The current strength in a homogeneous section of the circuit is directly proportional to the voltage at a constant resistance of the section and inversely proportional to the resistance of the section at a constant voltage.

WhereU - voltage in the area, R- resistance of the area.

Ohm's law for an arbitrary section of a circuit containing a direct current source.

Whereφ 1 - φ 2 + ε = U voltage at a given section of the circuit,R - electrical resistance of a given section of the circuit.

Ohm's law for a complete circuit.

The current strength in a complete circuit is equal to the ratio of the electromotive force of the source to the sum of the resistances of the external and internal sections of the circuit.

WhereR - electrical resistance of the external section of the circuit,r - electrical resistance of the internal section of the circuit.

Short circuit.

From Ohm's law for a complete circuit it follows that the current strength in a circuit with a given current source depends only on the resistance of the external circuit R.

If a conductor with a resistance R is connected to the poles of a current source<< r, то тогда только ЭДС источника тока и его сопротивление будут определять значение силы тока в цепи. Такое значение силы тока будет являться предельным для данного источника тока и называется током короткого замыкания.

Electrical resistance (R) is a physical quantity numerically equal to the ratio
voltage at the ends of the conductor to the current passing through the conductor.
The resistance value for a section of a circuit can be determined from the formula of Ohm's law for a section of a circuit.

However, the resistance of a conductor does not depend on the current in the circuit and voltage, but is determined only by the shape, size and material of the conductor.

where l is the length of the conductor (m), S is the cross-sectional area (sq.m),
r (ro) - resistivity (Ohm m).

Resistivity

Shows the resistance of a conductor made of a given substance,
1 m long and with a cross section of 1 sq. m.

SI unit of resistivity: 1 ohm m

However, in practice, the thickness of the wires is significantly less than 1 sq. m.
Therefore, a non-system unit of measurement of resistivity is more often used:

System resistance unit in SI:

The resistance of a conductor is 1 Ohm if, with a potential difference at its ends of 1 V,
a current of 1 A flows through it.

The reason for the presence of resistance in a conductor is the interaction of moving electrons with ions of the crystal lattice of the conductor. Due to the difference in the structure of the critical lattice for conductors made of different substances, their resistances differ from each other.

N39

Serial and parallel connections in electrical engineering - two main ways of connecting elements of an electrical circuit. In a series connection, all elements are connected to each other in such a way that the section of the circuit that includes them does not have a single node. In a parallel connection, all elements included in the chain are united by two nodes and have no connections with other nodes, unless this contradicts the condition.

When conductors are connected in series, the current in all conductors is the same.

In a parallel connection, the voltage drop between the two nodes connecting the elements of the circuit is the same for all elements. In this case, the reciprocal value of the total resistance of the circuit is equal to the sum of the reciprocal values ​​of the resistances of parallel-connected conductors.

When conductors are connected in series, the current strength in any part of the circuit is the same:

The total voltage in the circuit in a series connection, or the voltage at the poles of the current source, is equal to the sum of the voltages in individual sections of the circuit:

N40

Electromotive force(EMF) is a scalar physical quantity that characterizes the work of external (non-potential) forces in direct or alternating current sources. In a closed conducting circuit, the EMF is equal to the work of these forces to move a single positive charge along the circuit.

EMF can be expressed in terms of the electric field strength of external forces (). In a closed loop () then the EMF will be equal to:

, where is the element of the contour length.

EMF, like voltage, is measured in volts. We can talk about electromotive force at any part of the circuit. This is the specific work of external forces not throughout the entire circuit, but only in a given area. The EMF of a galvanic cell is the work of external forces when moving a single positive charge inside the element from one pole to another. The work of external forces cannot be expressed through a potential difference, since external forces are non-potential and their work depends on the shape of the trajectory. So, for example, the work of external forces when moving a charge between current terminals outside the source itself is zero.

[edit] Induction emf

The cause of the electromotive force can be a change in the magnetic field in the surrounding space. This phenomenon is called electromagnetic induction. The magnitude of the induced emf in the circuit is determined by the expression

where is the magnetic field flux through a closed surface bounded by a contour. The “−” sign before the expression shows that the induced current created by the induced emf prevents a change in the magnetic flux in the circuit

n41

The work done by an electric current shows how much work was done by the electric field when moving charges along a conductor.

Knowing two formulas:
I = q/t ..... and..... U = A/q
You can derive a formula for calculating the work of electric current:

The work done by electric current is equal to the product of current and voltage
and for the duration of current flow in the circuit.

Unit of measurement of electric current work in the SI system:
[A] = 1 J = 1A. B. c

The power of an electric current shows the work done by the current per unit time.
and is equal to the ratio of the work done to the time during which this work was done.

(power in mechanics is usually denoted by the letter N, in electrical engineering - the letter R)
because A = IUt, then the power of the electric current is equal to:

Unit of electric current power in the SI system:

[P] = 1 W (watt) = 1 A. B

N42

Semiconductor- a material that, in terms of its specific conductivity, occupies an intermediate position between interconductors and dielectrics and differs from conductors in the strong dependence of the specific conductivity on the concentration of impurities, temperature and exposure to various types of radiation. The main property of a semiconductor is an increase in electrical conductivity with increasing temperature.

Semiconductors are substances whose band gap is on the order of several electron volts (eV). For example, a diamond can be classified as wide bandgap semiconductors, and indium arsenide - to narrow-gap. Semiconductors include many chemical elements (germanium, silicon, selenium, tellurium, arsenic and others), a huge number of alloys and chemical compounds (gallium arsenide, etc.). Almost all inorganic substances in the world around us are semiconductors. The most common semiconductor in nature is silicon, making up almost 30% of the earth's crust.

Depending on whether the impurity atom gives up an electron or captures it, impurity atoms are called donor or acceptor. The nature of the impurity can vary depending on which atom of the crystal lattice it replaces and into which crystallographic plane it is embedded.

The conductivity of semiconductors is highly dependent on temperature. Near absolute zero temperature, semiconductors have the properties of dielectrics.

N43

Magnetic phenomena were known in the ancient world. The compass was invented more than 4,500 years ago. It appeared in Europe around the 12th century AD. However, it was only in the 19th century that the connection between electricity and magnetism was discovered and the idea of magnetic field .

The first experiments (carried out in 1820) that showed that there is a deep connection between electrical and magnetic phenomena were the experiments of the Danish physicist H. Oersted. These experiments showed that a magnetic needle located near a current-carrying conductor is acted upon by forces that tend to turn it. In the same year, the French physicist A. Ampere observed the force interaction of two conductors with currents and established the law of interaction of currents.

According to modern concepts, current-carrying conductors exert a force on each other not directly, but through the magnetic fields surrounding them.

The sources of the magnetic field are moving electric charges (currents). A magnetic field arises in the space surrounding current-carrying conductors, just as an electric field arises in the space surrounding stationary electric charges. The magnetic field of permanent magnets is also created by electric microcurrents circulating inside the molecules of a substance (Ampere's hypothesis).

Scientists of the 19th century tried to create a theory of the magnetic field by analogy with electrostatics, introducing into consideration the so-called magnetic charges two signs (for example, north N and southern S poles of the magnetic needle). However, experience shows that isolated magnetic charges do not exist.

The magnetic field of currents is fundamentally different from the electric field. A magnetic field, unlike an electric field, has a force effect only on moving charges (currents).

To describe the magnetic field, it is necessary to introduce a field strength characteristic similar to the electric field strength vector. This characteristic is magnetic induction vector which determines the forces acting on currents or moving charges in a magnetic field.

For the positive vector direction the direction is taken from the south pole S to the north pole N of the magnetic needle, freely oriented in the magnetic field. Thus, by studying the magnetic field created by a current or a permanent magnet using a small magnetic needle, it is possible to determine the direction of the vector at each point in space. Such research allows us to visualize the spatial structure of the magnetic field. Similar to the lines of force in electrostatics, one can construct magnetic induction lines , at each point of which the vector is directed along a tangent.

N44

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 expression for 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 , in the International System of Units (SI) has the form:

.

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 direction of the force is determined by the rule for calculating the vector product, which is convenient to remember using the left-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 conductor element with current is located perpendicular to the lines of magnetic induction ():

N45

Let us consider a current-carrying circuit formed by fixed wires and a movable jumper of length sliding along them l(Fig. 2.17). This circuit is in an external uniform magnetic field perpendicular to the plane of the circuit. With the current direction shown in the figure I, the vector is codirectional with .

Per current element I(movable wire) length l The Ampere force acts to the right:

Let the conductor l will move parallel to itself at a distance d x. This will do the following:

,

(2.9.1)

Job , performed by a current-carrying conductor when moving, numerically equal to the product of current and magnetic flux, crossed by this conductor.

The formula remains valid if a conductor of any shape moves at any angle to the lines of the magnetic induction vector.

Lorentz force

The force exerted by a magnetic field on a moving electrically charged particle.

where q is the charge of the particle;
V - charge speed;
B - magnetic field induction;
a is the angle between the charge velocity vector and the magnetic induction vector.

The direction of the Lorentz force is determined Byleft hand rule:

If you place your left hand so that the component of the induction vector perpendicular to the speed enters the palm, and the four fingers are located in the direction of the speed of movement of the positive charge (or against the direction of the speed of the negative charge), then the bent thumb will indicate the direction of the Lorentz force

.

Since the Lorentz force is always perpendicular to the speed of the charge, it does not do work (that is, it does not change the value of the charge speed and its kinetic energy).

If a charged particle moves parallel to the magnetic field lines, then Fl = 0, and the charge in the magnetic field moves uniformly and rectilinearly.

If a charged particle moves perpendicular to the magnetic field lines, then the Lorentz force is centripetal

and creates a centripetal acceleration equal to

In this case, the particle moves in a circle.

.

According to Newton's second law: the Lorentz force is equal to the product of the mass of the particle and the centripetal acceleration

then the radius of the circle

and the period of charge revolution in a magnetic field is

Since electric current represents the ordered movement of charges, the effect of a magnetic field on a conductor carrying current is the result of its action on individual moving charges.

MAGNETIC PROPERTIES OF SUBSTANCE

The magnetic properties of matter are explained according to Ampere’s hypothesis by closed currents circulating inside any substance:

Inside atoms, due to the movement of electrons in orbits, there are elementary electric currents that create elementary magnetic fields.
That's why:
1. if the substance does not have magnetic properties, the elementary magnetic fields are unoriented (due to thermal motion);

2. if a substance has magnetic properties, the elementary magnetic fields are equally directed (oriented) and the substance’s own internal magnetic field is formed.

Electromagnetic induction- the phenomenon of the occurrence of electric current in a closed circuit when the magnetic flux passing through it changes.

Electromagnetic induction was discovered by Michael Faraday on August 29 [ source not specified 253 days] 1831. He discovered that the electromotive force arising in a closed conducting circuit is proportional to the rate of change of the magnetic flux through the surface bounded by this circuit. The magnitude of the electromotive force (EMF) does not depend on what is causing the flux change - a change in the magnetic field itself or the movement of the circuit (or part of it) in the magnetic field. The electric current caused by this emf is called induced current.

According to Faraday's law of electromagnetic induction, when the magnetic flux passing through an electrical circuit changes, a current called induction is excited in it. The magnitude of the electromotive force responsible for this current is determined by the equation:

where the minus sign means that the induced emf acts in such a way that the induced current prevents a change in flux. This fact is reflected in Lenz's rule.

N48

So far we have considered changing magnetic fields without paying attention to what their source is. In practice, magnetic fields are most often created using various types of solenoids, i.e. multi-turn circuits with current.

There are two possible cases here: when the current in the circuit changes, the magnetic flux changes: a ) the same circuit ; b ) adjacent circuit.

The induced emf arising in the circuit itself is called Self-induced emf, and the phenomenon itself – self-induction.

If the induced emf occurs in the adjacent circuit, then they talk about the phenomenon mutual induction.

It is clear that the nature of the phenomenon is the same, but different names are used to emphasize the place where the induced emf occurs.

Self-induction phenomenon discovered by the American scientist J. Henry.

According to the law of electromagnetic induction

But ΔФ=LΔI, hence:

N49

An electric motor is simply a device for efficiently converting electrical energy into mechanical energy.

The basis of this transformation is magnetism. Electric motors use permanent magnets and electromagnets and also use the magnetic properties of various materials to create these amazing devices.

There are several types of electric motors. Let's note two main classes: AC and DC.

AC (Alternating Current) class electric motors require an alternating current or voltage source to operate (you can find such a source in any electrical outlet in the house).

Electric motors of the DC (Direct Current) class require a source of direct current or voltage to operate (you can find such a source in any battery).

Universal motors can be powered by any type of source.

Not only are the designs of the motors different, the methods for controlling speed and torque are different, although the principle of energy conversion is the same for all types.

The section of the circuit in which external forces do not act, leading to the occurrence of EMF (Fig. 1), is called homogeneous.

Ohm's law for a homogeneous section of the 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. To the station MN the circuits alternately include conductors with different resistances.

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 ( R A ≈ 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): .

The current strength in a closed circuit consisting of a current source with internal resistance and a load with resistance is equal to the ratio of the source emf to the sum of the internal resistance of the source and the load resistance.

8.RESISTANCE AND CONDUCTIVITY OF CONDUCTORS. DEPENDENCE OF CONDUCTOR RESISTANCE ON PHYSICAL CONDITIONS

When an electrical circuit is closed, at the terminals of which there is a potential difference, an electric current occurs. Free electrons, under the influence of electric field forces, move along the conductor. In their movement, free electrons collide with the atoms of the conductor and give them a supply of their kinetic energy.

Thus, electrons passing through a conductor encounter resistance to their movement. When electric current passes through a conductor, the latter heats up.

The electrical resistance of a conductor (denoted by the Latin letter r) is responsible for the phenomenon of converting electrical energy into heat when an electric current passes through the conductor. In the diagrams, electrical resistance is indicated as shown in Fig. 18.

The unit of resistance is taken to be 1 ohm. Om is often represented by the Greek capital letter Ω (omega). Therefore, instead of writing: “The resistance of the conductor is 15 ohms,” you can simply write: r = 15 Ω.

1000 ohms is called 1 kiloohm (1 ohm, or 1 kΩ).

1,000,000 ohms is called 1 megohm (1 mg ohm, or 1 MΩ).

Serial and parallel connections in electrical engineering - two main ways of connecting elements of an electrical circuit. In a series connection, all elements are connected to each other in such a way that the section of the circuit that includes them does not have a single node. In a parallel connection, all elements included in the chain are united by two nodes and have no connections with other nodes, unless this contradicts the condition.

When conductors are connected in series, the current in all conductors is the same.

In a parallel connection, the voltage drop between the two nodes connecting the elements of the circuit is the same for all elements. In this case, the reciprocal value of the total resistance of the circuit is equal to the sum of the reciprocal values ​​of the resistances of parallel-connected conductors.

How to determine the total resistance of a circuit, if we already know all the resistances included in it in series? Using the position that the voltage U at the terminals of the current source is equal to the sum of the voltage drops in the sections of the circuit, we can write:

U = U1 + U2 + U3

U1 = IR1 U2 = IR2 and U3 = IR3

IR = IR1 + IR2 + IR3

Taking the equality I out of brackets on the right side, we obtain IR = I(R1 + R2 + R3).

Having now divided both sides of the equality by I, we will finally have R = R1 + R2 + R3

Thus, we came to the conclusion that when resistances are connected in series, the total resistance of the entire circuit is equal to the sum of the resistances of the individual sections.

Electromotive force.

If an electric field is created in a conductor and measures are not taken to maintain it, then the movement of current carriers will very quickly lead to the fact that the field inside the conductor will disappear and the current will stop. In order to maintain the current for a long time, it is necessary to continuously remove the positive charges brought here by the current from the end of the conductor with a lower potential j 2 and transfer them to the end with a higher potential (Fig. 56.1).

The electric field created in a conductor cannot carry out such a transfer of charges. In order for a constant current to exist, the action of some other forces (not Coulomb forces) is necessary, moving charges against electric forces and maintaining the constancy of electric fields. These can be magnetic forces, charges can be separated due to chemical reactions, diffusion of charge carriers in an inhomogeneous medium, etc. To emphasize the difference between these forces and the Coulomb interaction forces, it is customary to denote them by the term outside forces. Devices in which free charges move under the influence of external forces are called current sources. These include electromagnetic generators, thermoelectric generators, and solar panels. A separate group consists of chemical power sources: galvanic cells, batteries and fuel cells.

The action of external forces can be characterized by introducing the concept of field strength of external forces: .

The work of external forces to move the charge q on the site dl can be expressed as follows:

along the entire length of the section l:

. (56.1)

The value equal to the ratio of the work done by external forces to move a charge to this charge is called electromotive force(EMF):

. (56.2)

In a conductor through which current flows, the electric field strength is the sum of the field strengths of Coulomb forces and external forces:

Then for the current density we can write

Let's replace the vectors with their projections onto the direction of the closed loop and multiply both sides of the equation by dl:

Having made the substitution , , we reduce the resulting equation to the form

We integrate the resulting expression over the length of the electrical circuit:

The integral on the left side of the equation represents the resistance R sections 1-2. On the right side of the equation, the value of the first integral is numerically equal to the work of Coulomb forces to move a unit charge from point 1 to point 2 - this is the potential difference. The value of the second integral is numerically equal to the work of external forces to move a unit charge from point 2 to point 1 - this is electromotive force. In accordance with this, equation (56.3) is reduced to the form

Magnitude IR, equal to the product of the current strength and the resistance of the circuit section, is called voltage drop on a section of the chain. Voltage drop is numerically equal to the work done when moving a unit charge by external forces and electric field forces (Coulomb).

The section of the circuit containing the EMF is called a non-uniform section. We find the current strength in such a section from formula (56.4):

Considering that the current source can be connected to a section of the circuit in two ways, we replace the sign in front of the EMF with “±”:

Expression (56.5) is Ohm's law for a non-uniform section of a chain. The signs “+” or “-” take into account how external forces influence the flow of current in the indicated direction: they promote or hinder (Fig. 56.2).

If a section of the circuit does not contain an EMF, i.e. it is homogeneous, then from formula (56.5) it follows that

From formula (56.5) it follows

Where IR- voltage drop on the external section of the circuit, Ir- voltage drop on the internal section of the circuit.

Hence, The emf of the current source is equal to the sum of the voltage drops in the external and internal sections of the circuit.

.

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.

When connecting conductors 1, 2, 3 in parallel (Fig. 5), their beginnings and ends have common connection points to the current source.

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.

Charge accumulation cannot occur in the nodes of a DC circuit. This leads to Kirchhoff's first rule:

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

In a branched chain, it is always possible to distinguish a certain number of closed paths, consisting of homogeneous and heterogeneous sections. Such closed paths are called contours . Different currents can flow in different parts of the selected circuit. In Fig. Figure 7 shows a simple example of a branched chain. The circuit contains two nodes a and d, in which identical currents converge; therefore only one of the nodes is independent (a or d).

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.

Let us write down a generalized Ohm's law for the sections that make up one of the contours of the circuit shown in Fig. 8, for example abcd. To do this, at each site you need to set positive direction of current And positive direction of circuit bypass. When writing the generalized Ohm's law for each of the sections, it is necessary to observe certain “sign rules”, which are explained in Fig. 8.

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

for sectionbc:

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.

Rice. 3 Charge movement in these areas is possible only with the help of forces

non-electrical origin(external forces): chemical processes, diffusion of charge carriers, vortex electric fields. Analogy: a pump pumping water into a water tower operates due to non-gravitational forces (electric motor).

External forces can be characterized by the work they do on moving charges.

The quantity equal to the work of external forces to move a unit positive charge is called electromotive force. E.D.S. acting in the circuit.

It is clear that the dimension of E.M.F. coincides with the dimension of the potential, i.e. measured in volts.

The external force acting on the charge can be represented as:

	= ∫ F st. d l		Q ∫ Est . d l ,
	ε 12					= ∫ Est . dl.
For a closed circuit: ε = ∑ ε i							= ∫ Est . dl.

The circulation of the tension vector of external forces is equal to the E.M.F. acting in a closed circuit (the algebraic sum of the E.M.F.).

It must be remembered that the field of external forces is not potential, and the term potential difference or voltage cannot be applied to it.

7.5. Ohm's law for a non-uniform section of a circuit.

Let's consider a non-uniform section of the circuit, a section containing a source of E.M.F.

(i.e. the area where non-electric forces act). The field strength E at any point in the chain is equal to the vector sum of the field of Coulomb forces and the field of external forces, i.e.

E = Eq + Est. .

The value numerically equal to the work of transferring a single positive charge by the total field of Coulomb and external forces in the circuit section (1 – 2) is called the voltage in this section U12 (Fig. 4)

	2 r
	U 12 = ∫ E q d l +	∫ Est . d l ;
	Eq d l = − dφ and ∫ Eq d l			= φ 1 − φ 2 ;
	U 12 = (φ 1 – φ 2) + ε 12
The voltage at the ends of the circuit section coincides with the potential difference only in
if there is no E.M.F. in this area, i.e. on a homogeneous section of the chain.
	I R12 = (φ1 – φ2) + ε 12

This is a generalized Ohm's law. The generalized Ohm's law expresses the law of conservation of energy as applied to a section of a direct current circuit. It is equally valid for both passive sections (not containing E.M.F.) and active ones.

In electrical engineering the term is often used voltage drop - change in voltage due to charge transfer through resistance

	In a closed circuit: φ 1 = φ 2;
	I RΣ = ε
				R∑

Where R Σ =R + r; r – internal resistance of the active section of the circuit (Fig. 5).

Then Ohm's law for a closed section of the circuit containing E.M.F. sign up for

		R+r

7.6. Ohm's law in differential form.

Ohm's law in integral form for a homogeneous section of the circuit (not containing E.M.F.)

	I = U
	For a homogeneous linear conductor, we express R in terms of ρ
	R = ρ

ρ – volumetric resistivity; [ρ] = [Ohm m].

Let's find the connection between j and E in an infinitesimal volume of a conductor - Ohm's law in

differential form.

In an isotropic conductor (in this case with constant resistance), charge carriers (Fig. 6) move in the direction of the force, i.e. current density

j E , therefore, the vectors are collinear.

And we know that: j =

E, i.e.

E j or

j = σE

This is a representation of Ohm's law in differential form.

Here σ is the specific electrical conductivity. Dimension j – [ Ohm − 1 m − 1 ]; The current density can be expressed in terms of charge, n and v r dr. .

j = en vr dr .

denote: b = v E dr . , then v r others = b E ;

j = enb E,

and if σ = enb,

where n is the number of ion pairs, b is the distance. j = jE

– Ohm’s law in differential form.

7.7. Work and current power. Joule-Lenz law.

Let's consider an arbitrary section of the circuit, to the ends of which a voltage U is applied. During the time dt, a charge passes through each section of the conductor

It is useful to remember other formulas for power and work:

	N=RI2
	A = RI2t
	In 1841 English physicist James Joule and Russian physicist
	Emilius Lenz established the law of thermal action of electric
	JOULE James Presscott (Fig. 6)
	(12/24/1818 – 10/11/1889) – English physicist, one
	one of the discoverers of the law of conservation of energy.
	His first lessons in physics were given to him by J. Dalton, under
	the influence of which Joule began his experiments.
	The works are devoted to electromagnetism, kinetic
	theory of gases.
	LENZ Emilius Christianovich (Fig. 7) (24.2.1804
– 10.2.1865) – Russian physicist. Main works in the field
electromagnetism. In 1833 he established the rule for determining
electromotive force of induction (Lenz's law), and in 1842 (independently

from J. Joule) – the law of thermal action of electric current (Joule-Lenz law). Discovered the reversibility of electric machines. Studied the dependence of the resistance of metals on temperature. The works also relate to geophysics.

Independently of each other, Joule and Lenz showed that when current flows in a conductor, the amount of heat released is:

(7.7.7) is the Joule–Lenz law in integral form.

Consequently, heating occurs due to the work done by field forces on the charge (heat release power N = RI2).

Let us obtain the Joule–Lenz law in differential form.

dQ = RI 2 dt = ρ dS dl (jdS ) 2 dt = ρj2 dldSdt = ρj2 dldSdt = ρj2 dVdt,