Ohm's law is a non-uniform section of a circuit. Ohm's law for a homogeneous section of a chain

The section of the circuit in which no external forces act, leading to the emergence of an electromotive force (Fig. 1), is called homogeneous.

Ohm's law for a homogeneous section of a chain was established experimentally in 1826 by G. Ohm.

According to this law, the current strength I in a homogeneous metal conductor is directly proportional to the voltage U at the ends of this conductor and inversely proportional to the resistance R of this conductor:

Figure 2 shows an electrical circuit diagram that allows you to experimentally test this law. Conductors with different resistances are alternately included in the MN section of the circuit.

Rice. 2

The voltage at the ends of the conductor is measured by a voltmeter and can be varied using a potentiometer. The current strength is measured with an ammeter, the resistance of which is negligible (RA ≈ 0). A graph of the dependence of the current in a conductor on the voltage on it - the current-voltage characteristic of the conductor - is shown in Figure 3. The angle of inclination of the current-voltage characteristic depends on the electrical resistance of the conductor R (or its electrical conductivity G): .

Rice. 3

The resistance of conductors depends on its size and shape, as well as on the material from which the conductor is made. For a homogeneous linear conductor, resistance R is directly proportional to its length l and inversely proportional to its cross-sectional area S:

where r is a proportionality coefficient characterizing the material of the conductor and called electrical resistivity. The unit of electrical resistivity is ohm×meter (Ohm×m).

30. Ohm's law for a non-uniform section of a circuit and for a closed circuit.

When an electric current passes in a closed circuit, free charges are subject to forces from a stationary electric field and external forces. In this case, in certain sections of this circuit, the current is created only by a stationary electric field. Such sections of the chain are called homogeneous. In some sections of this circuit, in addition to the forces of a stationary electric field, external forces also act. The section of the chain on which external forces act is called a non-uniform section of the chain.

In order to find out what the current strength in these areas depends on, it is necessary to clarify the concept of voltage.

Rice. 1

Let us first consider a homogeneous section of the chain (Fig. 1, a). In this case, the work of moving the charge is performed only by the forces of a stationary electric field, and this section is characterized by the potential difference Δφ. Potential difference at the ends of the section , where AK is the work done by the forces of a stationary electric field. The inhomogeneous section of the circuit (Fig. 1, b) contains, in contrast to the homogeneous section, a source of EMF, and the work of the electrostatic field forces in this section is added to the work of external forces. By definition, , where q is the positive charge that moves between any two points in the chain; - potential difference between points at the beginning and end of the section under consideration; . Then they talk about tension for tension: Estatic. e. n. = Ee/stat. n. + Estor. Voltage U in a section of a circuit is a physical scalar quantity equal to the total work of external forces and electrostatic field forces to move a single positive charge in this section:

From this formula it is clear that in the general case, the voltage in a given section of the circuit is equal to the algebraic sum of the potential difference and the emf in this section. If only electric forces act on the section (ε = 0), then . Thus, only for a homogeneous section of the circuit the concepts of voltage and potential difference coincide.

Ohm's law for a non-uniform section of a chain has the form:

where R is the total resistance of the inhomogeneous section.

Electromotive force (EMF ) ε can be either positive or negative. This is due to the polarity of inclusion electromotive force ( EMF ) into the section: if the direction created by the current source coincides with the direction of the current passing in the section (the direction of the current in the section coincides inside the source with the direction from the negative pole to the positive), i.e. EMF promotes the movement of positive charges in a given direction, then ε > 0, otherwise, if the EMF prevents the movement of positive charges in a given direction, then ε< 0.

31. Ohm's law in differential form.

Ohm's law for a homogeneous section of a chain, all points of which have the same temperature, is expressed by the formula (in modern notation):

In this form, the formula of Ohm's law is valid only for conductors of finite length, since the quantities I and U included in this expression are measured by devices connected in this section.

The resistance R of a section of a circuit depends on the length l of this section, the cross section S and the resistivity of the conductor ρ. The dependence of resistance on the conductor material and its geometric dimensions is expressed by the formula:

which is valid only for conductors of constant cross-section. For conductors of variable cross-section, the corresponding formula will not be so simple. In a conductor of variable cross-section, the current strength in different sections will be the same, but the current density will be different not only in different sections, but even at different points of the same section. The tension and, consequently, the potential difference at the ends of various elementary sections will also have different meanings. The averaged values ​​of I, U and R over the entire volume of the conductor do not provide information about the electrical properties of the conductor at each point.

To successfully study electrical circuits, it is necessary to obtain an expression of Ohm's law in differential form so that it is satisfied at any point on a conductor of any shape and any size.

Knowing the relationship between the electric field strength and the potential difference at the ends of a certain section , the dependence of the conductor resistance on its size and material and using Ohm’s law for a homogeneous section of the circuit in integral form let's find:

Designating where σ is the specific electrical conductivity of the substance from which the conductor is made, we obtain:

where is the current density. Current density is a vector whose direction coincides with the direction of the velocity vector of positive charges. The resulting expression in vector form will look like:

It is performed at any point on a conductor through which electric current flows. For a closed circuit, one should take into account the fact that in it, in addition to the field strength of the Coulomb forces, there are external forces that create a field of external forces, characterized by the intensity Est. Taking this into account, Ohm's law for a closed circuit in differential form will have the form:

32. Branched electrical circuits. Kirchhoff's rules.

The calculation of branched circuits is simplified if you use Kirchhoff's rules. The first rule applies to the nodes of the chain. A node is a point where more than two currents converge. Currents flowing to a node are considered to have one sign (plus or minus), while currents flowing from a node are considered to have a different sign (minus or plus).

Kirchhoff's first rule is an expression of the fact that in the case of a steady direct current, electric charges should not accumulate at any point of the conductor and in any section of it and is formulated as follows: the algebraic sum of the currents converging at a node is equal to zero

Kirchhoff's second rule is a generalization of Ohm's law to branched electrical circuits.

Consider an arbitrary closed circuit in a branched circuit (circuit 1-2-3-4-1) (Fig. 1.2). Let's set the circuit to traverse clockwise and apply Ohm's law to each of the unbranched sections of the circuit.

Let's add these expressions, while the potentials are reduced and we get the expression

In any closed circuit of an arbitrary branched electrical circuit, the algebraic sum of the voltage drops (products of currents and resistance) of the corresponding sections of this circuit is equal to the algebraic sum of the emfs entering the circuit.

33. DC operation and power. Joule-Lenz law.

Current work is the work of an electric field to transfer electric charges along a conductor;

The work done by the current on a section of the circuit is equal to the product of the current, voltage and time during which the work was performed.

Using the formula of Ohm's law for a section of a circuit, you can write several versions of the formula for calculating the work of the current:

According to the law of conservation of energy:

work is equal to the change in energy of a section of the circuit, therefore the energy released by the conductor

equal to the work of the current.

In the SI system:

JOULE-LENZ LAW

When current passes through a conductor, the conductor heats up and heat exchange occurs with the environment, i.e. the conductor gives off heat to the bodies surrounding it.

The amount of heat released by a conductor carrying current into the environment is equal to the product of the square of the current strength, the resistance of the conductor and the time the current passes through the conductor.

According to the law of conservation of energy, the amount of heat released by a conductor is numerically equal to the work done by the current flowing through the conductor during the same time.

In the SI system:

DC POWER

The ratio of the work done by the current during time t to this time interval.

In the SI system:

34. Direct current magnetic field. Power lines. Magnetic field induction in vacuum .

35. Biot-Savart-Laplace law. Superposition principle.

The Biot-Savart-Laplace law for a conductor with current I, the element dl of which creates an induction field dB at some point A (Fig. 1), is equal to

(1)

where dl is a vector equal in modulus to the length dl of the conductor element and coinciding in direction with the current, r is the radius vector, which is drawn from the conductor element dl to point A of the field, r is the modulus of the radius vector r. The direction dB is perpendicular to dl and r, i.e. perpendicular to the plane in which they lie, and coincides with the direction of the tangent to the magnetic induction line. This direction can be found by the right-hand screw rule: the direction of rotation of the screw head gives the direction dB if the forward motion of the screw coincides with the direction of the current in the element.

The magnitude of the vector dB is given by the expression

(2)

where α is the angle between the vectors dl and r.

Similar to the electric field, for the magnetic field it is true superposition principle: the magnetic induction of the resulting field created by several currents or moving charges is equal to the vector sum of the magnetic induction of the added fields created by each current or moving charge separately:

Using these formulas to calculate the characteristics of the magnetic field (B and H) in the general case is quite complicated. However, if the current distribution has any symmetry, then the application of the Biot-Savart-Laplace law together with the superposition principle makes it possible to simply calculate some fields.

36. Magnetic field of a straight conductor carrying current.

The lines of magnetic induction of the magnetic field of a rectilinear current are concentric circles located in a plane perpendicular to the conductor, with the center on the axis of the conductor. The direction of the induction lines is determined by the right-hand screw rule: if you turn the screw head so that the translational movement of the screw tip occurs along the current in the conductor, then the direction of rotation of the head indicates the direction of the magnetic induction field lines of a straight conductor with current.

In Figure 1, a straight conductor with current is located in the plane of the figure, the induction line is in a plane perpendicular to the figure. Figure 1, b shows a cross-section of a conductor located perpendicular to the plane of the picture, the current in it is directed away from us (this is indicated by a cross “x”), the induction lines are located in the plane of the picture.

As calculations show, the modulus of magnetic induction of the rectilinear current field can be calculated using the formula

where μ is the magnetic permeability of the medium, μ0 = 4π·10-7 H/A2 is the magnetic constant, I is the current strength in the conductor, r is the distance from the conductor to the point at which the magnetic induction is calculated.

Magnetic permeability of a medium is a physical quantity that shows how many times the magnetic induction module B of a field in a homogeneous medium differs from the magnetic induction module B0 at the same field point in a vacuum:

The magnetic field of a straight conductor carrying current is a non-uniform field.

37. Magnetic field of a circular coil with current.

According to the Biot-Savart-Laplace law, the induction of the magnetic field created by a current element dl at a distance r from it is

where α is the angle between the current element and the radius vector drawn from this element to the observation point; r is the distance from the current element to the observation point.

In our case, α = π/2, sinα = 1; , where a is the distance measured from the center of the coil to the point in question on the coil axis. The vectors form a cone at this point with an opening angle at the apex 2 = π - 2β, where β is the angle between the segments a and r.

From symmetry considerations, it is clear that the resulting magnetic field on the coil axis will be directed along this axis, that is, only those components that are parallel to the coil axis contribute to it:

The resulting value of the magnetic field induction B on the coil axis is obtained by integrating this expression over the length of the circuit from 0 to 2πR:

or, substituting the value of r:

In particular, at a = 0 we find the magnetic field induction in the center of a circular coil with current:

This formula can be given a different form using the definition of the magnetic moment of a coil with current:

The last formula can be written in vector form (see Fig. 9.1):

38. The effect of a magnetic field on a current-carrying conductor. Ampere's law.

A magnetic field acts with some force on any current-carrying conductor located in it.

If a conductor through which an electric current flows is suspended in a magnetic field, for example, between the poles of a magnet, then the magnetic field will act on the conductor with some force and deflect it.

The direction of movement of the conductor depends on the direction of the current in the conductor and on the location of the magnet poles.

The force with which a magnetic field acts on a current-carrying conductor is called the Ampere force.

The French physicist A. M. Ampere was the first to discover the effect of a magnetic field on a current-carrying conductor. True, the source of the magnetic field in his experiments was not a magnet, but another conductor with current. By placing current-carrying conductors next to each other, he discovered the magnetic interaction of currents (Fig. 67) - the attraction of parallel currents and the repulsion of antiparallel ones (that is, flowing in opposite directions). In Ampere's experiments, the magnetic field of the first conductor acted on the second conductor, and the magnetic field of the second conductor acted on the first. In the case of parallel currents, Ampere's forces turned out to be directed towards each other and the conductors were attracted; in the case of antiparallel currents, Ampere's forces changed their direction and the conductors repelled each other.

The direction of the Ampere force can be determined using the left-hand rule:

if you position the left palm of your hand so that the four extended fingers indicate the direction of the current in the conductor, and the magnetic field lines enter the palm, then the outstretched thumb will indicate the direction of the force acting on the current-carrying conductor (Fig. 68).

This force (Ampere force) is always perpendicular to the conductor, as well as to the lines of force of the magnetic field in which this conductor is located.

The Ampere force does not act for any orientation of the conductor. If the current-carrying conductor is placed along the

Ampere's law is the law of interaction of electric currents. It was first installed by André Marie Ampère in 1820 for direct current. From Ampere's law it follows that parallel conductors with electric currents flowing in one direction attract, and in opposite directions they repel. Ampere's law is also the law that determines the force with which a magnetic field acts on a small segment of a conductor carrying current. The force with which the magnetic field acts on a volume element of a conductor with current density located in a magnetic field with induction:

.

If current flows through a thin conductor, then , where is the “element of length” of the conductor - a vector that is equal in magnitude and coincides in direction with the current. Then the previous equality can be rewritten as follows:

The force with which the magnetic field acts on an element of a current-carrying conductor located in a magnetic field is directly proportional to the current strength in the conductor and the vector product of the conductor length element and the magnetic induction:

.

The direction of the force is determined by the rule for calculating the vector product, which is convenient to remember using the right-hand rule.

The ampere force modulus can be found using the formula:

where is the angle between the magnetic induction and current vectors.

The force is maximum when the current-carrying conductor element is perpendicular to the magnetic induction lines

39. Interaction of rectilinear parallel currents.

Ampere's law is used to find the force of interaction between two currents. Consider two infinite rectilinear parallel currents I1 and I2; (the directions of the currents are given in Fig. 1), the distance between which is R. Each of the conductors creates a magnetic field around itself, which acts according to Ampere’s law on the adjacent conductor with current. Let us find the force with which the magnetic field of current I1 acts on element dl of the second conductor with current I2. The magnetic field of current I1 is the lines of magnetic induction, which are concentric circles. The direction of vector B1 is given by the rule of the right screw, its modulus is

The direction of the force dF1 with which the field B1 acts on the section dl of the second current is found according to the left-hand rule and is indicated in the figure. The force modulus, using (2), taking into account the fact that the angle α between the elements of the current I2 and the straight vector B1, will be equal to

substituting the value for B1, we find

Arguing similarly, it can be shown that the force dF2 with which the magnetic field of current I2 acts on the element dl of the first conductor with current I1 is directed in the opposite direction and is equal in magnitude

Comparison of expressions (3) and (4) gives that

that is, two parallel currents of the same direction are attracted to each other with a force equal to

(5)

If the currents have opposite directions, then, using the left-hand rule, we determine that there is a repulsive force between them, determined by expression (5).

Fig.1

40. Magnetic field of a moving electric charge.

Any conductor carrying current creates a magnetic field in the surrounding space. In this case, electric current is the ordered movement of electric charges. This means that we can assume that any charge moving in a vacuum or medium generates a magnetic field around itself. As a result of generalizing numerous experimental data, a law was established that determines the field B of a point charge Q moving with a constant non-relativistic speed v. This law is given by the formula

where r is the radius vector drawn from the charge Q to the observation point M (Fig. 1). According to (1), vector B is directed perpendicular to the plane in which the vectors v and r are located: its direction coincides with the direction of translational motion of the right screw as it rotates from v to r.

Fig.1

The magnitude of the magnetic induction vector (1) is found by the formula

(2)

where α is the angle between vectors v and r.

Comparing the Biot-Savart-Laplace law and (1), we see that a moving charge is equivalent in its magnetic properties to a current element:

The given laws (1) and (2) are satisfied only at low speeds (v<<с) движущихся зарядов, когда электрическое поле движущегося с постоянной скорость заряда можно считать электростатическим, т. е. создаваемым неподвижным зарядом, который находится в той точке, где в данный момент времени находится движущийся заряд.

Formula (1) specifies the magnetic induction of a positive charge moving with speed v. When a negative charge moves, Q is replaced by -Q. Speed ​​v - relative speed, i.e. speed relative to the observer's frame of reference. Vector B in a given reference frame depends on both time and the location of the observer. Therefore, it should be noted the relative nature of the magnetic field of a moving charge.

41. Theorem on the circulation of the magnetic field induction vector.

Suppose that in the space where the magnetic field is created, some conditional closed circuit (not necessarily flat) is selected and the positive direction of the circuit is indicated. On each individual small section Δl of this contour, it is possible to determine the tangent component of the vector at a given location, that is, determine the projection of the vector onto the direction of the tangent to a given section of the contour (Fig. 4.17.2). 2

Figure 4.17.2. Closed loop (L) with a specified bypass direction. The currents I1, I2 and I3 are shown, creating a magnetic field.

The circulation of a vector is the sum of products Δl taken over the entire contour L:

Some currents creating a magnetic field may penetrate the selected circuit L, while other currents may be away from the circuit. The circulation theorem states that the circulation of the magnetic field vector of direct currents along any circuit L is always equal to the product of the magnetic constant μ0 by the sum of all currents passing through the circuit:

As an example in Fig. 4.17.2 shows several conductors with currents creating a magnetic field. Currents I2 and I3 penetrate the circuit L in opposite directions; they must be assigned different signs - currents that are associated with the selected direction of traversing the circuit by the rule of the right screw (gimlet) are considered positive. Therefore, I3 > 0, and I2< 0. Ток I1 не пронизывает контур L. Теорема о циркуляции в данном примере выражается соотношением:

The circulation theorem in general follows from the Biot-Savart law and the superposition principle. The simplest example of the application of the circulation theorem is the determination of the magnetic induction field of a straight conductor carrying current. Taking into account the symmetry in this problem, it is advisable to choose the contour L in the form of a circle of some radius R lying in a plane perpendicular to the conductor. The center of the circle is located at some point on the conductor. Due to symmetry, the vector is directed along a tangent (), and its magnitude is the same at all points of the circle. Application of the circulation theorem leads to the relation:

whence follows the formula for the modulus of magnetic induction of the field of a straight conductor with current, given earlier. This example shows that the theorem on the circulation of the magnetic induction vector can be used to calculate magnetic fields created by a symmetrical distribution of currents, when, from symmetry considerations, the overall structure of the field can be “guessed.” There are many practically important examples of calculating magnetic fields using the circulation theorem. One such example is the problem of calculating the field of a toroidal coil (Fig. 4.17.3).

Figure 4.17.3. Application of the circulation theorem to a toroidal coil.

It is assumed that the coil is wound tightly, that is, turn to turn, on a non-magnetic toroidal core. In such a coil, the lines of magnetic induction are closed inside the coil and are concentric circles. They are directed in such a way that, looking along them, we would see the current in the turns circulating clockwise. One of the induction lines of some radius r1 ≤ r< r2 изображена на рис. 4.17.3. Применим теорему о циркуляции к контуру L в виде окружности, совпадающей с изображенной на рис. 4.17.3 линией индукции магнитного поля. Из соображений симметрии ясно, что модуль вектора одинаков вдоль всей этой линии. По теореме о циркуляции можно записать:B ∙ 2πr = μ0IN,

where N is the total number of turns, and I is the current flowing through the turns of the coil. Hence,

Thus, the magnitude of the magnetic induction vector in a toroidal coil depends on the radius r. If the coil core is thin, that is, r2 – r1<< r, то магнитное поле внутри катушки практически однородно. Величина n = N / 2πr представляет собой число витков на единицу длины катушки. В этом случае B = μ0In.

42. Magnetic field of an infinite straight conductor with current and an infinitely long solenoid.

Each part of the toroidal coil can be considered as a long straight coil. Such coils are called solenoids. Far from the ends of the solenoid, the magnetic induction module is expressed by the same ratio as in the case of a toroidal coil. In Fig. Figure 4.17.4 shows the magnetic field of a coil of finite length. It should be noted that in the central part of the coil the magnetic field is almost uniform and much stronger than outside the coil. This is indicated by the density of magnetic induction lines. In the limiting case of an infinitely long solenoid, the uniform magnetic field is entirely concentrated inside the solenoid.

Figure 4.17.4. Magnetic field of a coil of finite length. In the center of the solenoid, the magnetic field is almost uniform and significantly exceeds in magnitude the field outside the coil.

In the case of an infinitely long solenoid, the expression for the magnetic induction modulus can be obtained directly using the circulation theorem, applying it to the rectangular loop shown in Fig. 4.17.5.

Ohm's law for a homogeneous section of a chain:

A section of a circuit is called homogeneous if it does not include a current source. I=U/R, 1 Ohm – the resistance of a conductor in which a force of 1A flows at 1V.

The amount of resistance depends on the shape and properties of the conductor material. For a homogeneous cylindrical conductor, its R=ρl/S, ρ is a value depending on the material used - the resistivity of the substance, from ρ=RS/l it follows that (ρ) = 1 Ohm*m. The reciprocal of ρ is the specific conductivity γ=1/ρ.

It has been experimentally established that with increasing temperature, the electrical resistance of metals increases. At not too low temperatures, the resistivity of metals increases ~ absolute temperature p = α*p 0 *T, p 0 is the resistivity at 0 o C, α is the temperature coefficient. For most metals α = 1/273 = 0.004 K -1. p = p 0 *(1+ α*t), t – temperature in o C.

According to the classical electronic theory of metals, in metals with an ideal crystal lattice, electrons move without experiencing resistance (p = 0).

The reason that causes the appearance of electrical resistance is foreign impurities and physical defects in the crystal lattice, as well as the thermal movement of atoms. The amplitude of atomic vibrations depends on t. The dependence of resistivity on t is a complex function:

p(T) = p rest + p id. , p rest – residual resistivity, p ID. - ideal metal resistance.

The ideal resistance corresponds to an absolutely pure metal and is determined only by the thermal vibrations of the atoms. Based on general considerations, resistance id. metal should tend to 0 at T → 0. However, resistivity as a function is composed of the sum of independent terms, therefore, due to the presence of impurities and other defects in the crystal lattice of resistivity with a decrease in t → to some increase in DC. p rest. Sometimes for some metals the temperature dependence of p passes through a minimum. Res. value beat resistance depends on the presence of defects in the lattice and the impurity content.

j=γ*E – Ohm’s law in differentiated form, describing the process at each point of the conductor, where j is the current density, E is the electric field strength.

The circuit includes a resistor R and a current source. In a non-uniform section of the circuit, current carriers are acted upon by external forces in addition to electrostatic forces. External forces can cause ordered movement of current carriers, such as electrostatic ones. In a non-uniform section of the circuit, the field of external forces created by the EMF source is added to the field of electric charges. Ohm's law in differentiated form: j=γE. Generalizing the formula to the case of a non-uniform conductor j=γ(E+E*)(1).

From Ohm's law in differentiated form for an inhomogeneous section of a chain, one can move on to the integral form of Ohm's law for this section. To do this, consider a heterogeneous area. In it, the cross-section of the conductor may be variable. Let us assume that inside this section of the circuit there is a line, which we will call a current circuit, satisfying:

1. In each section perpendicular to the contour, the quantities j, γ, E, E* have the same values.

2. j, E and E* at each point are directed tangent to the contour.

Let us arbitrarily choose the direction of movement along the contour. Let the chosen direction correspond to movement from 1 to 2. Take a conductor element with area S and contour element dl. Let us project the vectors included in (1) onto the contour element dl: j=γ(E+E*) (2).

I along the contour is equal to the projection of the current density onto the area: I=jS (3).

Specific conductivity: γ=1/ρ. Replacing in (2) I/S=1/ρ(E+E*). Multiply by dl and integrate along the contour ∫Iρdl/S=∫Eedl+∫E*edl. Let's take into account that ∫ρdl/S=R, and ∫Eedl=(φ 1 -φ 2), ∫E*edl= ε 12, IR= ε 12 +(φ 1 -φ 2). ε 12, like I, is an algebraic quantity, therefore it was agreed that when ع promotes the movement of positive current carriers in the chosen direction 1-2, consider ε 12 >0. But in practice, this is the case when, when going around a section of the circuit, a negative pole is first encountered, then a positive one. If ع prevents the movement of positive carriers in the chosen direction, then ε 12<0.

From the last formula I=(φ 1 -φ 2)+(-)ε 12 /R. This formula expresses Ohm's law for a non-uniform section of the chain. Based on it, one can obtain Ohm's law for an inhomogeneous section of the chain. In this case, ε 12 =0, therefore, I=(φ 1 -φ 2)/R, I=U/R, as well as Ohm’s law for a closed circuit: φ 1 =φ 2, which means I=ع/R, where R is the total resistance of the entire circuit: I=ع/ R 0 +r.

Electric current is the ordered movement of an uncompensated electric charge. If this movement occurs in a conductor, then the electric current is called conduction current. Electric current can be caused by Coulomb forces. The field of these forces is called Coulomb and is characterized by intensity E coul.

The movement of charges can also occur under the influence of non-electric forces, called external forces (magnetic, chemical). E st is the field strength of these forces.

The ordered movement of electric charges can occur without the action of external forces (diffusion, chemical reactions in the current source). For generality of reasoning, in this case we will introduce an effective external field E st.

Total work done to move a charge along a section of a circuit:

Let us divide both sides of the last equation by the amount of charge moved along this area.

.

Potential difference across a section of a circuit.

The voltage on a section of a circuit is a value equal to the ratio of the total work done when moving a charge in this section to the amount of charge. Those. VOLTAGE AT A CIRCUIT SECTION IS THE TOTAL WORK TO MOVE A SINGLE POSITIVE CHARGE AROUND THE SECTION.

EMF in a given area is called a value equal to the ratio of the work done by non-electric energy sources when moving a charge to the value of this charge. EMF IS THE WORK OF EXTERNAL FORCES TO MOVE A SINGLE POSITIVE CHARGE ON A SECTION OF A CIRCUIT.

Third-party forces in an electrical circuit operate, as a rule, in current sources. If there is a current source on a section of the circuit, then such a section is called inhomogeneous.

The voltage on a non-uniform section of the circuit is equal to the sum of the potential difference at the ends of this section and the emf of the sources in it. In this case, the EMF is considered positive if the direction of the current coincides with the direction of action of external forces, i.e. from minus source to plus.

If there are no current sources in the area of ​​interest to us, then in this and only in this case the voltage is equal to the potential difference.

In a closed circuit, for each of the sections forming a closed loop, we can write:

Because the potentials of the starting and ending points are equal, then .

Therefore, (2),

those. the sum of the voltage drops in a closed loop of any electrical circuit is equal to the sum of the emf.

Let us divide both sides of equation (1) by the length of the section.

Where is the total field strength, is the external field strength, is the Coulomb field strength.

For a homogeneous chain section.

Current density means Ohm's law in differential form. THE CURRENT DENSITY IN A HOMOGENEOUS SECTION OF THE CIRCUIT IS DIRECTLY PROPORTIONAL TO THE ELECTROSTATIC FIELD STRENGTH IN THE CONDUCTOR.

If a Coulomb and external field (inhomogeneous section of the circuit) acts on a given section of the circuit, then the current density will be proportional to the total field strength:

. Means, .

Ohm's law for a non-uniform section of a circuit: THE CURRENT STRENGTH IN A INHOMOGENEOUS SECTION OF THE CIRCUIT IS DIRECTLY PROPORTIONAL TO THE VOLTAGE IN THIS SECTION AND INVERSE PROPORTIONAL TO ITS RESISTANCE.

If the direction of E c t and E cool coincide, then the emf and the potential difference have the same sign.

In a closed circuit V=O, because the Coulomb field is conservative.

From here: ,

where R is the resistance of the external part of the circuit, r is the resistance of the internal part of the circuit (i.e., current sources).

Ohm's law for a closed circuit: THE CURRENT STRENGTH IN A CLOSED CIRCUIT IS DIRECTLY PROPORTIONAL TO THE EMF OF THE SOURCES AND INVERSE PROPORTIONAL TO THE COMPLETE RESISTANCE OF THE CIRCUIT.

KIRCHHOFF'S RULES.

Kirchhoff's rules are used to calculate branched electrical circuits.

The point in a circuit where three or more conductors intersect is called a node. According to the law of conservation of charge, the sum of the currents entering and leaving the node is zero. . (Kirchhoff's first rule). THE ALGEBRAIC SUM OF CURRENTS PASSING THROUGH THE NODE IS EQUAL TO ZERO.

The current entering the node is considered positive, leaving the node is considered negative. The directions of currents in sections of the circuit can be chosen arbitrarily.

From equation (2) it follows that WHEN BYPASSING ANY CLOSED CIRCUIT, THE ALGEBRAIC SUM OF THE VOLTAGE DROP IS EQUAL TO THE ALGEBRAIC SUM OF THE EMF IN THIS CIRCUIT , - (Kirchhoff's second rule).

The direction of traversing the contour is chosen arbitrarily. The voltage in a section of the circuit is considered positive if the direction of the current in this section coincides with the direction of bypassing the circuit. The EMF is considered positive if, when going around the circuit, the source passes from the negative pole to the positive one.

If the chain contains m nodes, then m-1 equations can be made using the first rule. Each new equation must include at least one new element. The total number of equations compiled according to Kirchhoff’s rules must coincide with the number of sections between the nodes, i.e. with the number of currents.

Differential form of Ohm's law. Let's find the connection between the current density j and field strength E at the same point on the conductor. In an isotropic conductor, the ordered movement of current carriers occurs in the direction of the vector E. Therefore, the directions of the vectors j And E match up. Let us consider an elementary volume in a homogeneous isotropic medium with generators parallel to the vector E, length limited by two equipotential sections 1 and 2 (Fig. 4.3).

Let us denote their potentials by and, and the average cross-sectional area by. Using Ohm's law, we obtain for the current, or for the current density, therefore

Let us move to the limit at , then the volume under consideration can be considered cylindrical, and the field inside it is uniform, so that

Where E- electric field strength inside the conductor. Considering that j And E coincide in direction, we get

.

This ratio is differential form of Ohm's law for a homogeneous section of the circuit. The quantity is called specific conductivity. In a non-uniform section of the circuit, current carriers are acted upon, in addition to electrostatic forces, by external forces, therefore, the current density in these sections turns out to be proportional to the sum of the voltages. Taking this into account leads to differential form of Ohm's law for a non-uniform section of the circuit.

.

When an electric current passes in a closed circuit, free charges are subject to forces from a stationary electric field and external forces. In this case, in certain sections of this circuit, the current is created only by a stationary electric field. Such sections of the chain are called homogeneous. In some sections of this circuit, in addition to the forces of a stationary electric field, external forces also act. The section of the circuit where external forces act is called non-uniform section of the chain.

In order to find out what the current strength in these areas depends on, it is necessary to clarify the concept of voltage.

Let us first consider a homogeneous section of the chain (Fig. 1, a). In this case, the work to move the charge is performed only by the forces of a stationary electric field, and this section is characterized by the potential difference Δ φ . Potential difference at the ends of the section Δ φ =φ 1−φ 2=AKq, Where A K is the work done by the forces of a stationary electric field. The inhomogeneous section of the circuit (Fig. 1, b) contains, in contrast to the homogeneous section, a source of EMF, and the work of the electrostatic field forces in this section is added to the work of external forces. A-priory, Aelq=φ 1−φ 2, where q- a positive charge that moves between any two points in the chain; φ 1−φ 2 - potential difference between points at the beginning and end of the section under consideration; Astq=ε . Then we talk about tension for tension: E static e. n. = E e/stat. n. + E side Voltage U in a section of the circuit is a physical scalar quantity equal to the total work of external forces and the forces of the electrostatic field to move a single positive charge in this section:

U=AKq+Astorq=φ 1−φ 2+ε .

From this formula it is clear that in the general case, the voltage in a given section of the circuit is equal to the algebraic sum of the potential difference and the emf in this section. If only electrical forces act on the site ( ε = 0), then U=φ 1−φ 2. Thus, only for a homogeneous section of the circuit the concepts of voltage and potential difference coincide.

Ohm's law for a non-uniform section of a chain has the form:

I=UR=φ 1−φ 2+εR,

Where R- the total resistance of the heterogeneous area.

EMF ε can be either positive or negative. This is due to the polarity of the inclusion of the EMF in the section: if the direction created by the current source coincides with the direction of the current passing in the section (the direction of the current in the section coincides inside the source with the direction from the negative pole to the positive), i.e. EMF promotes the movement of positive charges in a given direction, then ε > 0, otherwise, if the EMF prevents the movement of positive charges in a given direction, then ε < 0.

Conductors that obey Ohm's law are called linear.

Graphical dependence of current on voltage (such graphs are called volt-ampere characteristics, abbreviated as CVC) is depicted by a straight line passing through the origin of coordinates. It should be noted that there are many materials and devices that do not obey Ohm's law, for example, a semiconductor diode or a gas-discharge lamp. Even for metal conductors, at sufficiently high currents, a deviation from Ohm’s linear law is observed, since the electrical resistance of metal conductors increases with increasing temperature.

1.5. Series and parallel connection of conductors

Conductors in DC electrical circuits can be connected in series or in parallel.

When connecting conductors in series, the end of the first conductor is connected to the beginning of the second, etc. In this case, the current strength is the same in all conductors , A the voltage at the ends of the entire circuit is equal to the sum of the voltages at all series-connected conductors. For example, for three series-connected conductors 1, 2, 3 (Fig. 4) with electrical resistances , we get:

Rice. 4.

According to Ohm's law for a section of a circuit:

U 1 = IR 1, U 2 = IR 2, U 3 = IR 3 and U = IR (1)

where is the total resistance of a section of a circuit of series-connected conductors. From expression and (1) we have . Thus,

R = R 1 + R 2 + R 3 . (2)

When conductors are connected in series, their total electrical resistance is equal to the sum of the electrical resistances of all conductors.

From relations (1) it follows that the voltages on series-connected conductors are directly proportional to their resistances:

Rice. 5.

In this case, the voltage on all conductors is the same, and the current in an unbranched circuit is equal to the sum of the currents in all parallel-connected conductors . For three parallel-connected conductors with resistances, and based on Ohm’s law for a section of the circuit, we write

Denoting the total resistance of a section of an electrical circuit of three parallel-connected conductors through , for the current strength in an unbranched circuit we obtain

, (5)

then from expressions (3), (4) and (5) it follows that:

. (6)

When connecting conductors in parallel, the reciprocal of the total resistance of the circuit is equal to the sum of the reciprocals of the resistances of all parallel-connected conductors.

The parallel connection method is widely used to connect electric lighting lamps and household electrical appliances to the electrical network.

1.6. Resistance measurement

What are the features of resistance measurement?

When measuring small resistances, the measurement result is influenced by the resistance of the connecting wires, contacts and contact thermo-emf. When measuring large resistances, it is necessary to take into account volumetric and surface resistances and take into account or eliminate the influence of temperature, humidity and other reasons. Measurement of the resistance of liquid conductors or conductors with high humidity (grounding resistance) is carried out using alternating current, since the use of direct current is associated with errors caused by the phenomenon of electrolysis.

The resistance of solid conductors is measured using direct current. Since this, on the one hand, eliminates errors associated with the influence of the capacitance and inductance of the measurement object and the measuring circuit, on the other hand, it becomes possible to use magnetoelectric system devices with high sensitivity and accuracy. Therefore, megohmmeters are produced with direct current.

1.7. Kirchhoff's rules

Kirchhoff's rules– relationships that hold between currents and voltages in sections of any electrical circuit.

Kirchhoff's rules do not express any new properties of a stationary electric field in current-carrying conductors compared to Ohm's law. The first of them is a consequence of the law of conservation of electric charges, the second is a consequence of Ohm’s law for a non-uniform section of the circuit. However, their use greatly simplifies the calculation of currents in branched circuits.

Kirchhoff's first rule

Nodal points can be identified in branched chains ( nodes ), in which at least three conductors converge (Fig. 6). The currents flowing into the node are considered to be positive; flowing from the node - negative.

the algebraic sum of current strengths converging at a node is equal to zero:

Or in general:

In other words, as much current flows into a node, as much flows out of it. This rule follows from the fundamental law of conservation of charge.

Kirchhoff's second rule

The circuit contains one independent node (a or d) and two independent circuits (for example, abcd and adef)

In the circuit, three circuits abcd, adef and abcdef can be distinguished. Of these, only two are independent (for example, abcd and adef), since the third does not contain any new regions.

Kirchhoff's second rule is a consequence of the generalized Ohm's law.

For contour sections abcd, the generalized Ohm's law is written as:

for section bc:

for section da:

Adding the left and right sides of these equalities and taking into account that , we get:

Similarly, for the adef contour one can write:

According to Kirchhoff's second rule:

in any simple closed circuit, arbitrarily chosen in a branched electrical circuit, the algebraic sum of the products of the current strengths and the resistance of the corresponding sections is equal to the algebraic sum of the emfs present in the circuit:

,

where is the number of sources in the circuit, is the number of resistances in it.

When drawing up a stress equation for a circuit, you need to choose the positive direction of traversing the circuit.

If the directions of the currents coincide with the selected direction of bypassing the circuit, then the current strengths are considered positive. EMF are considered positive if they create currents co-directed with the direction of bypassing the circuit.

A special case of the second rule for a circuit consisting of one circuit is Ohm's law for this circuit.

The procedure for calculating branched DC circuits

The calculation of a branched DC electrical circuit is performed in the following order:

· arbitrarily choose the direction of currents in all sections of the circuit;

· write independent equations according to Kirchhoff’s first rule, where is the number of nodes in the chain;

· choose arbitrarily closed contours so that each new contour contains at least one section of the circuit that is not included in the previously selected contours. Write down Kirchhoff's second rule for them.

In a branched chain containing nodes and sections of the chain between adjacent nodes, the number of independent equations corresponding to the contour rule is .

Based on Kirchhoff's rules, a system of equations is compiled, the solution of which allows one to find the current strengths in the branches of the circuit.

Example 1:

Kirchhoff's first and second rules, written down for everyone independent nodes and circuits of a branched circuit, together give the necessary and sufficient number of algebraic equations for calculating the values ​​of voltages and currents in an electrical circuit. For the circuit shown in Fig. 7, the system of equations for determining three unknown currents has the form:

,

,

.

Thus, Kirchhoff's rules reduce the calculation of a branched electrical circuit to solving a system of linear algebraic equations. This solution does not cause any fundamental difficulties, however, it can be very cumbersome even in the case of fairly simple circuits. If, as a result of the solution, the current strength in some area turns out to be negative, then this means that the current in this area goes in the direction opposite to the selected positive direction.