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Файл:Steady electric current. Tutorial
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21
lightning entry point. The soil is considered homogeneous in all directions.
Lightning parameters: current 0.5 MA, channel thickness equals 20 cm.
Given
SI
Solution
I = 0.5 MA
d = 20 cm
r = 10 m
5∙105 A
2∙10−1 m
j = f(r)?
The figure shows a photograph of lightning (left) and a diagram of the
current distribution in the earth's soil (right). The problem statement implies
that the electrical properties of the soil are isotropic. Consequently, the current lines in the ground will be spherically symmetrical relative to the point
of entry of lightning and the current density will be the same at all points of
an arbitrary hemisphere of radius r.
The area of an arbitrary hemisphere with radius r is equal to
2
2
sphere
r2π
2
r4π
2
S
S* ===)1
,
current density in the soil at a distance r from the entry point
2
r2π
I
*S
I
j* ==)2
.
Let's compare it with the current density in the lightning channel
2
4
d
π
I
S
I
j ==)3
.
Performing a substitution of numerical data.
Answer: j* = 800 a/m2; j = 16 MA/m2.
Problem 4. The steady current density in a stationary metal conductor
of constant cross-section is 1 A/mm2. The diameter of the conductor wire is
0.5 mm, the potential difference at its ends is 20 V. What is the work of the
Coulomb forces in the conductor in 0.5 hour?
Given
SI
Solution
d = 0.08 mm
t = 0.5 hr
j = 35 A/mm2
Δφ = 220 V
8∙10−5 m
1800 s
35∙106 A/m2
AC – ?
The work of Coulomb forces on the movement of current carriers in
1,21,2
qA φΔ=
C
)1
tIq =)2
SjI =)3
4
dπ
S
2
=)4
tΔφ
4
dπ
jA
2
C
=)5
a conductor is determined by formula (1.1):
.
The magnitude of the displaced charge, i. e. the charge that has passed
through the cross-section of the conductor, is determined by formula (1.14).
For steady current we write:
.
We find the current strength in a conductor knowing the current density
and the cross-sectional area of the conductor, using formulas (1.15)–(1.17):
.
We express the cross-sectional area through the wire diameter:
.
rent, and current in a vacuum?
The final formula for calculating the work of Coulomb forces:
.
Performing a substitution of numerical data.
Answer: AC = 70 kJ.
S e l f - c h e c k q u e s t i o n s
1. What movement of charges is called electric current?
2. What is the difference between conduction current, convection cur-
22
3. What current is called conduction current? Name the charge carriers
that create it.
4. What current is called a transfer current? Give examples.
5. What current is called current in a vacuum? Give examples.
6. What conductors are called conductors of the first and second kind?
Give examples.
7. What conditions are necessary for the long-term existence of an
electric conduction current?
8. Why can't Coulomb forces alone provide continuous current?
9. What forces are called extraneous forces? Give examples.
10. What is called potential difference? In what units is it measured?
11. What is called electromotive force? Specify the units of measure-
ment of EMF.
12. What is called voltage in this section of the circuit? In what units
is voltage measured?
13. What section of the circuit is called homogeneous? Is it true that
the voltage and potential difference in such a section are equal to each other?
14. What electric current is called steady current?
15. What is called current strength? In what units is current strength
measured?
16. How is the direction of steady current determined?
17. What is called a current density vector? What characterizes current
density? Specify the units of measurement of current density.
18. How is the current density vector field represented graphically?
19. How can one calculate the magnitude of the current through
a given cross-section of a conductor using a known current density vector
field?
20. Why is the flux of the steady current density vector through
a closed surface equal to zero?
23
24
2 . L A W S O F S T E A D Y C U R R E N T
Steady electric current is a current whose magnitude and direction do
not change over time. The basic laws of steady current are Ohm's law, which
establishes a linear dependence of current on voltage, and the Joule–Lenz
law, which determines energy losses caused by the release of heat when current flows in a conducting medium.
In all sections of an unbranched circuit the current strength is the same.
In cases of parallel or series connection of resistors or series current sources,
the problem can be reduced to an equivalent circuit involving one resistor
and one emf source by reducing the sections of the circuit via the series and
parallel rules for resistors or EMF.
Calculation of currents and voltages in branched steady current circuits is performed using Kirchhoff's rules.
2.1. O h m ' s l a w
Georg Ohm experimentally (1826) and theoretically (1827) discovered the fundamental law of the electric circuit, which was later named after
him. Also, the unit of measurement of electrical resistance in the SI system,
ohm (Ω), is named in his honor.
The current flowing through a metal conductor (fig. 2.1) is proportional to the voltage drop across the conductor: I ~ U.
Fig. 2.1. Metalic conductor
Let us write the expression for Ohm's law using the proportionality coefficients:
25
GU
R
U
I ==
, (2.1)
where R is the electrical resistance; G is the electrical conductance of the
conductor.
The electrical resistance of an object is a measure of its opposition to
the flow of electric current. Its reciprocal is electrical conductance, which
measures the ease with which electric current passes. The SI unit of electrical
resistance is ohm (Ω), electrical conductance—Siemens (S)1.
A conductor is a homogeneous section of a circuit; it does not contain
current sources. Ohm's law in the form (2.1) is called Ohm's law for a homogeneous section of a circuit. In general, the value of resistance R depends on
the shape and size of the conductor, and on the properties of the conducting
material. It was experimentally established that the resistance of a homogeneous conductor of constant cross-section and composition is equal to:
S
l
R ρ=
, (2.2)
where l is the length of the conductor, S is the cross-sectional area of the
conductor (fig. 2.2), and ρ is the coefficient depending on the properties of
the material and called the specific electrical resistance or resistivity of the
substance:
Fig. 2.2. Metalic conductor of constant cross-section
The value σ reciprocal of resistivity is called specific electrical conductivity or simply conductivity.
. (2.3)
The SI unit of resistivity is ohm∙m (Ω∙m), conductivity—siemens per
metre (S/m).
1
Ernst Werner Siemens (1816–1892) was a German physicist, electrical engineer, in-
ventor. Designed an electric telegraph, built the first tram.
26
Let us obtain Ohm's law in differential form, namely, we will find the
relationship between the current density and the electric field strength at any
point of the conductor. Let us consider an elementary small element of a conductor, in the form of a small cylinder (fig. 2.3) oriented in the direction of
the current density vector.
Fig. 2.3. An elementary small element of a conductor in the form
of a small cylinder, oriented in the direction of the current density vector
The length of this conductor element is dl, the cross-sectional area is
dS, the voltage drop across it is dU, the current density vector
j
is directed in
the direction of the resulting field strength
extr
EEE
+=
C
.
The modulus of the current density vector
j
is equal to:
dS
dI
j =||
, (2.4)
where the current dI flowing through the conductor element, according to
Ohm's law (2.1), is equal to:
dR
dU
dI =
. (2.5)
The voltage drop dU across a given small element of a conductor of
length dl, according to expression (1.12), is equal to:
dlEld,Ecos(dlEldE)ldEEdU
extrC
===+= )()((
)
. (2.6)
The electrical resistance of a given small element, taking into account
formula (2.2), is equal to:
Sd
ld
dR ρ=
. (2.7)
27
Substituting expressions (3.5)–(3.7) into formula (3.4), and taking into
account the relationship between resistivity and conductivity (2.3), we obtain:
σE
ρ
E
ρdl
dS
dS
Edl
dS
dI
j ====||
. (2.8)
Consequently, the magnitude of the current density vector is proportional to the strength of the resulting field of Coulomb and external forces at
a given point.
Since
Eldj
, then expression (2.8) in vector form will look
like:
Eσ
ρ
E
j
==
. (2.9)
The resulting expression is a differential form of Ohm's law. In this
expression (2.9), the strength of the resulting force field
E
is equal to the
sum of the strengths of the Coulomb field
C
E
and the field of extraneous
forces
extr
E
.
We obtain Ohm's law for a non-uniform section of a circuit (fig. 2.4).
The section of the circuit on which both Coulomb and external forces act
(containing current sources) is called a non-uniform section.
Fig. 2.4. Non-uniform section of the circuit
Let us multiply the left and right parts of expression (2.9) in a scalar
manner by the length of the element:
. (2.10)
Since
ldj
, their scalar product (on the left) is equal to the product
of their moduli:
dljld,jdlcosjldj == )(
)(
. (2.11)
28
The steady current density in a homogeneous conductor of constant
cross-section has the same value both inside and on the surface of the conductor1. It will change only with a change in the cross-sectional area or resistivity of a conductor with a heterogeneous composition. Therefore, for a section of a homogeneous conductor of constant cross-section, formula (2.4) can
be written as:
S
I
j =||
. (2.12)
Let us substitute expressions (2.11) and (2.12) into (2.10):
)) ldEldE
S
dlρ
I
extrC
+= ((
, (2.13)
here
dR
S
dlρ
=
is the resistance of a section of a conductor with a cross-sec-
tional area S, specific resistance ρ and length dl.
Let us integrate expression (2.13) along the conductor from point 1 to 2:
+
=
2
1
extr
2
1
C
2
1
ldEldE
S
dlρ
I )()(
. (2.14)
The integral on the left side of expression (2.11) represents the resistance of the circuit section from point 1 to 2:
==
2
1
1,2
2
1
RdR
S
dlρ
. (2.15)
The integrals on the right side of expression (2.11) represent the potential difference Δφ and the EMF of the current source in a given section,
respectively:
1,2
2
1
C
ldE φΔ=
)(
, (2.16)
1,2
2
1
extr
ldE ε=
)(
. (2.17)
1
Unlike direct current, the density of alternating current is not constant across the
cross-section of the conductor. Alternating current is forced onto the surface of the
conductor—this phenomenon is called the skin effect. At very high frequencies, current flows through a thin surface layer. The depth of the skin layer depends on the conductivity, frequency of the alternating current and the condition of the conductor surface. Conductors for high-frequency currents are silver-plated and made hollow.
29
Consequently, the current strength in the section of the circuit where
there are both Coulomb and extraneous forces will be equal to:
1,2
1,21,2
R
Δφ
I
ε+
=
. (2.18)
Expression (2.18) is called Ohm's law for a non-uniform section of
a circuit.
Let us finally obtain an expression for Ohm's law for a closed circuit.
A closed circuit (fig. 2.5) has a current source and a resistor: the current
source's own resistance or its internal resistance is usually designated by the
symbol r, the resistor's resistance or external resistance is R.
Fig. 2.5. Closed electrical circuit
Let us apply formula (2.18) to this closed circuit. The work of the
Coulomb forces along a closed circuit is zero, and, consequently, Δφ = 0;
the total resistance of the circuit is equal to the sum of the external R and
internal r resistances. Then, the current in the closed circuit will be equal to:
rR
I
+
=
ε
. (2.19)
Expression (2.19) is called Ohm's law for a closed circuit.
2.2. B r a n c h e d e l e c tr i c c i r c u i t s ,
K i r c h h o f f ' s r u l e s
Ohm's law allows finding current in unbranched steady current circuits.
Circuit sections can be either homogeneous (not containing current sources)
or non-homogeneous (containing current sources), and can contain parallel or
series-connected resistors and current sources. The problem of calculating currents in branched electrical circuits can be solved using two Kirchhoff rules.
The meaning of Kirchhoff's rules is very simple, as are the rules themselves.
30
Kirchhoff's first rule is a consequence of the law of conservation of electric
charge. Kirchhoff's second rule is a consequence of Ohm's law.
Let's introduce the terms used to describe branched electric circuits.
A node of a branched circuit is a point where three or more conductors converge. A branch of an electric circuit is a section of the circuit along which
the same current passes. A loop is any closed path that can be bypassed by
moving along any of the branches of the circuit.
Kirchhoff's first rule. The algebraic sum of currents converging at
a node is zero:
0I
N
1k
k
=
=
. (2.20)
Currents flowing into a node are taken with a plus sign “+”. Currents flowing
out of a node are taken with a minus sign “−”.
Kirchhoff's first rule is a consequence of the law of conservation of
electric charge. Indeed, equation (2.20), taking into account the definition of
current strength (1.14), takes the form:
0
dt
dq
N
1k
i
=
=
. (2.21)
Changing the order of summation and differentiation, we obtain:
0q
dt
d
N
1k
i
=
=
. (2.22)
Expression (2.22) means that the algebraic sum of charges introduced into
a node (by currents flowing to the node) and carried away from the node (by
currents flowing from the node) does not change over time.
Let us write the equation according to Kirchhoff's first rule for node C
(fig. 2.6).
Fig. 2.6. Branched circuit node
Currents I1, I4, flowing into the node, are taken with a plus sign, currents I2,
I3, I5, flowing out of the node, are taken with a minus sign:
I1 – I2 – I3 + I4 – I5 = 0. (2.23)
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