4.5. Kirchhoff's rules - Direct electric current: current, voltage, emf,

In practice, complex (branched) electric circuits are very often encountered, for the calculation of which it is convenient to use Kirchhoff's rules (Fig. 4.22).

Fig. 4.22. G. Kirchhoff (1824–1887) — German physicist

Kirchhoff's first rule is a consequence of the law of conservation of charge and of the natural requirement that in stationary processes charge should not accumulate or decrease at any point of a conductor. This rule applies to nodes, that is, to those points in a branched circuit at which at least three conductors meet.

Kirchhoff's first rule states:

Here the currents flowing toward a node and away from it have opposite signs (Fig. 4.23).

Fig. 4.23. The sum of the currents meeting at a node is zero

Kirchhoff's second rule is a generalization of Ohm's law and applies to any closed loop of a branched circuit.

Kirchhoff's second rule states:

Fig. 4.24. Example of a branched electric circuit.
The circuit contains one independent node (a or d) and two independent loops (for example, abcd and adef)

Kirchhoff's rules make it possible to determine the magnitude and direction of the current in any part of a branched circuit, provided the resistances of its sections and the emfs included in them are known. The number of equations written from Kirchhoff's first and second rules must equal the number of unknown quantities sought. Using the first rule for a branched circuit containing m nodes and n branches (sections), one can write (m – 1) independent equations, and using the second rule, (n – m + 1) independent equations.

Let us give an example of calculating the currents in a branched circuit (Fig. 4.25).

Fig. 4.25. Example of a branched circuit

The directions of the emfs are shown by blue arrows. In this circuit we have two nodes — points b and d (m = 2), and three branches — the section b–a–d with current I₁, the section b–d with current I₂, and the section b–c–d with current I₃ (n = 3). Hence, we can write one (m – 1 = 2 – 1 = 1) equation from the first rule and two (n – m + 1 = 3 – 2 + 1 = 2) equations from the second rule. How is this done in practice?

Step one. Let us choose directions for the currents flowing in each of the branches of the circuit. It is entirely unimportant how these directions are chosen. If we guessed correctly, the value of this current will come out positive in the final result; if not, and the direction should have been reversed, the value of this current will come out negative. In our example we chose the current directions as shown in the figure. It is important to stress that the directions of the emfs are not arbitrary — they are determined by the way the poles of the current sources are connected (see Fig. 4.25).

Step two. We write Kirchhoff's first rule for all nodes except one (at the last node, whose choice is arbitrary, this rule will be satisfied automatically). In our case we can write the equation for node b, into which current I₂ flows and out of which currents I₁ and I₃ flow

Step three. It remains for us to write the equations (in our case – two) for the second rule. To do this we need to choose two independent closed loops. In the example under consideration there are three such possibilities: the path around the left loop b–a–d–b, the path around the right loop b–c–d–b, and the path around the whole circuit b–a–d–c–b. It is enough to take any two of them, and then the second rule will be satisfied automatically for the third loop. The direction of traversal of the loop does not matter, but during traversal a current is taken with a plus sign if it flows in the direction of traversal, and with a minus sign if it flows in the opposite direction. The same applies to the signs of the emfs.

Let us first take the loop b–a–d–b. We start from point b and move counterclockwise. Along our path we meet two currents, I₁ and I₂, whose directions coincide with the chosen direction of traversal. The emf also acts in this same direction. Therefore the second rule for this part of the circuit is written as

As the second closed path, for variety let us choose the path b–a–d–c–b around the whole circuit. Along this path we meet two currents I₁ and I₃, of which the first enters with a plus sign and the second — with a minus sign. We also meet two emfs, of which enters the equation with a plus sign, and — with a minus sign. The equation for this closed path has the form

Step four. We have found three equations for the three unknown currents in the circuit. The solution of an arbitrary system of linear equations is described in a course of mathematics. For our purposes (the circuit is simple enough) we can simply express I₃ in terms of I₁ from equation (4.47)

I₂ in terms of I₁ by means of equation (4.46)

and substitute (4.48), (4.49) into the equation of the first rule (4.45). This equation contains only the unknown I₁, which is found without difficulty

Substituting this expression into (4.48), (4.49), we find respectively the currents I₂, I₃

Suppose that at the ends of a section of conductor there is a potential difference Moving from point 2 at higher potential to point 1, where the potential is lower, a positive charge loses energy

By definition, for direct current

then

and the energy lost (or the work done by the electric field forces) equals

Where does this energy go? It does not turn into kinetic energy of the charge, since at constant current the drift velocity of the charges is unchanged. Recall that the charge is not accelerated because of collisions with the atoms of the conductor's crystal lattice. So, if a current flows through a conductor and the conductor is stationary, the work done by the electric field forces is spent heating the conductor. Colliding with the particles of the conductor, the charge carrier transfers to them the energy it receives from the field. Hence the work done by the field on the charges ultimately turns into the energy of the thermal (chaotic) motion of the atoms of the conductor, that is, the conductor heats up (Figs. 27, 28, 29).

Fig. 4.27. Release of heat in an electric discharge

Fig. 4.28. Release of heat in a resistor and an incandescent lamp

Fig. 4.29. Release of heat in a heating element

Video 4.7. Release of heat in series-connected conductors of different resistivity.

Thus, the work A done over time is released in the conductor as heat

This formula is called the Joule–Lenz law. The law was established by J. Joule in 1841 (Fig. 4.30) and, independently of him, by the Russian physicist E.Kh. Lenz in 1842.

Fig. 4.30. J. Joule (1818–1889) — English physicist

At the beginning of the twentieth century it was proven experimentally that free electrons are the charge carriers in metals. Proceeding from these concepts, the German physicist Drude created (1900) the classical electron theory of metallic conductivity, later improved by other physicists. The internal structure of metals is characterized by a crystal lattice, such as, for example, the one shown in (Fig. 4.33).

Fig. 4.33. Diagram of the crystal lattice of metals

Positive ions, which are metal atoms stripped of one or several valence electrons and therefore positively charged, are located at the sites of the lattice. These positive ions are capable of only small thermal oscillations about their equilibrium positions at the lattice sites. In the space between the ions, the valence electrons, torn away from the atoms and «shared» by the crystal, move practically freely, forming what is called the electron gas. According to Drude's theory, the electrons in the crystal lattice behave in many respects like an ideal gas, so the well-known formulas of the kinetic theory of gases can be used to describe their behavior (Fig. 4.34).

Fig. 4.34. Gas of free electrons in the crystal lattice of a metal.
The trajectory of one of the electrons is shown

In the absence of an external field, all directions of the velocity of electrons in chaotic thermal motion are equally probable; consequently, the average current density is zero, and one can say that the electron gas as a whole is at rest relative to the positive ions of the lattice. According to classical thermodynamics, the average energy of the translational thermal motion of the molecules of any gas depends only on the temperature T, but not on the chemical nature or molecular weight of the gas, and equals

From this we find the root-mean-square velocity of the chaotic motion of the particles

Remark: Note that relations (4.62), (4.63) and those following from them hold only when the distribution of the particles (in this case, electrons) over velocities is Maxwellian, which in turn is the case when the translational motion is entirely classical. The electron gas in a metal is a quantum system and is not described by the Maxwell distribution. Therefore the presentation of the classical Drude theory undertaken below is of limited interest, as a way of obtaining relations that are qualitatively (dimensionally) correct but cannot in any way claim quantitative agreement with experiment.

We saw (see Sec. 4.1) that for room temperatures v_T = 10⁶ m/s. When an external electric field is present, the electrons in the metal will also possess some average (drift) velocity v of directed motion against the external field E. According to the estimates given above, the velocity v is many orders of magnitude smaller than the velocity v_T (Fig. 4.35).

Fig. 4.35. Motion of a free electron in a crystal lattice:
a — chaotic motion of an electron in the crystal lattice of a metal;
b — chaotic motion with a drift caused by the electric field.
The scale of the drift is greatly exaggerated

If the electron gas in a metal is treated as an ideal gas, then the thermal motion of the electrons in the crystal lattice can be characterized by the mean free path , that is, the average distance traveled by freely moving electrons in the metal between two successive collisions with lattice ions. The average time between two collisions will be

Since v << v_T, we can assume that neither nor changes when an electric field E is applied.

Ohm's law

When an external electric field E is present, a force F = eE acts on the electron, as a result of which it acquires an acceleration

(m — the mass of the electron). Let us assume that upon colliding with an ion, the electron completely loses its directed-motion velocity: v_min = 0. It then begins to accelerate under the electric field and by the next collision acquires a velocity

Consequently, the average velocity of directed motion v over the period between two collisions, that is, over the time is

Hence the current density will be equal to

Comparing the resulting expression for the current density j with Ohm's law (4.9), we obtain the formula for the conductivity s and for the associated resistivity

Let us give some numerical estimates. As will be shown further on, the relation between the mean free path and the effective
cross-section of the scattering center (ion) and the concentration of these centers has the form

For metals, as we have already discussed, the concentration of ions approximately coincides with the concentration of free electrons

Substituting the expression for into formula (4.66), we obtain

An estimate for the cross-section can be obtained knowing the order of magnitude of atomic diameters

whence

The velocity of chaotic motion was estimated as v_T = 10⁵ m/s. We now obtain from (4.67) the value of the resistivity

The estimate obtained does indeed reproduce the order of magnitude of the resistivities of metals given in the table (see Sec. 4.3).

The Joule–Lenz law

Upon colliding with an ion, as we assumed, the electron transfers to it the drift energy it has accumulated over the time

Multiplying W_e by the electron concentration n and dividing by the time we obtain the thermal energy transferred to a unit volume of the conductor per unit time (that is, the power density)

The last equality in this chain is obtained using expression (14.66) for . As a result, we have derived the Joule–Lenz law (4.60) from the microscopic theory.

The Wiedemann–Franz law

Metals, along with high electrical conductivity, are also characterized by high thermal conductivity. In 1853, the German physicists G. Wiedemann and R. Franz established an empirical law, according to which the ratio of the thermal conductivity coefficient k to the electrical conductivity coefficient s is approximately the same for all metals and varies proportionally to temperature. The thermal conductivity of metals significantly exceeds the thermal conductivity of dielectrics, from which one can conclude that thermal conductivity in metals is mainly due to the electron gas rather than to its crystal lattice. For the thermal conductivity coefficient of the electron gas in a metal, an expression can be borrowed from the kinetic theory of gases

where the product nm of the electron concentration and their mass is substituted in place of the density of the electron gas.

The specific heat capacity of a monatomic gas equals

Substituting this value into the expression for the thermal conductivity coefficient k, we obtain

Dividing k by , we obtain

Since

we arrive at the relation

This relation expresses the Wiedemann–Franz law. Substituting the values k_B = 1.38·10^–23 J/K and e = 1.6·10^–19 C, we obtain

However, the relation obtained does not agree sufficiently accurately with experiment, that is, the classical theory explained the Wiedemann–Franz law only qualitatively. The dependence of resistivity on temperature that follows from the classical theory

also does not agree with experiment. Only quantum theory was able to explain all the discrepancies.

Supplements

The relation between the directed-motion velocity and the charge-flux-density vector (current density) and the electric field strength can be obtained within the framework of the following, sufficiently simple, phenomenological model. When an electron collides with a heavy ion of the crystal lattice (for example, for aluminum ) the electron loses practically all of its directed-motion momentum. According to Newton's second law, the rate of change of momentum equals the force acting on the body. Since collisions of electrons with ions lead to a loss of momentum, it is natural, firstly, to call this force a friction force. Secondly, the greater the electron's velocity before the collision, the greater the momentum it will lose, so this friction force must be proportional to the directed-motion velocity of the electrons. Let us write it in the form

,

then the equation of directed motion of the electron — the equation of Newton's second law for this motion under the action of an electric field of strength has the form

We are looking for a stationary solution corresponding to a constant current, when the directed-motion velocity and . In this case it follows directly from the previous relation that

.

The minus sign in this relation is due to the fact that the electron is negatively charged. For the current density we immediately obtain:

It should be noted that the result does not depend on the sign of the charge carrier.