of energy into the energy of an electric field: the mechanical energy of
a rotating rotor, the thermal energy of plasma (fig. 1.2) in an MHD generator2, the energy of a chemical bond in galvanic cells, etc.
Fig. 1.2. Scheme of MHD generator
Oppositely charged ions of high-temperature plasma escape from the
nozzle at great speed and enter the magnetic field. In a magnetic field, the
Lorentz force acts on the ions, deflecting them in opposite directions.
The Lorentz force does not perform work, but only bends the trajectories of
the ions. The internal thermal energy of the plasma is converted into the
energy of the electric field.
1
The phenomenon of electromagnetic induction was discovered by Faraday (1831).
With every change in the flow of the magnetic induction vector through a surface limited
by a closed contour, an electric current arises in it. The reason for the appearance of an
induction electric current is the vortex (non-potential) electric field that arises with every
change in the magnetic field.
2
The idea of MHD energy conversion was expressed by Faraday back in 1831. The first
MHD generator was built in the USA in 1959, its power was 11.5 kW.
14
Coulomb and extraneous forces, moving charges in a circuit, do work.
The quantitative characteristic of the work performed is, accordingly, the potential difference and the electromotive force of the current source.
Potential difference. The work performed by electrostatic (Coulomb)
forces to move an electric charge from point 1 to point 2 of an electric circuit,
divided by the magnitude of the charge being moved, is called the potential
difference in a given section of the circuit.
q
A
1,2
1,2
C
=φΔ
. (1.1)
The SI unit of potential difference is volt (V), V = J/C.
Expressing sequentially:
)( ldFA
2
1
C
C
1,2
=
, (1.2)
EqF
C
=
, (1.3)
we receive:
)()()( ldEldEq
q
1
ldF
q
1
q
A
C
1,2
1,2
=
=
==
2
1
2
1
2
1
C
φΔ
. (1.4)
Coulomb forces are conservative forces, and therefore their work in
moving a charge along any closed path is zero. In other words, the circulation
of the electrostatic field strength vector along any closed trajectory is zero.
= 0dlE )(
. (1.5)
After the charges are redistributed under the action of Coulomb forces, the
potentials are equalized and the current in the circuit stops. Coulomb forces
cannot support a constant electric current in a closed circuit.
Electromotive force. The work performed by extraneous (non-Coulomb) forces
extr
F
to move an electric charge q from point 1 to point 2 of an
electric circuit, divided by the magnitude of the charge being moved, is called
the electromotive force (EMF) in a given section of the circuit.
q
A
1,2
1,2
extr
=ε
. (1.6)
The unit of measurement of electromotive force in the SI system, like potential difference, is the volt (V).
15
Extraneous forces
extr
F
form a vector force field, characterized by
a vector quantity, which is called the external force field strength
extr
E
.
By analogy with the Coulomb force field, we can write:
q
F
E
extr
extr
=
. (1.7)
Taking into account (1.7), the work of extraneous forces to move charge q in
an electric circuit from point 1 to point 2 will be equal to:
)()( ldEqldFA
2
1
extr
2
1
extr
extr
1,2
=
=
, (1.8)
and electromotive force (EMF):
)()( ldEldEq
q
1
q
A
extrextr
1,2
1,2
=
==
2
1
2
1
extr
ε
. (1.9)
Extraneous forces are non-conservative forces; their work on an arbitrary
closed circuit is not equal to zero. The work of extraneous forces in a closed
circuit, related to the charge being moved, is equal to the EMF acting in this
circuit. Taking into account expression (1.8), the EMF in a closed circuit is
equal to the circulation of the field strength vector of external forces.
= )( ldE
extr
ε
. (1.10)
Voltage, voltage drop. At some section of the electric circuit, both
Coulomb and extraneous forces may be present. Their combined action is
described by the concept of voltage or voltage drop. The sum of the work
performed by Coulomb and extraneous forces to move an electric charge
from point 1 to point 2 of an electric circuit, divided by the magnitude of the
charge being moved, is called the voltage in a given section of the circuit.
2121
2
1
2
1
q
1
,,
)()( εφΔ +=
+
=
+
= ldFldF
q
1
q
AA
U
extrC
extr
1,2
C
1,2
1,2
. (1.11)
The unit of measurement of voltage in the SI system, like potential difference
and electromotive force, is volt (V).
Let us connect the Coulomb and extraneous forces with the intensities
of their fields:
)(
2
1
2
1
C
2
1
2
1
ldEldEEldEqldEq
q
1
U
extrextrC1,2
=
+=
+
= )()}()({
. (1.12)
16
Here
extr
EEE
+=
C
is the strength of the resulting field of Coulomb and
extraneous forces.
If there are no extraneous forces on a given section of the chain, then
such a section is called homogeneous. On a homogeneous section of the circuit,
the EMF is equal to zero, therefore the potential difference and voltage on it
are equal to each other:
1,21,21,2
U0 =→= φΔε
. (1.13)
1.3. S t e a d y c u r r e n t a n d c u r r e n t d ens i t y v e c t o r
A steady current (constant current, time-independent current, stationary current) is a type of direct current (DC) that does not change its intensity
with time; neither the magnitude nor the direction of such a current changes
over time. The magnitude of a steady current (current strength) or simply
current I is a scalar quantity numerically equal to the electric charge passing
through the cross-section of a conductor in 1 sесond:
dt
dq
I =
. (1.14)
The SI unit of current strength is ampere (A), is established on the
basis of the law of magnetic interaction of currents1 and is the basic unit of
the SI system. All other electromagnetic units are defined on the basis of the
four basic units of mass, length, time, and current: kg, m, s, A.
Electric current can be created by the movement of both positive and
negative charges. The direction of steady current is the direction of the ordered
movement of positive charges in a conductor. In metal conductors, current carriers are electrons with a negative charge; the direction of current is opposite
to the direction of their movement. Current flows from the positive pole of
a current source to the negative, from a higher potential to a lower one.
1
The unit of current is the basic unit in the SI. 1 A is the strength of such a constant
current which, passing through two infinitely long parallel rectilinear conductors of infinitely small cross-section, located at a distance of 1 m from each other in a vacuum,
causes an interaction force between them of 2‧10−7 Η for each meter of the length of the
conductors.
17
The distribution of current across the cross-section of a conductor is
described by the current density vector
j
. It is numerically equal to the cur-
rent dI through an elementary small area dS located perpendicular to the direction of movement of the current carriers, divided by its area. The direction
of the current density vector coincides with the direction of ordered movement of positive charges at a given point. The SI unit of current density is
ampere per square meter (A/m2).
. (1.15)
Current density vector is a differential characteristic of current distribution in a conductor cross-section. Current flowing through some arbitrary
conductor cross-section S can be found by integration.
=
==
S
n
SS
dSjcosαdSjSdjI )(
, (1.16)
where the conditional vector
ndSSd
=
is used,
n
is a unit vector perpendi-
cular to the surface dS. Current I is equal to the flux of the current density
vector
j
through the surface S.
If the current density
j
at all points of the cross-section S oriented
perpendicular to the charge movement is constant, then
I = j∙S. (1.17)
Since the current density vector is defined at each point of the conductor, the concept of the current density vector field can be introduced. This
field is graphically depicted using current lines by analogy with the E lines
of the electric field. The density of current lines or current density vector
lines is proportional to its magnitude, the tangent at each point of the current
line coincides with the direction of the current density vector. Current lines
do not intersect each other.
The flux of the steady current density vector
j
through a closed sur-
face S is zero (fig. 1.3):
0)( =
S
Sdj
. (1.18)
In a steady current circuits, the vector
j
has no sources, and the cur-
rent lines are closed on themselves. Let us consider, for example, a closed
circuit (fig. 1.4) containing a current source, a load (conductor), and connecting wires.
18
Fig. 1.3. The flux of the vector
j
through a closed surface S is zero.
The number of incoming current lines is equal to the number
of outgoing lines
On the section external to the current source (нагрузка и соединительные провода) the direction of the current density vector coincides with
the direction of movement of positive charges under the action of Coulomb
forces from high potential to low.
At the same time, inside the current source positive charges move under the action of extraneous forces from low potential to high potential.
As can be seen from (fig. 1.4), the number of current lines in the entire
circuit remains constant, their density changes in individual sections.
This means, on the one hand, that the current lines have no sources and are
closed on themselves, and on the other hand, that the current density is higher
in areas where the current lines are concentrated.
Fig. 1.4. Circulation of charges in a closed electric circuit:
current source – ε; load – R; connecting wires. Dashed lines
are the current density vector lines. The current lines are closed
on themselves
19
1.4. P r o b l e m s
Problem 1. In a Van de Graaff electrostatic generator, a rubberized
belt 0.2 m wide moves at a velocity of 25 m/s. At the bottom roller, the
belt is given a surface charge, creating an electric field with an intensity of
106 V/m on each side of the belt. What is the current in the generator?
Given
Solution
b = 0.2 m
V = 25 m/s
E = 1∙106 V/m
I – ?
The electric current in the Van de Graaff1 generator is a transport current,
created by mechanical transfer of charges by a dielectric belt, its basic diagram
and device are shown in the figure (see “Electrostatics” section 2.3).
The strength of such a transfer current will be equal to the charge transferred by the moving belt per unit time:
td
qd
I =)1
.
1
Robert Van de Graaff (1901–1967) was an American physicist. Inventor of a high-voltage electrostatic generator, works in the fields of nuclear physics and particle accelerator
technology. The figure on the right shows a 5 MeV Van de Graaff generator (20 m high)
built in 1937 by Westinghouse Electric in Forest Hills, Pennsylvania. It was used as
a high voltage source for particle acceleration in nuclear physics experiments.
20
The belt is driven at a constant speed V. Using special brushes connected to a low-voltage current source, the belt is electrified with a uniform
surface charge density σ. In time dt, the belt of width b is displaced by
a distance Vdt and carries a charge dq.
Vdtbσdq =)2
.
The electric field near the belt can be considered as a field of an infinite plane, the strength of which is determined only by the surface charge
density.
εε
σ
0
2
)3 =E
.
By successively substituting expressions (2) and (3) into the first formula, we arrive at the answer:
bVεE2εσbV
dt
dq
I
0
===)4
.
Performing a substitution of numerical data, taking the dielectric constant of air ε = 1, electrical constant ε0 = 8,85∙10
−12
F/m.
Answer: I = 88∙10−6 A = 88 μА.
Problem 2. Along with specific resistance, permissible current density is used in calculations of electrical wiring in everyday life and industry.
Estimate the cross-sectional area of copper and aluminum wire for wiring to
a household electric stove consuming a current of 10 A. Wire cross-sections
have standard values in mm2: 0.35; 0.5; 0.75; 1.0; 1.5; 2.5; 4.0 etc. The permissible current densities (for open wiring) are taken as 8 A/mm2 for copper
and 5 A/mm2 for aluminum.
Given
Solution
I = 10 A
j
cu
= 8 A/mm2
jal = 5 A/mm2
S
I
j =)1
→
j
I
S =)2
.
Performing a substitution of numerical data: Scu = 1.25 mm2;
Sal = 2 mm2. We select suitable values from the standard
ones.
Scu – ? Sal – ?
Answer: Scu = 1.5 mm2; Sal = 2.5 mm2.
Problem 3. Calculate the dependence of the current density in the
ground after a lightning strike and find its value on the distance 10m to the
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