# Law of Joule-Lenz. Work and power current. Joule-Lenz law

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# Law of Joule-Lenz. Work and power current. Joule-Lenz law

1.9 Work and power of electric current. Law of Joule-Lenz.

The work of the forces of an electric field that creates an ordered motion of charged particles in a conductor, i.e. an electric current, is called the work of a current.

The work done by the electric field on the displacement of the charge q on the chain segment is equal to:

A = q U = I U t = I 2 * R t = U 2 / R * t

where I is the current strength in the given section, U is the voltage in the circuit section, t is the time of passage of the current along the section of the circuit, q == It is the electric charge (quantity of electricity) flowing through the conductor cross-section during the time interval t. The unit of measurement of the work is the joule: 1 J = 1 A * 1 V * 1 s. 1 J is work

direct current

force of 1 A for 1 s on a section of 1 V voltage.

According to the law of conservation of energy, this work is equal to the change in the energy of the conductor.

The power of an electric current as it passes through a conductor with resistance R is equal to the work done by the current per unit time:

P = A / t = I * U = U 2 * R

The unit for measuring the electric current in the SI is watt: 1 W = 1 J / s. The current can also be determined as follows:

The unit of measurement of operation is also kilowatt-hour (kWh) or watt-hour (Wh):

1W * h = 3.6 * 10 2 J

In these units, work is usually expressed in electrical engineering. The total power developed by a current source with EMF and internal resistance r, when a load with resistance R is included in the external circuit, is determined by the formula:

P = I (R + r) = IR + Ir = I * I * (R + r) = Ie

The total power is used to produce heat in the external and internal resistance.

Useful power (the power allocated in external resistance) is equal to:

P useful = I 2 R = e 2 R / (R + r) 2

It is used in electric heating and lighting devices.

The lost power (the power allocated in the internal resistance) is:

P ter = I 2 r = e 2 r / (R + r) 2

It is not used.

The current in the entire external circuit for any connection is equal to the sum of the powers in individual sections of the circuit.

The operation of the electric field leads to the heating of the conductor, if the electric field does not occur in the section of the circuit

mechanical work

and chemical transformations of substances do not occur. Therefore, the energy (amount of heat) released at a given section of the circuit during the time t is equal to the work of the electric current:

The amount of heat released by a conductor when it is heated by a current is determined according to the Joule-Lenz law:

Q = I 2 Rt or

This law was established experimentally by the English scientist James Joule (1818-1889) and the Russian scientist Emil Khristianovich Lenz (1804-1865) and formulated as follows.

The amount of heat emitted by a conductor with current is equal to the product of the square of the current strength, the resistance of the conductor and the time of passage of current through the conductor.

When

serial connection

conductors with resistance R1 and R2, the amount of heat emitted by the current in each conductor is directly proportional to the resistance of these conductors:

Q 1 / Q 2 = R 1 / R 2, since I 1 = I 2 with serial connection

The amount of heat released by a current in parallel connected two sections of a circuit without an EMF with resistances 2 ^ and W ^ is inversely proportional to the resistance of these sections:

Q 1 / Q 2 = R 1 / R 2, since U 1 = U 2 for

parallel connection

1.10. Electric current in metals.

The passage of current through metals (conductors of the first kind) is not accompanied by a chemical change, therefore, the metal atoms do not move together with the current. According to the theory of electron theory, positively charged ions (or atoms) form the skeleton of a metal, forming its crystal lattice. Electrons, separated from atoms and wandering around the metal, are carriers of free charge. They participate in chaotic thermal motion. These free electrons under the action of the electric field begin to move in an orderly manner with a certain average velocity. Thus, the conductivity of metals is due to the motion of free electrons. Experimental evidence of these representations was the experiments performed for the first time in 1912 by the Soviet Academician Leonid Isaakovich Mandelshtam (1879-1944) and Nikolai Dmitrievich Papaleksi (1880-1947), but not published by them. Later in 1916 American physicists T. Stewart and Richard Chase Tollin (1881-1948) published the results of their experiments, which turned out to be similar to the experiments of Soviet scientists.

The ends of the wire wound on the coil are soldered to two metal disks isolated from each other. With the help of sliding contacts (brushes), a galvanometer is attached to the ends of the disks.

The coil is rotated and then stopped abruptly. If we assume that there are free charges in the metal, then after a sharp stopping of the coil, the free charged particles will move for some time relative to the conductor by inertia. Consequently, an electric current appears in the coil, which, because of the resistance of the conductor, will last a short time. The direction of this current will allow us to judge the sign of those particles that moved by inertia. Since the resulting current depends on the size and mass of the charges, this experiment allows us not only to assume the existence of free charges in the metal, but also to determine the charge sign, their mass and magnitude (more precisely, to determine the specific charge-charge-to-mass ratio).

Experience has shown that, after stopping the coil, a short-time electric current appears in the galvanometer. The direction of this current indicates that negatively charged particles move by inertia. Measuring the magnitude of the charge carried by this short-time current through the galvanometer, it was possible to determine the ratio of the amount of free charges to their mass. It turned out to be equal to e / m = 1.8 10 11 C / kg, which coincides with the value of such a ratio for an electron, found earlier by other methods.

Thus, experience shows that in metals there are free electrons, the ordered motion of which creates an electric current in metals.

Under the influence of a constant force from the side of an electric field, electrons in a metal acquire a certain speed of ordered motion, which is constant. The ordered motion of electrons in a metal can be considered as a uniform motion, since On the side of the ions of the crystal lattice, some inhibitory force acts on them. In collisions with ions, free electrons transmit to them kinetic energy acquired at free path under the action of an electric field. Consequently, the average velocity of the ordered motion of electrons is proportional to the electric field in the conductor v cm E. Given the relationship between the strength and the potential difference at the ends of the conductor (E = U / d), we can say that the electron velocity is proportional to the potential difference at the ends of the conductor v ~ U .

The speed of the ordered particle motion determines the current strength in the conductor: I = q 0 nv S, so the current is proportional to the potential difference at the ends of the conductor I ~ U, which gives a qualitative explanation of Ohm’s law based on the electronic theory of metal conductivity.

The heating of a conductor when a direct current passes through it can be explained by the fact that the kinetic energy of the electrons is transferred in the collision of the ions of the crystal lattice.

The quantitative theory of the motion of electrons in a metal can be constructed on the basis of the laws of quantum mechanics, Newton’s classical mechanics is inapplicable for describing this motion.

Define the work done by direct current in the conductor, which has resistance R and is under voltage. Since the current is the displacement of the charge q under the action of the field, the operation of the current can be determined from the formula. Taking into account the formula and Ohm’s law, we get, or, or, where t is the time of current flow. Dividing both sides of the equality by t, we obtain expressions for **capacity** direct current N

The current operation in the SI system is measured in doles (J), and power in watts (W). In practice, non-system units of current work are also used: watt-hour (Wt-hr) and kilowatt-hour (kWh). 1W × h – operation of a current of 1 W power for one hour. 1W × h = 3.6 × 10 3 J.

Experience shows that the current always causes some heating of the conductor. The heating is due to the fact that the kinetic energy of the electrons moving along the conductor (ie, the current energy) during each collision with the ions of the metal lattice passes into the heat Q. If the current passes through a fixed metallic conductor, then all the work of the current is expended on heating it and, following the law of conservation of energy, we can write . These relationships express **joule-Lenz law**. For the first time this law was established experimentally by D. Joule in 1843 and independently by E. Lentz in 1844. The application of the thermal action of current in engineering began with the discovery in 1873 of the Russian engineer A. Ladygin **incandescent lamps**.

On the thermal action of the current, a number of

electrical appliances

and installations: thermal electrical appliances, electric furnaces, electric welding equipment, household electric heaters – kettles, boilers, irons. In the food industry, the method of electrocontact heating is widely used, which consists in the fact that an electric current passing through a product with a certain resistance causes its uniform heating. For example, for the production of sausage products through the dispenser the forcemeat enters the molds, the end walls of which serve as electrodes. With this treatment, uniformity of heating throughout the product volume is ensured, the possibility of maintaining a certain

temperature conditions

, the highest biological value of the product, the shortest process duration and energy consumption.

Define **specific **

thermal power

** current**w, i.e. the amount of heat released per unit volume per unit time. Let us single out in the conductor an elementary cylindrical volume dV with a cross section dS and a length d *l* parallel to the current direction, and resistance,. According to the law of Joule-Lentz, during the time dt, heat will be released in this volume. Then and, using Ohm’s law for the current density and the relation, we obtain . These relations express **joule-Lenz law in differential form**.

Mechanical work ** – this****physical quantity****, which is****scalar**** quantitative measure of action****force**** or forces on a body or system that depends on the numerical value, the direction of the force (forces), and****displacement**** point (s), body or system**

__Electric current__The work done per unit time is called the power and is denoted by the letter *P*.

*A* = *P* × *t*.

Unit of measurement of capacity:

Power is measured by a wattmeter. __Law of Joule-Lenz__– the power of heat released per unit volume of the medium during the flow of electric current is proportional to the product of the electric current density by the strength of the electric field.

where – the power of heat generation per unit volume, – the density of the electric current, – the electric field strength, *σ * – the conductivity of the environment, and the dot denotes the scalar product.

.__In the integral form__ this law has the form (for the case of the flow of currents in thin wires)

: The amount of heat released per unit time in the section of the chain under consideration is proportional to the product of the square of the current in this section of the resistance of the section.

where *dQ* – the amount of heat released over a period of time *dt*, *I* – current strength, *R* – resistance, *Q* – the total amount of heat allocated for a period of time from *t* *1 * before *t* *2 * . In the case of constant current and resistance:

__The derivation of the Joule-Lenz law in a differential form:__

If the current passes through a fixed metal conductor, then all the work of the current goes to its heating and, according to the law of conservation of energy,

Thus, we get:

This expression is the law of Joule-Lentz.

## 35.Classical electronic theory of metals. Derivation of DC laws on the basis of this theory. The concept of the quantum theory of electrical conductivity of metals.

Current carriers in metals are free electrons, that is, electrons that are loosely coupled to the ions of the crystal lattice of the metal. This idea of the nature of current carriers in metals is based on the electronic theory of the conductivity of metals, created by the German physicist P. Drude

__The basic assumptions of the Drude theory__.1) in the absence of external electromagnetic fields, each electron moves at a constant speed along a straight line. Further, it is believed that in the presence of external fields the electron moves in accordance with Newton’s laws; In this case, only the influence of these fields is taken into account, neglecting the complex additional fields generated by other electrons and ions. approximation of free electrons. 2) In the Drude model, collisions are instantaneous events that suddenly change the speed of an electron. Drude attributed them to the fact that electrons bounce off impenetrable ion cores 3) per unit time, an electron collides with a probability equal to. In the simplest applications, the Drude model believes that the relaxation time does not depend on the spatial position of the electron and its velocity. 4) It is assumed that the electrons come to a state of thermal equilibrium with their environment solely due to collisions.

__The derivation of the basic laws of electric current in the classical theory of the electrical conductivity of metals__

__1. Ohm’s law__**. ** during a free run, the electrons move at the same acceleration, acquiring, by the end of the free path,

According to the Drude theory, at the end of the mean free path, an electron, colliding with the ions of the lattice, gives them the energy stored in the field, so the velocity of its ordered motion becomes zero. Consequently, the average velocity of the directed motion of the electron

The classical theory of metals does not take into account the velocity distribution of electrons, and therefore Om law was obtained in differential form

__2. The Law of Joule-Lenz.__ At the end of the free path, the electron under the action of the field acquires additional kinetic energy

When an electron collides with an ion, this energy is completely transferred to the lattice and goes to increase the internal energy of the metal, i.e., to heat it.

From this follows the expression-the Joule-Lenz cone in differential form.

__ Quantum theory of electrical conductivity of metals __ – the theory of electrical conductivity, based on quantum mechanics and quantum statistics of Fermi-Dirac,

*.*

The quantum theory of the electrical conductivity of metals, in particular, explains the dependence of the conductivity on temperature:

Quantum theory considers the motion of electrons taking into account their interaction with the crystal lattice. According to the corpuscular-wave dualism, the motion of an electron is correlated with the wave process. Ideal crystal lattice) behaves like an optically homogeneous medium – it does not scatter “electronic waves”. This corresponds to the fact that the metal does not have an electric current-the ordered motion of electrons-no resistance. “Electron waves,” propagating in an ideal crystal lattice, as it were, round the lattice sites and pass considerable distances.

In a real crystal lattice there are always inhomogeneities, which can be, for example, impurities, vacancies; The inhomogeneities are also due to thermal oscillations. In a real crystal lattice, scattering of “electronic waves” occurs at inhomogeneities, which is the cause of the electrical resistance of metals. The scattering of “electronic waves” by inhomogeneities associated with thermal vibrations can be regarded as collisions of electrons with phonons.

According to the classical theory, áu F ñ ~ ÖT, so she could not explain the true dependence *the*of the temperature. In the quantum theory, the average velocity áu F ñ * *does not depend on temperature, since it is proved that with a temperature change the Fermi level remains practically unchanged. However, as the temperature is raised, the scattering of “electronic waves” by thermal lattice vibrations (on phonons) increases, which corresponds to a decrease in the mean free path of the electrons. In the room temperature range b *l* F ñ ~ T -1, therefore, taking into account the independence of temperature, we find that the resistance of metals (R ~ l / g) * *in accordance with the experimental data, it increases in proportion to T *. *Thus, the quantum theory of electrical conductivity of metals has also removed this difficulty of the classical theory.

№36 **Work function of electrons from metals. Derivation of DC laws on the basis of form.**

Experiments show that free electrons do not practically leave the metal at ordinary temperatures. Consequently, in the surface layer of the metal there must be a retarding electric field that prevents the electrons from leaving the metal in the surrounding vacuum. The work that needs to be spent to remove an electron from a metal into a vacuum is called the work function.

Contact – called two different metals in contact, between them there are different potentials. The contact difference of potentials is due to the fact that when metals contact a part of the electrons from one metal passes into another.

where e is the electron charge, is the exit potential.

where m and e are the electron mass and charge, respectively, and the electron velocity before and after the metal exit. The contact potential difference between the first and second metals is equal to the difference in the work functions for the second and first metal, divided by the elementary charge.

**Thermoelectric** the phenomenon-between thermal and electrical processes in metals and semiconductors-is interconnected.

№37 **Semiconductors.**

**Semiconductor** – a material which in its specific conductivity occupies an intermediate place between conductors and dielectrics and differs from conductors by a strong dependence of the conductivity on the impurity concentration, temperature, and the effect of various types of radiation. The main property of the semiconductor is an increase in electrical conductivity with increasing temperature.

Semiconductors are substances whose width of the forbidden band is of the order of several electron-volts (eV). For example, a diamond can be attributed to *wide band semiconductors*, and indium arsenide – to *narrow-band*. Among the semiconductors are many chemical elements (germanium, silicon, selenium, tellurium, arsenic and others), a huge number of alloys and chemical compounds (gallium arsenide, etc.). Almost all the inorganic substances of the world around us are semiconductors. The most common semiconductor in nature is silicon, which accounts for almost 30% of the earth’s crust.

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

№ 38 The magnetic field. The power of Ampere. Induction of the magnetic field. The power of Lorentz. Motion of charged particles in a magnetic field.

**A magnetic field** – a force field acting on moving electrical charges and on bodies possessing a magnetic moment, regardless of the state of their motion; magnetic component of the electromagnetic field

The magnetic induction at a given point of a homogeneous magnetic field is determined by the maximum torque acting on the frame with a magnetic moment equal to unity, when the normal to the frame is perpendicular to the direction of the field. The magnetic field is force, then, by analogy with the electrical one, it is represented by lines of magnetic induction-lines, the tangents to which at each point coincide with the direction of the vector B. The direction is given by the rule of the right screw: the screw head screwed in the direction of the current rotates in the direction of the lines of magnetic induction.

The magnetic field exerts an orienting action on the frame with the current. Consequently, the torque experienced by the frame is the result of the action of forces on its individual elements. . The ampere established that the force dF, the magnetic field acting on the conductor element d / with the current in the magnetic field, is equal to

Where df-vector, modulo dl and coinciding in direction with current, B-vector of magnetic induction.

The direction of the vector d F can be found, according to the general rules of the vector product, from where the rule of the left hand follows:

Experience shows that the magnetic field acts not only on conductors with current, but also on individual charges moving in a magnetic field. The force acting on the electric charge Q moving in a magnetic field with a velocity v is called the Lorentz force and is expressed

Where B is the induction of the magnetic field in which the charge moves.

The direction of the Lorentz force is determined by the left hand rule:

№39 **The Biot-Savart-Laplace law. Magnetic field of direct and circular currents. Magnetic field of a moving charge.**

**The Biot-Savart Law**-Laplace for a conductor with drain I, whose element dl creates at some point A (Fig. 116) the induction of the field dB, is written in the form where dl is the vector modulo equal to the length dl of the conductor element and coincides in direction with the current; r is the radius vector drawn from the element dl of the conductor to the point A of the field, the r-modulus of the radius vector r. The direction of dB is perpendicular to df and r, i.e. perpendicular to the plane in which they lie and coincides with the tangent to the magnetic induction line. This direction can be specified by the rule of finding the lines of magnetic induction (right-screw rule): the direction of rotation of the screw head gives the direction dD if the translational motion of the screw corresponds to the direction of the current in the element . The modulus of the dB vector is given by where a is the angle between the vectors dl and r. For the magnetic field, as well as for the electric field, the superposition principle is valid: the vector of magnetic induction of the resultant field created by several currents or moving charges is equal to the vector sum of the magnetic inductions of the folded fields created by each current or moving charge separately: **Direct current magnetic field**a current flowing through a thin, straight wire of infinite length. At an arbitrary point A, distant from the axis of the conductor by a distance R, the vectors dB from all elements of the current have the same direction perpendicular to the plane of the drawing (“to us”). Therefore, the addition of the vectors dB can be replaced by adding their moduli. As the integration constant, we choose the angle a (the angle between the vectors d / and r), expressing through it all the remaining quantities.

**The magnetic field in the center of a circular conductor with a current**-As follows from the figure (1), all the elements of a circular conductor with current create in the center magnetic fields of the same direction-along the normal from the turn. Therefore, the addition of the dB vectors can be replaced by adding their modules. Since all the elements of the conductor are perpendicular to the radius vector (sina = 1) and the distance of all the elements of the conductor to the center of the circular current is the same and equal to R,

Consequently, the magnetic induction of the field in the center of the circular conductor with current. Each conductor with current creates in the surrounding space a magnetic

Field. The electric current is an ordered movement of electric charges, so we can say that any charge moving in a vacuum or medium creates around itself a magnetic field. As a result of generalization of the experimental data, a law was established that determines the field B of a point charge Q that moves freely with a nonrelativistic velocity v. Under the free motion of a charge, its motion is understood to be constant velocity. Formula 12 where r is the radius vector drawn from the charge Q to the observation point M. №40 **The law of total current. The magnetic field of a solenoid and a toroid. Magnetic flow.**The total current is the algebraic sum of the currents passing through a surface bounded by a closed contour. In our example, the total current Σ I is the sum of the currents I 1 and I 2:

Σ I = I 1 – I 2

The signs of currents are determined by the rule of a borer.

Now we find the magnetic voltage along the contour L. We break the contour into segments that can be considered straight, and the magnetic field at the location of the segments is homogeneous. The magnetic voltage U m for one such segment of length ΔL:

The magnetic stress along the entire contour L (see Magnetic voltage)

U L = Σ H L * ΔL

The total current is equal to the magnetic stress along the contour:

Σ I = Σ H L * ΔL The magnetic stress along a closed contour is often called *magnetomotive force*. Another name for the magnetic stress along a closed contour – *magnetizing force*.

Determination of the total current law: the magnetomotive force F along the closed contour L is equal to the total current Σ I penetrating the surface bounded by this contour. Formula of the law of total current:

F = Σ I **Magnetic flow** Φ through the surface S is the number of lines of the vector of magnetic induction B passing through the surface S.

The magnetic flux formula:

here α is the angle between the direction of the magnetic induction vector B and the normal to the surface S.

It can be seen from the magnetic flux formula that the maximum magnetic flux will be at cos α = 1, and this will happen when the vector B is parallel to the normal to the surface S. The minimal magnetic flux will be at cos α = 0, this will be when the vector B is perpendicular to the normal to the surface S, because in this case the lines of the vector B will slide along the surface S without intersecting it. And by the definition of the magnetic flux, only those lines of the vector of magnetic induction, which cross the given surface, are taken into account.

The magnetic flux in the webs (volts-seconds) is measured: 1 Vb = 1 V * s. In addition, Maxwell is used to measure the magnetic flux: 1 V = 10 8 μs. Accordingly, 1 μs = 10 -8 vb.

The magnetic flux is a scalar quantity.

Consider a homogeneous section of the circuit between the ends of which there is a voltage U. For a current I during a time t, a charge q = It passes through the circuit. Therefore, the work of the electric current in this section will be equal to:

A = Uq = IUt. (20.1)

By combining Ohm’s law for a homogeneous part of the chain U = IR, we can obtain two more expressions for the operation of the current:

A = IUt = t = I 2 Rt. (20.2)

Expression (20.2) is valid for a direct current in any case, for any part of the chain.

Current power, i.e. the work per unit time is:

P = IU = I 2 R. (20.3)

The formula (20.3) in the SI system is used to determine the unit of voltage. Voltage unit volt is

[U] = [P] / [I] = 1 W / A = 1 V.

Volt is the electrical stress causing in

electrical circuit

A constant current of 1 A at 1 watts.

If the current is expressed in Amps, the voltage is in Volts, the resistance in Ohms, then the current work is expressed in Joules, and the power is in Watts. In practice, non-system units of current work are also used: watt × hour (W × h) and kilowatt × hour (kWh). 1 W × h – operation of a current of 1 W power for 1 hour: 1 W × h = 3,600 W × s = 3,6 × 10 3 J; 1 kW × h = 10 3 W × h = 3.6 × 10 6 J.

In a homogeneous fixed conductor, in the absence of chemical transformations in it, the entire operation of the current is used to increase the internal energy of the conductor, as a result of which the conductor heats up. According to the law of conservation of energy, the quantity of heat Q, released in a fixed conductor, when the current is passed through the time t is equal to A, then from (20.2) we have

Q = IUt = t = I 2 Rt. (20.4)

Expression (20.4) is the Joule-Lenz law experimentally established independently by J. Joule and E. X. Lenz. Joule and Lenz established their law for a homogeneous section of the chain. However, it is also valid for an inhomogeneous section of the chain, provided that the external forces acting in it are of non-chemical origin.

We select an elementary cylindrical volume dV = dS × dl in the conductor (the axis of the cylinder coincides with the direction of the current), whose electrical resistance is R = r × dl / dS. According to the Joule-Lenz law, during the time dt, the heat in the volume dV

dQ = I 2 Rdt = (jdS) 2 dt = rj 2 × dV × dt.

The amount of heat released per unit of time per unit volume is called the specific thermal power of the current. It is equal to

*w* = = rj 2. (20.5)

Using the differential form of Ohm’s law (18.3) from (20.5), we obtain:

* w* = rj 2 = sE 2 = jE. (20.6)

Formulas (20.6) are a generalized expression of the Joule-Lenz law in a differential form, suitable for any conductor.

The thermal action of the current is widely used in technology, which began with the invention in 1873 of the Russian engineer AN Lodygin (1847-1923) incandescent lamp. On the heating of conductors

electric shock

The action of electric muffle furnaces, electric arc (opened by the Russian engineer VV Petrov (1761-1834)), contact electric welding, household electric heaters, etc., is based

Now consider the energy transformations in a closed circuit containing EMF (see Figure 23). In this case, we will take into account the relations (18.6), (18.7), and (18.8) obtained earlier. The power consumed by the circuit (that is, the power developed by the current source) is P = EI. The capacities allocated on the load P R and the internal resistance P r, respectively, are equal

P R = I 2 R = R = E 2, P r = I 2 r = E 2. (20.7)

According to the law of conservation of energy P = P R + P r, i.e.

EI = U R I + U r I = I 2 (R + r). (20.8)

The efficiency h of the current source is:

h = = = = =. (20.9)

It can be seen from expression (20.9) that h reaches the maximum value h = 1 in the case of an open circuit (R® ¥, while P R ®0) and vanishes (h = 0) for a short circuit (R = 0).

The dependence of the useful power on the load resistance R

P R (R) = E 2 (20.10)

is shown in Figure 26.

As can be seen from the graph of P R (R), the same power P 0 is extracted at two different load resistance values R 1 and R 2 (for R 1 R R 2, the efficiency values are different, i.e., h 1 ¹ h 2). If we substitute P 0 in place of P R in (20.10), we obtain a quadratic equation from which we can determine the values of R 1 and R 2:

P 0 (R) = P 0 = E 2 or R 2 + R + r 2 = 0. (20.11)

The value of the external resistance R max, at which the maximum power P max is extracted on it, we find, differentiating the expression P 0 (R) in R and equating the first derivative to zero:

P 0 (R) ¢ R = = E 2 = 0, (20.12)

whence, taking into account the fact that ru003e 0 and Ru003e 0, we obtain R max = r. Useful power, i.e. the power released in the external circuit reaches its maximum value if the resistance of the external circuit is equal to the internal resistance, i.e. at R = r. In this case, the current in the circuit is:

I max = = = I kz, (20.13)

those. half short-circuit current I ks. In this case, the efficiency of the current source is 0.5 (50%):

h max = = =. (20.14)

And the maximum power released on the load is

P max = E 2 = E 2 =. (20.15)

If we apply Viet’s theorem to the quadratic equation (20.11) for determining R 1 and R 2, then it gives formulas connecting the roots of this equation:

R 1 + R 2 = – r and R 1 × R 2 = r 2. (20.16)

Of practical interest is the second formula in (20.16); (R 1 × R 2 = r 2), which connects the internal resistance of the source r and the load resistances R 1 and R 2, at which the same power is allocated to the load (P 0).

**Control questions**

1 What is called current strength? current density? What are their units of measure? (Give definitions.)

2 Name the conditions for the occurrence and existence of an electric current.

3 What are third-party forces? What is their nature?

4 What is the physical meaning of the electromotive force acting in the circuit? tension? potential difference?

5 Why stress is a generalized concept of the potential difference?

6 What is the relationship between resistance and conductivity, resistivity and conductivity? What are their units of measure? (Give definitions.)

7 What is meant by the average, drift or ordered velocity of the current carriers?

8 What is understood by the intensity of the field of external forces?

9 Derive the Ohm and Joule-Lenz laws in a differential form.

10 What is the physical meaning of the specific thermal power of the current?

11 Analyze the generalized Ohm’s law. What particular laws can be obtained from it?

12 How are the Kirchhoff rules formulated? What are they based on?

13 How are the equations expressing the Kirchhoff rules drawn up? How to avoid unnecessary equations?

14 Which parts of the chain are called homogeneous (non-homogeneous)?

**Tests**

1. The expression is:

A) current in a closed circuit

B) power released in the external circuit

C) power released in the internal circuit of the current source

D) the voltage at the terminals of the current source

E) work of moving a unit positive charge along a closed circuit

2. A battery with an internal resistance of r = 0.08 Ω at a current of I 1 = 4 A gives an external circuit power P 1 = 8 W. What power P 2 will it give to the external circuit at the current I 2 = 6 A?

4. Two resistors with the same resistance each are connected to the network

constant voltage

the first time in parallel, and the second time consistently. Find the ratio between the power consumption in these cases.

6. An element with an EMF equal to 6 V gives a maximum current strength of 3 A. Find the greatest amount of heat that can be released by an external resistance in 2 minutes.

8. Calculate the resistance of the lamp spiral from the flashlight, if at a voltage of 3.5 V, the current in it is 280 mA.

10. What is the current required to pass through a conductor connected to a 220V mains network so that 6.6 kJ of heat is emitted every minute in it?

12. The electric iron is designed for a voltage of 220 V. The resistance of it

heating element

88 Ohm. What is the power of this iron?

14. The battery had an EMF of E 1 = 90 V before charging, after charging E 2 = 100 V. The current at the beginning of charging was I 1 = 10 A. What was the value of the current I 2 at the end of the charge, if the internal resistance of the battery r = 2 Ohm, and the voltage U, created by the charger, is constant.

A) 8 A | B) 9 A | C) 6 A | E) 5A | E) 4 A |

15. The efficiency of a current source can be calculated by the formula …

17. Two conductors connected in series have a resistance of 6.25 times greater than when they are connected in parallel. Find how many times the resistance of one conductor is greater than the resistance of the other.

21. What kind of work will be done if a voltage U = 12 V is applied to the ends of a conductor with a resistance R = 10 Ω for a time t = 20 s?

23. If the element with an EMF of 12 V and an internal resistance of 2 Ohms is short-circuited to a resistance of 10 Ohm, then the power released in the external circuit will be equal to …

26. The power of the electric heating device when the length of the heating spiral is halved and the voltage in the circuit is halved …

A) will decrease 8 times

B) will decrease 4 times

C) will decrease by a factor of 2

D) will increase 2 times

E) does not change

27. Two resistors, whose resistances differ by n = 4.8 times, are included in the dc circuit with a constant voltage in the circuit once in series and the other in parallel. What is the ratio of the thermal powers emitted by the resistors in the second (P 2) and in the first (P 1) cases?

29. How many times will increase the current flowing through the conductor, if the voltage at the ends of the conductor is increased by 2 times, and the length of the conductor is reduced by 4 times?

A) 2 times | B) 4 times | C) times | D) 8 times | E) 16 times |

30.

Physical quantity

, the dimension of which can be represented as, is

A) resistance

B) EMF power source

C) resistivity

D) current intensity

E) conductivity

The correct answers in the tasks are marked in red.

Chuyên mục: Kiến thức