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Файл:Steady electric current. Tutorial
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31
Kirchhoff's second rule. The algebraic sum of the products of currents and resistances in the branches of any closed circuit is equal to the algebraic sum of the EMFs encountered in this circuit.
=
==
n
1k
k
n
1i
ii
εRI
. (2.24)
The signs of the terms in the left and right sums are determined taking into
account an arbitrarily chosen conditional positive direction of the circuit traversal (for example, counterclockwise). If the direction of the current, that is,
the direction of movement of positive charges in a given section of the circuit
coincides with the direction of the circuit traversal, then the corresponding
term IiRi or εk is taken with the plus sign “+”. And if the direction of the
current, that is, the direction of movement of positive charges in a given section of the circuit does not coincide with the direction of the circuit traversal,
then the corresponding term IiRi or εk is taken with the minus sign “−”.
Let us consider an arbitrary closed loop in a branched electric circuit
(fig. 2.7). The loop contains four branches: AB, BC, CD, DA with currents
I1, I2, I3, I4, respectively. The branches contain resistors and current
sources. Next, we designate the currents in the branches of the circuit and
arbitrarily indicate their directions. We also indicate the direction of the
positive circuit bypass.
Fig. 2.7. Arbitrary closed contour ABCD in a branched circuit.
Nodes of the circuit: A, B, C, D. Branches of the contour: AB, BC, CD, DA.
The positive direction of the contour traversal is indicated
32
The equation written for the ABCD circuit, in accordance with Kirchhoff's second rule, will look like:
432144332211
RIRIRIRI εεεε −+−=−+−
. (2.25)
Kirchhoff’s second rule is a consequence of Ohm’s law. Indeed, ap-
plying Ohm's law (2.18) to individual non-uniform sections of the ABCD
contour, taking into account the directions of currents and the polarities of
current sources, we obtain a system of equations (2.26):
1ВА11
RI εφφ +−=
2СВ22
RI εφφ −−=−
(2.26)
3DC33
RI εφφ +−=
4AD44
RI εφφ +−=−
.
After adding these equations together we arrive at equation (2.25).
When composing equations according to Kirchhoff's rules, the following restrictions are applied:
a) if there are N nodes in the circuit, then the number of independent
equations (according to Kirchhoff's first rule) is equal to N − 1;
b) if the total number of all possible circuits is equal to M, then the
number of independent equations (according to Kirchhoff's second rule) is
equal to M − 1.
As an example of the application of Kirchhoff's rules, let us consider
a simple, branched electrical circuit (fig. 2.8a). Let us designate the currents
flowing in individual branches of the circuit as I1, I2, I3 and indicate their
directions. The directions of the currents are indicated arbitrarily. Let us mark
nodes A and B and also indicate the positive direction of bypass of circuits I
and II. We can also indicate the positive poles of current sources. Thus,
we have supplemented the original diagram (fig. 2.8a) with the notations
necessary for the solution (fig. 2.8b).
Let us compose equations according to the first and second Kirchhoff
rules for node A and two circuits I and II:
0III
321
=+−
2
( εε +−=−+−
122411
RIRRI )
(2.27)
3253322
RRIRI εε +−=++ )(
The resulting system of three independent equations is sufficient to find the
values of currents I1, I2, I3 in the branches of the circuit. After substituting the
33
numerical values of Ri and εi, a system of linear algebraic equations is obtained. The solution of the system is easy to solve if you use the Cramer1
method (determinant method).
a b
Fig. 2.8. Branched circuit
For example, for this circuit (fig. 2.8b) the current I1 will be equal to:
532
241
53232
221
1
RRR0
0R)R(R
111
RRRεε
0Rεε
110
I
+
−+−
−
++−
−+−
−
=
, (2.28)
where the determinant in the denominator is composed of coefficients for
unknown currents I1, I2, I3. In the determinant in the numerator, the column
of coefficients for the sought current (in this case, current I1) is replaced by
free terms. If the values found for some currents turn out to be negative, then
the directions of these currents indicated in the diagram should be changed
1
Gabriel Cramer (1704–1752) was a Swiss mathematician and one of the founders of
linear algebra. Cramer's method is a method for solving systems of linear algebraic
equations in which the number of unknown variables is equal to the number of equations
and the determinant of the underlying matrix is nonzero.
34
to the opposite. Note that in the example considered (to simplify the notations) the internal resistances of the current sources were assumed to be equal
to zero, ri = 0 (ideal current sources).
2.3. S e r i e s a n d p a r al l e l c o n n e c t i o n
of r e s i s t o r s a n d c u r r e n t s o u r c e s
In electrical circuits there are sections of resistors or current sources
connected in series or in parallel. In any case, according to Ohm's law (2.1),
the total resistance of a section of a circuit is determined by the ratio of the
voltage drop across it to the current.
Let's consider a series connection of resistors (fig. 2.9). If the resistors
are connected in series, the total voltage drop across a section is equal to
=
=
n
1i
i
UU
, (2.29)
i. e., the total voltage drop is equal to the sum of the voltage drops across individual resistors; with the same current I in all resistors.
Fig. 2.9. Series connection of resistors
Dividing both sides of the equation (2.29) by I we get
=
=n1i
i
I
U
I
U
, (2.30)
that is, we received
=
=
n
1i
i
RR
. (2.31)
Thus, the total resistance of a section of a circuit consisting of resistors connected in series is equal to the algebraic sum of their resistances.
35
Let's consider a parallel connection of resistors (fig. 2.10).
Fig. 2.10. Parallel connection of resistors
When resistors are connected in parallel, the total current I is divided
into currents Ii flowing through individual resistors:
=
=
n
1i
i
II
, (2.32)
i. e., the total current is equal to the sum of the currents through individual
resistors; with the same voltage drop U in all resistors.
Dividing both sides of the equation (2.32) by U we get
=
=
n
1i
i
U
I
U
I
, (2.33)
that is, we received
=
=
n
1i
i
R
1
R
1
. (2.34)
The reciprocal value of the total resistance of parallel-connected resistors is
equal to the algebraic sum of the values of their reciprocal resistances.
Let us consider a closed electric circuit containing series-connected
current sources (fig. 2.11).
Fig. 2.11. Closed circuit containing series-connected current sources
36
When several sources are connected in series, the resulting EMF is
equal to the sum of the εi of all current sources, and the resulting internal
resistance r of such a battery will be equal, according to (2.31), to the sum of
the internal resistances ri of each current source. The current in a closed circuit (fig. 2.11) according to Ohm's law (2.20) is equal to:
+
=
=
=
n
1i
i
n
1i
i
rR
I
ε
. (2.35)
Let us consider a closed electric circuit containing the same parallel
connected current sources (fig. 2.12).
Fig. 2.12. Closed circuit containing parallel connected current sources
When current sources are connected in parallel, the battery EMF is
equal to the EMF of one source. The battery resistance when n sources are
connected in parallel is determined by (2.34) and, accordingly, the current
through resistor R according to Ohm's law (2.20) will be equal to:
n
r
R
I
+
=
ε
. (2.36)
When n identical current sources are connected in parallel in a battery, the
EMF does not change, but the internal resistance decreases by n times.
2.4. T h e r m a l e f f e c t o f c u r r e n t , J o u l e – L e n z l a w
Coulomb and external forces perform work, transforming electrical
energy or other types of energy into kinetic energy of the ordered motion of
current carriers. Let the voltage on some section of the circuit be equal to U.
37
If a charge dq passes through it during time dt, then the work of the Coulomb
and external forces on this section of the circuit, taking into account (1.11) and
(1.14), will be equal to:
IUdtUdqdAdAdA
extrC
==+=
. (2.37)
The kinetic energy of the ordered motion of current carriers in an electric
circuit is further transformed either into the internal energy of the conductor
(electric heating devices) or into the mechanical energy of its motion (electric
motor). The work performed by the current carriers, or simply, the work of
the current, will be determined by expression (2.37).
The current power, the speed at which work is performed, is equal to:
IU
dt
dA
P ==
. (2.38)
If the circuit conductors are stationary, then the work of the current
is entirely spent on heating the conductors, and is converted into internal
energy. Since the transfer of kinetic energy of current carriers occurs at the
level of interaction of particles (electrons, solvent molecules in electrolytes,
lattice ions in metals), we speak of the released heat.
Joule's law, also known as the Joule–Lenz law1, states that the heat
released in a conductor is equal to the product of its resistance and the square
of the current and the time of its passage.
Q = I2Rt. (2.39)
In general, if the current and the resistance of the conductor change over time,
then the amount of heat released during time t is determined by integration:
Q
=
t
0
2
RdtIQ
. (2.40)
Let us obtain the Joule–Lenz law in differential form. Let us consider
an elementary small element of a conductor (fig. 2.13) oriented in the direc-
tion of the current density vector
j
. The conductor element has a length of
dl, a cross-sectional area of dS, and a resistance of dR.
1
The law of the thermal action of electric current was established in 1841 by James
Joule and independently in 1842 by Emil Lenz.
38
Fig. 2.13. Elementary small element of a conductor with resistance dR
The heat released in a given element of the conductor during time dt
is equal to
dtdRdIdQ
2
=
, (2.41)
where the current dI in the conductor element and its resistance dR are equal,
respectively
dSjdI =
, (2.42)
dS
dl
dR ρ=
. (2.43)
Substituting expressions (4.6) and (4.7) into (4.5), we obtain
dtdSdljdt
dS
dl
jdSdQ
22
ρρ == )(
, (2.44)
where dl dS = dV is the volume of the conductor element.
The heat released per unit volume of a conductor per unit time is called
the unit thermal power of a current
dtdV
dQ
Q
u
=
. (2.45)
Based on (2.44) and definition (2.45), the unit thermal power of the
current is equal to
2
u
jQ ρ=
, (2.46)
or, taking into account the relationship (2.3) between resistivity ρ and conductivity σ and Ohm's law (2.9) in differential form, we write
2
u
EQ σ=
. (2.47)
The obtained equations (2.46) or (2.47) determine the unit thermal
power of the current and represent a differential form of the Joule–Lenz law.
39
Let us pay attention to some practical aspects of the thermal effect of
electric current. Joule heating is referred to as ohmic heating because of its
relationship to Ohm's Law. It forms the basis for the large number of practical
applications involving electric heating. The use of high voltages in power
transmission lines makes it possible to reduce heat losses in wires by operating
with significantly lower currents. Joule heating does not occur in superconducting materials, because superconductors have zero electrical resistance.
Fuses are used to protect electrical circuits from short circuits and excessively
high currents. This is a conductor of a relatively small cross-section and made
of such an alloy that at permissible currents, heating of the conductor does not
lead to its overheating, and at excessively high currents, the overheating of the
conductor is so significant that the conductor melts and opens the circuit.
2.5. T e m p e r a t u r e c o e f f i c i e n t o f r e s i s t a nc e
o f m e t a l s
The intensification of ion oscillations in the crystal lattice of metals
leads to the fact that the resistance of metals increases with temperature.
The dependence of the resistance of metals on temperature over a wide temperature range is linear and obeys the expression
)( αt1RR
0
+=
, (2.48)
where R0 ‒ conductor resistance at 0 °C; α ‒ temperature coefficient of resistance.
The temperature coefficient of resistance α is by definition equal to
dt
dR
R
1
α =
. (2.49)
The temperature coefficient of electrical resistance is a value equal to the
relative change in the electrical resistance of a section of an electrical circuit or the specific resistance of a substance with a change in temperature
by one unit.
In other words, the temperature coefficient of resistance of metals is
a number that shows how much each unit of resistance of a conductor will
change when the temperature changes by 1 °C
40
. (2.50)
where R0 is the resistance of the conductor at 0 °C; R is the resistance of the
conductor at t °C.
To compare the properties of different materials, the value of the
temperature coefficient of resistance is usually determined in the range of
0–100 °С
, (2.51)
where R0 and R
100
are resistances at 0 and 100 °С, respectively.
For metals, α depends very weakly on temperature and is on average
0.004–0.006 deg−1.
2.6. P r o b l e m s
Problem 1. A galvanic element with ε = 1.6 V has an internal re-
sistance of 0.12 ohm. Find the efficiency of the element with a current in
the circuit of 4 A.
Given
Solution
ε = 1.6 V
r = 0.12 Ω
I = 4 A
Closed circuit:
η – ?
The efficiency of a current source is equal to the ratio of the work of
the Coulomb forces AC in the external circuit to the work of extraneous forces A
extr
inside the current element to separate unlike charges. If the circuit
conductors are stationary, then the work of the Coulomb forces AC is equal
to the heat Q released in the external circuit on the resistance R.
Using expressions (1.6), (1.14), (2.19) and (2.39) we obtain
rR
R
ε
IR
εIt
RtI
εq
RtI
A
Q
η
22
extr
+
=====)1
.
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