Ampere's law of interaction of parallel currents. Ampere power

Let's consider a wire located in a magnetic field and through which current flows (Fig. 12.6).

For each current carrier (electron), acts Lorentz force. Let us determine the force acting on a wire element of length d l

The last expression is called Ampere's law.

Ampere force modulus is calculated by the formula:

.

The Ampere force is directed perpendicular to the plane in which the vectors dl and B lie.

Distance between conductors - b. Let us assume that conductor I 1 creates a magnetic field by induction

According to Ampere's law, a force acts on the conductor I 2 from the magnetic field

, taking into account that (sinα =1)

Therefore, per unit length (d l=1) conductor I 2, force acts

.

The direction of the Ampere force is determined by the left hand rule: if the palm of the left hand is positioned so that the magnetic induction lines enter it, and the four extended fingers are placed in the direction of the electric current in the conductor, then the extended thumb will indicate the direction of the force acting on the conductor from the field .

12.4. Circulation of the magnetic induction vector (total current law). Consequence.

A magnetic field, in contrast to an electrostatic one, is a non-potential field: the circulation of the vector In the magnetic induction of the field along a closed loop is not zero and depends on the choice of the loop. In vector analysis, such a field is called a vortex field.

The lines of magnetic induction of this field are circles, the planes of which are perpendicular to the conductor, and the centers lie on its axis (in Fig. 12.8 these lines are shown as dotted lines). At point A of contour L, vector B of the magnetic induction field of this current is perpendicular to the radius vector.

From the figure it is clear that

Where - length of the vector projection dl onto the vector direction IN. At the same time, a small segment dl 1 tangent to a circle of radius r can be replaced by a circular arc: , where dφ is the central angle at which the element is visible dl contour L from the center of the circle.

Then we obtain that the circulation of the induction vector

At all points of the line the magnetic induction vector is equal to

integrating along the entire closed contour, and taking into account that the angle varies from zero to 2π, we find the circulation

The following conclusions can be drawn from the formula:

1. The magnetic field of a rectilinear current is a vortex field and is not conservative, since there is vector circulation in it IN along the magnetic induction line is not zero;

2. vector circulation IN The magnetic induction of a closed loop covering the field of a straight-line current in a vacuum is the same along all lines of magnetic induction and is equal to the product of the magnetic constant and the current strength.

If a magnetic field is formed by several current-carrying conductors, then the circulation of the resulting field

This expression is called total current theorem.

The interaction of stationary charges is described by Coulomb's law. However, Coulomb's law is insufficient for analyzing the interaction of moving charges. Ampere's experiments first reported that moving charges (currents) create a certain field in space, leading to the interaction of these currents. It was found that currents of opposite directions repel, and currents of the same direction attract. Since it turned out that the current field acts on the magnetic needle in exactly the same way as the field of a permanent magnet, this current field was called magnetic. The current field is called a magnetic field. It was subsequently established that these fields have the same nature.

Interaction of current elements .

The law of interaction of currents was discovered experimentally long before the creation of the theory of relativity. It is much more complex than Coulomb's law, which describes the interaction of stationary point charges. This explains that many scientists took part in his research, and significant contributions were made by Biot (1774 - 1862), Savard (1791 - 1841), Ampère (1775 - 1836) and Laplace (1749 - 1827).

In 1820, H. K. Oersted (1777 - 1851) discovered the effect of electric current on a magnetic needle. In the same year, Biot and Savard formulated a law for the force d F, with which the current element I D L acts on a magnetic pole at a distance R from the current element:

D F I d L (16.1)

Where is the angle characterizing the mutual orientation of the current element and the magnetic pole. The function was soon found experimentally. Function F(R) Theoretically, it was derived by Laplace in the form

F(R) 1/r. (16.2)

Thus, through the efforts of Biot, Savart and Laplace, a formula was found that describes the force of the current on the magnetic pole. The Biot-Savart-Laplace law was formulated in its final form in 1826. In the form of a formula for the force acting on the magnetic pole, since the concept of field strength did not yet exist.

In 1820 Ampere discovered the interaction of currents - the attraction or repulsion of parallel currents. He proved the equivalence of a solenoid and a permanent magnet. This made it possible to clearly set the research goal: to reduce all magnetic interactions to the interaction of current elements and to find a law that plays a role in magnetism similar to Coulomb’s law in electricity. Ampère, by his education and inclinations, was a theorist and mathematician. Nevertheless, when studying the interaction of current elements, he performed very scrupulous experimental work, constructing a number of ingenious devices. Ampere machine for demonstrating the forces of interaction of current elements. Unfortunately, neither in the publications nor in his papers there is a description of the path by which he came to the discovery. However, Ampere's formula for force differs from (16.2) in the presence of a total differential on the right side. This difference is not significant when calculating the interaction strength of closed currents, since the integral of the total differential along a closed loop is zero. Considering that in experiments it is not the strength of interaction of current elements that is measured, but the strength of interaction of closed currents, we can rightfully consider Ampere the author of the law of magnetic interaction of currents. The currently used formula for the interaction of currents. The formula currently used for the interaction of current elements was obtained in 1844. Grassmann (1809 - 1877).

If you enter 2 current elements and , then the force with which the current element acts on the current element will be determined by the following formula:

, (16.2)

In the same way you can write:

(16.3)

Easy to see:

Since the vectors and have an angle between themselves that is not equal to 180°, it is obvious , i.e. Newton’s III law is not satisfied for current elements. But if we calculate the force with which the current flowing in a closed loop acts on the current flowing in a closed loop:

, (16.4)

And then calculate , then, i.e. for currents, Newton’s third law is satisfied.

Description of the interaction of currents using a magnetic field.

In complete analogy with electrostatics, the interaction of current elements is represented by two stages: the current element at the location of the element creates a magnetic field that acts on the element with a force. Therefore, the current element creates a magnetic field with induction at the point where the current element is located

. (16.5)

An element located at a point with magnetic induction is acted upon by a force

(16.6)

Relationship (16.5), which describes the generation of a magnetic field by a current, is called the Biot-Savart law. Integrating (16.5) we get:

(16.7)

Where is the radius vector drawn from the current element to the point at which induction is calculated.

For volumetric currents, the Bio-Savart law has the form:

, (16.8)

Where j is the current density.

From experience it follows that the principle of superposition is valid for the induction of a magnetic field, i.e.

Example.

Given a direct infinite current J. Let us calculate the magnetic field induction at point M at a distance r from it.

= .

= = . (16.10)

Formula (16.10) determines the induction of the magnetic field created by direct current.

The direction of the magnetic induction vector is shown in the figures.

Ampere force and Lorentz force.

The force acting on a current-carrying conductor in a magnetic field is called the Ampere force. In fact this power

Or , Where

Let's move on to the force acting on a conductor with a current of length L. Then = and .

But the current can be represented as , where is the average speed, n is the concentration of particles, S is the cross-sectional area. Then

, Where . (16.12)

Because , . Then where - Lorentz force, i.e. the force acting on a charge moving in a magnetic field. In vector form

When the Lorentz force is zero, that is, it does not act on a charge that moves along the direction. At , i.e. the Lorentz force is perpendicular to the speed: .

As is known from mechanics, if the force is perpendicular to the speed, then the particles move in a circle of radius R, i.e.

The magnetic field (see § 109) has an orienting effect on the current-carrying frame. Consequently, the torque experienced by the frame is the result of the action of forces on its individual elements. Summarizing the results of a study of the effect of a magnetic field on various current-carrying conductors, Ampere established that the force d F, with which the magnetic field acts on the conductor element d l with current in a magnetic field is directly proportional to the current strength I in the conductor and the cross product of an element of length d l conductor for magnetic induction B:

d F = I. (111.1)

Direction of vector d F can be found, according to (111.1), using the general rules of vector product, which implies left hand rule: if the palm of the left hand is positioned so that vector B enters it, and the four extended fingers are positioned in the direction of the current in the conductor, then the bent thumb will show the direction of the force acting on the current.

Ampere force modulus (see (111.1)) is calculated by the formula

dF = I.B. d l sin, (111.2)

where a is the angle between vectors dl and B.

Ampere's law is used to determine the strength of interaction between two currents. Consider two infinite rectilinear parallel currents I 1 And I 2 (current directions are indicated in Fig. 167), the distance between which is R. Each of the conductors creates a magnetic field, which acts according to Ampere's law on the other conductor with current. Let's consider the strength with which the magnetic field of the current acts I 1 per element d l second conductor with current I 2. Current I 1 creates a magnetic field around itself, the lines of magnetic induction of which are concentric circles. Vector direction b 1 is given by the rule of the right screw, its module according to formula (110.5) is equal to

Direction of force d F 1, from which the field B 1 acts on section d l the second current is determined by the left-hand rule and is indicated in the figure. The force module, according to (111.2), taking into account the fact that the angle  between the current elements I 2 and vector B 1 straight line, equal

d F 1 =I 2 B 1 d l, or, substituting the value for IN 1 , we get

Using similar reasoning, it can be shown that the force d F 2, with which the magnetic field of the current I 2 acts on element d l first conductor with current I 1 , is directed in the opposite direction and is equal in magnitude

Comparison of expressions (111.3) and (111.4) shows that

i.e. two parallel currents of the same direction attract each other with force

If currents have opposite directions, then, using the left-hand rule, we can show that between them there is repulsive force, defined by formula (111.5).

45.Faraday's law and its derivation from the law of conservation of energy

Summarizing the results of his numerous experiments, Faraday came to the quantitative law of electromagnetic induction. He showed that whenever there is a change in the magnetic induction flux coupled to the circuit, an induced current arises in the circuit; the occurrence of an induction current indicates the presence of an electromotive force in the circuit, called electromotive force of electromagnetic induction. The value of the induction current, and therefore e. d.s, electromagnetic induction ξ i are determined only by the rate of change of magnetic flux, i.e.

Now we need to find out the sign of ξ i . In § 120 it was shown that the sign of the magnetic flux depends on the choice of the positive normal to the contour. In turn, the positive direction of the normal is related to the current by the rule of the right screw (see § 109). Consequently, by choosing a certain positive direction of the normal, we determine both the sign of the magnetic induction flux and the direction of the current and emf. in the circuit. Using these ideas and conclusions, we can accordingly arrive at the formulation Faraday's law of electromagnetic induction: whatever the reason for the change in the flux of magnetic induction, covered by a closed conducting circuit, arising in the emf circuit.

The minus sign shows that an increase in flow (dФ/dt>0) causes emf.

ξξ i<0, т. е. поле индукционного тока на­правлено навстречу потоку; уменьшение

flow (dФ/dt<0) вызывает ξ i >0,

i.e., the directions of flow and induced current fields coincide. The minus sign in formula (123.2) is a mathematical expression of Lenz's rule - a general rule for finding the direction of induction current, derived in 1833.

Lenz's rule: the induced current in the circuit always has such a direction that the magnetic field it creates prevents the change in the magnetic flux that caused this induced current.

Faraday's law (see (123.2)) can be directly derived from the law of conservation of energy, as was first done by G. Helmholtz. Consider a conductor carrying current I, which is placed in a uniform magnetic field perpendicular to the plane of the circuit and can move freely (see Fig. 177). Under the influence of Ampere force F, the direction of which is shown in the figure, the conductor moves to a segment dx. Thus, the Ampere force produces work (see (121.1)) d A=I dФ, where dФ is the magnetic flux crossed by the conductor.

If the loop impedance is equal to R, then, according to the law of conservation of energy, the work of the current source during the time dt (ξIdt) will consist of work on Joule heat (I 2 Rdt) and work on moving a conductor in a magnetic field ( I dФ):

where-dФ/dt=ξ i is nothing more than Faraday's law (see (123.2)).

Faraday's law can also be formulated this way: emf. ξ i electromagnetic induction in a circuit is numerically equal and opposite in sign to the rate of change of magnetic flux through the surface bounded by this circuit. This law is universal: e.m.f. ξ i does not depend on the way the magnetic flux changes.

E.m.f. Electromagnetic induction is expressed in volts. Indeed, given that the unit of magnetic flux is weber(Wb), we get

What is the nature of emf. electromagnetic induction? If the conductor (the movable jumper of the circuit in Fig. 177) moves in a constant magnetic field, then the Lorentz force acting on the charges inside the conductor, moving along with the conductor, will be directed opposite to the current, i.e. it will create an induced current in the conductor in the opposite direction (the direction of the electric current is taken to be the movement of positive charges). Thus, the excitation of the emf. induction when the circuit moves in a constant magnetic field is explained by the action of the Lorentz force that arises when the conductor moves.

According to Faraday's law, the occurrence of emf. electromagnetic induction is also possible in the case of a stationary circuit located in variable magnetic field. However, the Lorentz force does not act on stationary charges, so in this case it cannot explain the occurrence of emf. induction. Maxwell to explain the emf. induction in stationary conductors suggested that any alternating magnetic field excites an electric field in the surrounding space, which is the cause of the appearance of induced current in the conductor. Vector circulation E IN this field along any fixed contour L conductor represents the emf. electromagnetic induction:

47.. Loop inductance. Self-induction

An electric current flowing in a closed circuit creates a magnetic field around itself, the induction of which, according to the Biot-Savart-Laplace law (see (110.2)), is proportional to the current. The magnetic flux Ф coupled to the circuit is therefore proportional to the current I in the outline:

Ф=LI, (126.1)

where is the proportionality coefficient L called circuit inductance.

When the current in the circuit changes, the magnetic flux associated with it will also change; therefore, an emf will be induced in the circuit. Emergence of e.m.f. induction in a conducting circuit when the current strength changes in it is called self-induction.

From expression (126.1) the unit of inductance is determined Henry(H): 1 H - the inductance of such a circuit, the self-induction magnetic flux of which at a current of 1 A is equal to 1 Wb:

1 Gn=1 Wb/A=1B s/A.

Let's calculate the inductance of an infinitely long solenoid. According to (120.4), the total magnetic flux through the solenoid

(flux linkage) is equal to 0( N 2 I/ l)S. Substituting this expression into formula (126.1), we obtain

i.e. the inductance of the solenoid depends on the number of turns of the solenoid N, its length l, area S and magnetic permeability  of the substance from which the solenoid core is made.

It can be shown that the inductance of a circuit in the general case depends only on the geometric shape of the circuit, its size and the magnetic permeability of the medium in which it is located. In this sense, the inductance of the circuit is an analogue of the electrical capacitance of a solitary conductor, which also depends only on the shape of the conductor, its dimensions and the dielectric constant of the medium (see §93).

Applying Faraday's law to the phenomenon of self-induction (see (123.2)), we obtain that the emf. self-induction

If the circuit is not deformed and the magnetic permeability of the medium does not change (later it will be shown that the last condition is not always satisfied), then L=const and

where the minus sign, due to Lenz's rule, shows that the presence of inductance in the circuit leads to slowing down change current in it.

If the current increases over time, then

dI/dt>0 and ξ s<0, т. е. ток самоиндукции

is directed towards the current caused by an external source and inhibits its increase. If the current decreases over time, then dI/dt<0 и ξ s > 0, i.e. induction

the current has the same direction as the decreasing current in the circuit and slows down its decrease. Thus, the circuit, having a certain inductance, acquires electrical inertia, which consists in the fact that any change in current is inhibited the more strongly, the greater the inductance of the circuit.

59.Maxwell's equations for the electromagnetic field

Maxwell's introduction of the concept of displacement current led him to the completion of his unified macroscopic theory of the electromagnetic field, which made it possible from a unified point of view not only to explain electrical and magnetic phenomena, but also to predict new ones, the existence of which was subsequently confirmed.

Maxwell's theory is based on the four equations discussed above:

1. The electric field (see § 137) can be either potential ( e q), and vortex ( E B), therefore the total field strength E=E Q+ E B. Since the circulation of the vector e q is equal to zero (see (137.3)), and the circulation of the vector E B is determined by expression (137.2), then the circulation of the total field strength vector

This equation shows that the sources of the electric field can be not only electric charges, but also time-varying magnetic fields.

2. Generalized vector circulation theorem N(see (138.4)):

This equation shows that magnetic fields can be excited either by moving charges (electric currents) or by alternating electric fields.

3. Gauss's theorem for the field D:

If the charge is distributed continuously inside a closed surface with volume density , then formula (139.1) will be written in the form

4. Gauss’s theorem for field B (see (120.3)):

So, the complete system of Maxwell's equations in integral form:

The quantities included in Maxwell’s equations are not independent and the following relationship exists between them (isotropic non-ferroelectric and non-ferromagnetic media):

D= 0 E,

B= 0 N,

j=E,

where  0 and  0 are the electric and magnetic constants, respectively,  and  - dielectric and magnetic permeability, respectively,  - specific conductivity of the substance.

From Maxwell's equations it follows that the sources of the electric field can be either electric charges or time-varying magnetic fields, and magnetic fields can be excited either by moving electric charges (electric currents) or by alternating electric fields. Maxwell's equations are not symmetrical with respect to electric and magnetic fields. This is due to the fact that in nature there are electric charges, but no magnetic charges.

For stationary fields (E= const and IN=const) Maxwell's equations will take the form

i.e., in this case, the sources of the electric field are only electric charges, the sources of the magnetic field are only conduction currents. In this case, the electric and magnetic fields are independent of each other, which makes it possible to study separately permanent electric and magnetic fields.

Using the Stokes and Gauss theorems known from vector analysis

one can imagine a complete system of Maxwell's equations in differential form(characterizing the field at each point in space):

If charges and currents are distributed continuously in space, then both forms of Maxwell’s equations are integral

and differential are equivalent. However, when there are fracture surface- surfaces on which the properties of the medium or fields change abruptly, then the integral form of the equations is more general.

Maxwell's equations in differential form assume that all quantities in space and time vary continuously. To achieve mathematical equivalence of both forms of Maxwell's equations, the differential form is supplemented boundary conditions, which the electromagnetic field at the interface between two media must satisfy. The integral form of Maxwell's equations contains these conditions. They were discussed earlier (see § 90, 134):

D 1 n = D 2 n , E 1  = E 2  , B 1 n = B 2 n , H 1  = H 2 

(the first and last equations correspond to cases when there are neither free charges nor conduction currents at the interface).

Maxwell's equations are the most general equations for electric and magnetic fields in quiescent environments. They play the same role in the doctrine of electromagnetism as Newton's laws in mechanics. From Maxwell's equations it follows that an alternating magnetic field is always associated with the electric field generated by it, and an alternating electric field is always associated with the magnetic field generated by it, i.e., the electric and magnetic fields are inextricably linked with each other - they form a single electromagnetic field.

Maxwell's theory, being a generalization of the basic laws of electrical and magnetic phenomena, was able to explain not only already known experimental facts, which is also an important consequence of it, but also predicted new phenomena. One of the important conclusions of this theory was the existence of a magnetic field of displacement currents (see § 138), which allowed Maxwell to predict the existence electromagnetic waves- an alternating electromagnetic field propagating in space with a finite speed. Subsequently, it was proven that the speed of propagation of a free electromagnetic field (not associated with charges and currents) in a vacuum is equal to the speed of light c = 3 10 8 m/s. This conclusion and theoretical study of the properties of electromagnetic waves led Maxwell to the creation of the electromagnetic theory of light, according to which light is also electromagnetic waves. Electromagnetic waves were experimentally obtained by the German physicist G. Hertz (1857-1894), who proved that the laws of their excitation and propagation are completely described by Maxwell's equations. Thus, Maxwell's theory was experimentally confirmed.

Only Einstein’s principle of relativity is applicable to the electromagnetic field, since the fact of the propagation of electromagnetic waves in a vacuum in all reference systems with the same speed With is not compatible with Galileo's principle of relativity.

According to Einstein's principle of relativity, mechanical, optical and electromagnetic phenomena in all inertial reference systems proceed in the same way, i.e. they are described by the same equations. Maxwell's equations are invariant under Lorentz transformations: their form does not change during the transition

from one inertial frame of reference to another, although the quantities E, B,D, N they are converted according to certain rules.

It follows from the principle of relativity that separate consideration of electric and magnetic fields has a relative meaning. So, if an electric field is created by a system of stationary charges, then these charges, being stationary relative to one inertial reference system, move relative to another and, therefore, will generate not only an electric, but also a magnetic field. Similarly, a conductor with a constant current, stationary relative to one inertial reference frame, excites a constant magnetic field at each point in space, moves relative to other inertial frames, and the alternating magnetic field it creates excites a vortex electric field.

Thus, Maxwell's theory, its experimental confirmation, as well as Einstein's principle of relativity lead to a unified theory of electrical, magnetic and optical phenomena, based on the concept of an electromagnetic field.

44.. Dia- and paramagnetism

Every substance is magnetic, that is, it is capable of acquiring a magnetic moment (magnetization) under the influence of a magnetic field. To understand the mechanism of this phenomenon, it is necessary to consider the effect of a magnetic field on electrons moving in an atom.

For the sake of simplicity, let us assume that the electron in the atom moves in a circular orbit. If the electron’s orbit is oriented relative to vector B in an arbitrary manner, making an angle a with it (Fig. 188), then it can be proven that it begins to move around B in such a way that the magnetic moment vector R m, keeping angle a constant, rotates around direction B with a certain angular velocity. In mechanics this kind of movement is called precession. Precession around a vertical axis passing through the fulcrum is carried out, for example, by the disk of a top when it slows down.

Thus, the electron orbits of an atom under the influence of an external magnetic field undergo precessional motion, which is equivalent to a circular current. Since this microcurrent is induced by an external magnetic field, then, according to Lenz’s rule, the atom has a magnetic field component directed opposite to the external field. The induced components of the magnetic fields of atoms (molecules) add up and form the substance’s own magnetic field, which weakens the external magnetic field. This effect is called diamagnetic effect, and substances that are magnetized in an external magnetic field against the direction of the field are called Diamagnets.

In the absence of an external magnetic field, a diamagnetic material is nonmagnetic, since in this case the magnetic moments of the electrons are mutually compensated, and the total magnetic moment of the atom (it is equal to the vector sum of the magnetic moments (orbital and spin) of the electrons composing the atom) is zero. Diamagnets include many metals (for example, Bi, Ag, Au, Cu), most organic compounds, resins, carbon, etc.

Since the diamagnetic effect is caused by the action of an external magnetic field on the electrons of the atoms of a substance, diamagnetism is characteristic of all substances. However, along with diamagnetic substances, there are also paramagnetic- substances that are magnetized in an external magnetic field in the direction of the field.

In paramagnetic substances, in the absence of an external magnetic field, the magnetic moments of the electrons do not compensate each other, and the atoms (molecules) of paramagnetic materials always have a magnetic moment. However, due to the thermal motion of molecules, their magnetic moments are randomly oriented, therefore paramagnetic substances do not have magnetic properties. When a paramagnetic substance is introduced into an external magnetic field, preferential orientation of atomic magnetic moments on the field(full orientation is prevented by the thermal movement of atoms). Thus, the paramagnetic material is magnetized, creating its own magnetic field, which coincides in direction with the external field and enhances it. This Effect called paramagnetic. When the external magnetic field is weakened to zero, the orientation of the magnetic moments due to thermal motion is disrupted and the paramagnet is demagnetized. Paramagnetic materials include rare earth elements, Pt, Al, etc. The diamagnetic effect is also observed in paramagnetic materials, but it is much weaker than the paramagnetic one and therefore remains unnoticeable.

From an examination of the phenomenon of paramagnetism, it follows that its explanation coincides with the explanation of the orientational (dipole) polarization of dielectrics with polar molecules (see §87), only the electric moment of atoms in the case of polarization must be replaced by the magnetic moment of atoms in the case of magnetization.

Summarizing the qualitative consideration of dia- and paramagnetism, we note once again that the atoms of all substances are carriers of diamagnetic properties. If the magnetic moment of the atoms is large, then the paramagnetic properties prevail over the diamagnetic ones and the substance is paramagnetic; if the magnetic moment of the atoms is small, then diamagnetic properties predominate and the substance is diamagnetic.

Ferromagnets and their properties

In addition to the two classes of substances considered - dia- and paramagnets, called weakly magnetic substances, there are still highly magnetic substances - ferromagnets- substances that have spontaneous magnetization, i.e. they are magnetized even in the absence of an external magnetic field. In addition to their main representative - iron (from which the name “ferromagnetism” comes) - ferromagnets include, for example, cobalt, nickel, gadolinium, their alloys and compounds.

Animation

Description

In 1820, Ampere discovered the interaction of currents - the attraction or repulsion of parallel currents. This made it possible to set the research task: to reduce all magnetic interactions to the interaction of current elements and to find the law of their interaction as a fundamental law that plays a role in magnetism similar to Coulomb’s law in electricity. The currently used formula for the interaction of current elements was obtained in 1844 by Grassmann (1809-1877) and has the form:

, (in "SI") (1)

, (in the Gaussian system)

where d F 12 is the force with which the current element I 1 d I 1 acts on the current element I 2 d I 2 ;

r 12 - radius vector drawn from the element I 1 d I 1 to the current element I 2 d I 2 ;

c =3H 108 m/s - the speed of light.

Interaction of current elements

Rice. 1

The force d F 12 with which the current element I 2 d I 2 acts on the current element I 1 d I 1 has the form:

. (in "SI") (2)

The forces d F 12 and d F 21, generally speaking, are not collinear to each other, therefore, the interaction of current elements does not satisfy Newton’s third law:

d F 12 + d F 21 No. 0.

Law (1) has an auxiliary meaning, leading to correct, experimentally confirmed force values ​​only after integrating (1) over the closed contours L 1 and L 2.

The force with which the current I 1 flowing through the closed circuit L 1 acts on the closed circuit L 2 with the current I 2 is equal to:

. (in "SI") (3)

The force d F 21 has a similar form.

For the forces of interaction of closed circuits with current, Newton’s third law is satisfied:

dF 12 +d F 21 =0

In complete analogy with electrostatics, the interaction of current elements is represented as follows: the current element I 1 d I 1 at the location of the current element I 2 d I 2 creates a magnetic field, the interaction with which the current element I 2 d I 2 leads to the emergence of a force d F 12.

, (4)

. (5)

Relationship (5), which describes the generation of a magnetic field by a current, is called the Biot-Savart law.

The force of interaction between parallel currents.

The induction of the magnetic field created by a straight-line current I 1 flowing along an infinitely long conductor at the point where the current element I 2 dx 2 is located (see Fig. 2) is expressed by the formula:

. (in "SI") (6)

Interaction of two parallel currents

Rice. 2

Ampere's formula, which determines the force acting on a current element I 2 dx 2 located in a magnetic field B 12, has the form:

, (in "SI") (7)

. (in Gaussian system)

This force is directed perpendicular to the conductor with current I 2 and is an attractive force. A similar force is directed perpendicular to the conductor with current I 1 and is an attractive force. If currents in parallel conductors flow in opposite directions, then such conductors repel.

André Marie Ampère (1775-1836) - French physicist.

Timing characteristics

Initiation time (log to -15 to -12);

Lifetime (log tc from 13 to 15);

Degradation time (log td from -15 to -12);

Time of optimal development (log tk from -12 to 3).

Diagram:

Technical implementations of the effect

Installation diagram for “weighing” measurement currents

Implementation of a 1A unit using a force acting on a current-carrying coil.

Inside a large fixed coil is a “measuring coil” that is subjected to the force to be measured. The measuring coil is suspended from the beam of a sensitive analytical balance (Fig. 3).

Installation diagram for “weighing” measurement currents

Rice. 3

Applying an effect

Ampere's law of interaction of currents, or, which is the same thing, magnetic fields generated by these currents, is used to design a very common type of electrical measuring instruments - magnetoelectric devices. They have a light frame with wire, mounted on an elastic suspension of one design or another, capable of rotating in a magnetic field. The ancestor of all magnetoelectric devices is the Weber electrodynamometer (Fig. 4).

Weber electrodynamometer

Rice. 4

It was this device that made it possible to conduct classical studies of Ampere's law. Inside the fixed coil U, a moving coil C, supported by a fork ll, hangs on a bifilar suspension, the axis of which is perpendicular to the axis of the fixed coil. When current passes sequentially through the coils, the moving coil tends to become parallel to the stationary one and rotates, twisting the bifilar suspension. The rotation angles are measured using a mirror f attached to the frame ll ў.

Literature

1. Matveev A.N. Electricity and magnetism. - M.: Higher School, 1983.

2. Tamm I.E. Fundamentals of the theory of electricity. - M.: State Publishing House of Technical and Theoretical Literature, 1954.

3. Kalashnikov S.G. Electricity. - M.: Nauka, 1977.

4. Sivukhin D.V. General course of physics. - M.: Nauka, 1977. - T.3. Electricity.

5. Kamke D., Kremer K. Physical foundations of units of measurement. - M.: Mir, 1980.

Keywords

• Ampere power
• a magnetic field
• Biot-Savart's law
• magnetic field induction
• interaction of current elements
• interaction of parallel currents

Sections of natural sciences:

Relativistic form of Coulomb's law: Lorentz force and Maxwell's equations. Electromagnetic field.

Coulomb's law:

Lorentz force: LORENTZ FORCE - a force acting on a charged particle moving in an electromagnetic field. If the left hand is positioned so that the component of magnetic induction B, perpendicular to the speed of the charge, enters the palm, and the four fingers are directed along the movement of the positive charge (against the movement of the negative), then the thumb bent 90 degrees will show the direction of the Lorentz force acting on the charge.

Maxwell's equations: is a system of differential equations that describe the electromagnetic field and its relationship with electric charges and currents in vacuum and continuous media.

Electromagnetic field: is a fundamental physical field that interacts with electrically charged bodies, representing a combination of electric and magnetic fields that can, under certain conditions, generate each other.

Stationary magnetic field. Magnetic field induction, superposition principle. Bio-Savart's Law.

Constant (or stationary) magnetic field: is a magnetic field that does not change over time. M\G is a special type of matter through which interaction occurs between moving electrically charged particles.

Magnetic induction: - vector quantity, which is the force characteristic of the magnetic field at a given point in space. Determines the force with which the magnetic field acts on a charge moving at speed .

Superposition principle:- In its simplest formulation, the superposition principle states:

the result of the influence of several external forces on a particle is the vector sum of the influence of these forces.
Bio-Savart's Law: is a law that determines the strength of the magnetic field created by electric current at an arbitrary point in space around a conductor carrying current.

Ampere power. Interaction of parallel conductors with current. The work of magnetic field forces to move a coil with current.