Interaction of parallel currents is the formula for the interaction force. Ampere force. Interaction of parallel currents. Units of amperage. The work of moving a conductor and a circuit with a current in a magnetic field

We determine the force with which the conductors with currents I 1 and I 2 interact (attract or repel) (Figure 3.19)

The interaction of currents is carried out through a magnetic field. Each current creates a magnetic field that acts on another wire (current).

Suppose that both currents I 1 and I 2 flow in the same direction. Current I 1 creates a magnetic field with induction B 1 (see 3.61) at the location of the second wire (with current I 2), which acts on I 2 with force F:

(3.66)

Using the left-hand rule (see Ampere's law), you can establish:

a) parallel currents of one direction are attracted;

b) parallel currents of the opposite direction are repelled;

c) non-parallel currents tend to become parallel.

A circuit with a current in a magnetic field. Magnetic flux

Let in a magnetic field with induction B there is a contour with area S, the normal to which makes an angle α with the vector (Figure 3.20). To calculate the magnetic flux Ф, we divide the surface S into infinitesimal elements so that within one element dS the field can be considered uniform. Then the elementary magnetic flux through an infinitely small area dS will be:

where B n is the projection of the vector to normal .

If the site dS is located perpendicular to the magnetic induction vector, then α = 1, cos α = 1 and dФ = BdS;

The magnetic flux through an arbitrary surface S is equal:

If the field is uniform and the surface S is flat, then the value B n = const and:

(3.67)

For a flat surface located along a uniform field, α = π / 2 and Ф = 0. The induction lines of any magnetic field are closed curves. If there is a closed surface, then the magnetic flux entering this surface and the magnetic flux leaving it are numerically equal and opposite in sign. Therefore, the magnetic flux through an arbitrary closed surface is zero:

(3.68)

Formula (3.68) is Gauss's theorem for a magnetic field, reflecting its vortex character.

The magnetic flux is measured in Weber (Wb): 1Vb = Tm 2 .

The work of moving a conductor and a circuit with a current in a magnetic field

If a conductor or a closed loop with a current I move in a uniform magnetic field under the action of the Ampere force, then the magnetic field does the work:

A = IΔФ, (3.69)

where ΔФ is the change in magnetic flux through the area of ​​the contour or the area described by a straight conductor when moving.

If the field is non-uniform, then:

.

The phenomenon of electromagnetic induction. Faraday's law

The essence of the phenomenon of electromagnetic induction is as follows: for any change in the magnetic flux through the area bounded by a closed conducting loop, in the latter there is an EMF. and, as a consequence, induction electric current.

Induction currents always oppose the process that causes them. This means that the magnetic field they create tends to compensate for the change in magnetic flux that this current has caused.

It has been experimentally established that the value of E.D.S. induction ε i induced in the circuit depends not on the magnitude of the magnetic flux Ф, but on the rate of its change dФ / dt through the area of ​​the circuit:

(3.70)

The minus sign in the formula (3.70) is a mathematical expression Lenz rules: the induction current in the circuit is always in such a direction that the magnetic field it creates prevents the change in the magnetic flux that causes this current.

Formula (3.70) is an expression of the basic law of electromagnetic induction.

Using formula (3.70), you can calculate the strength of the induction current I, knowing the resistance of the circuit R, and the amount of charge Q elapsed during the time t in the loop:

If in a uniform magnetic field a segment of a straight conductor of length ℓ moves at a speed V, then the change in the magnetic flux is taken into account through the area described by the segment during movement, i.e.

Faraday's law can be derived from the law of conservation of energy. If a conductor with a current is in a magnetic field, then the work of the current source εIdt for time dt will be spent on Lenz-Joule heat (see formula 3.48) and the work of moving the conductor in the field IdF (see 3.69) can be determined:

εIdt = I 2 Rdt + IdФ (3.71)

then
,

where
is the EMF of induction (3.70)

those. with a change in Ф in the circuit, an additional EMF ε i arises in accordance with the law of conservation of energy.

It can also be shown that ε i arises in a metal conductor due to the action of the Lorentz force on the electrons. However, this force does not act on stationary charges. Then it is necessary to assume that the alternating magnetic field creates an electric field, under the influence of which the induction current I i arises in a closed loop.

The interaction of stationary charges is described by Coulomb's law. However, Coulomb's law is insufficient to analyze the interaction of moving charges. In the experiments of Ampere, a message appeared for the first time that moving charges (currents) create a certain field in space, leading to the interaction of these currents. It was found that currents in opposite directions are repelled, and currents of one direction are attracted. Since it turned out that the current field acts on the magnetic needle in the same way as the field of a permanent magnet, this current field was called magnetic. The current field is called the magnetic field. It was subsequently found 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 complicated than Coulomb's law, which describes the interaction of stationary point charges. This explains the fact that many scientists took part in his research, and a significant contribution was made by Biot (1774 - 1862), Savard (1791 - 1841), Ampere (1775 - 1836) and Laplace (1749 - 1827).

In 1820 H.K. Oersted (1777 - 1851) discovered the action electric current on the magnetic needle. In the same year, Bio 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) It was theoretically derived by Laplace in the form

F(R) 1 / r. (16.2)

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

In 1820. Ampere discovered the interaction of currents - attraction or repulsion parallel currents... He proved the equivalence of a solenoid and a permanent magnet. This made it possible to clearly formulate the research task: 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. Ampere was a theoretician and mathematician by his education and inclinations. Nevertheless, in the study of the interaction of current elements, he carried out 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 was left a description of the way in which he came to the discovery. However, Ampere's formula for force differs from (16.2) by the presence of a full differential on the right-hand side. This difference is insignificant when calculating the force of interaction of closed currents, since the integral of the total differential over a closed loop is zero. Taking into account that in the experiments it is not the strength of interaction of the elements of the current that is measured, but the strength of the interaction of closed currents, it is possible with good reason to consider Ampère as the author of the law of the magnetic interaction of currents. Currently used formula for the interaction of currents. The currently used formula for the interaction of current elements was obtained in 1844. Grassmann (1809 - 1877).

If we introduce 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)

Similarly, you can write:

(16.3)

It is easy to see:

Since the vectors and have an angle not equal to 180 °, it is obvious , that is, the III-th Newton's law for the current elements is not fulfilled. But if you 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, that is, for currents III, Newton's law is fulfilled.

Description of the interaction of currents using a magnetic field.

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

. (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-Savard law. By integrating (16.5) we get:

(16.7)

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

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

, (16.8)

Where j is the current density.

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

Example.

A direct infinite current J is given. Let us calculate the magnetic induction at point M at a distance r from it.

= .

= = . (16.10)

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

Direction of the magnetic induction vector Shown in the figures.

Ampere force and Lorentz force.

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

Or , where

We turn to the force acting on a conductor with a current of length L. Then = and .

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

, where . (16.12)

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

When the Lorentz force is equal to 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 velocity:.

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

Experience shows that electric currents interact with each other. For example, two thin straight parallel conductors through which currents flow (we will call them forward currents) attract each other if the currents in them have the same direction, and repel if the currents are opposite. The force of interaction per unit length of each of the parallel conductors is proportional to the values ​​of the currents in them and is inversely proportional to the distance b between them:

For reasons that will become clear in the future, we denoted the coefficient of proportionality through.

The law of interaction of currents was established in 1820 by Ampere. A general expression of this law, suitable for conductors of any shape, will be given in § 44.

Based on relationship (39.1), the unit of current strength is established in SI and in the absolute electromagnetic system of units (CGSM-system). The unit of current strength in SI - ampere - is defined as the strength of a constant current, which, passing through two parallel rectilinear conductors of infinite length and negligible circular cross-section, located at a distance of 1 m from one another in a vacuum, would cause a force equal to N between these conductors for every meter of length.

A unit of charge, called a coulomb, is defined as a charge passing in 1 s through the cross-section of a conductor through which it flows D.C. force 1 A. In accordance with this, the pendant is also called an ampere-second (A s).

In a rationalized form, formula (39.1) is written as follows:

where is the so-called magnetic constant (compare with formula (4.1)).

To find the numerical value, we use the fact that, according to the definition of ampere at, the force is equal to. Substitute these values ​​into formula (39.2):

The coefficient k in formula (39.1) can be made equal to unity by choosing the unit of current strength. This is how the absolute electromagnetic unit of current strength (CGSM-unit of current strength) is established, which is defined as the strength of such a current that, flowing through a thin rectilinear infinitely long wire, acts on an equal and parallel direct current, spaced 1 cm apart, with a force of 2 dyne for every centimeter of length.

In the CGSE-system, k turns out to be a dimensional quantity different from unity. According to formula (39.1), the dimension k is determined by the following expression:

We took into account that the dimension is the dimension of the force divided by the dimension of the length; therefore, the dimension of the product is equal to the dimension of the force. According to formulas (3.2) and (31.7)

Substituting these values ​​into expression (39.4), we find that

Therefore, in the CGSE system, k can be represented in the form

where c - has the dimension of velocity, called the electrodynamic constant. To find its numerical value, we use the relation (3.3) between the coulomb and the CGSE unit of charge, which was established empirically. The force is equivalent. According to formula (39.1), currents of CGSE-units (i.e., 1 A) interact with such a force each at the way,

The value of the electrodynamic constant coincides with the value of the speed of rkta in vacuum. From Maxwell's theornes follows the existence electromagnetic waves, the speed of which in vacuum is equal to the electrodynamic constant c. The coincidence with the speed of light in a vacuum gave Maxwell reason to assume that light is an electromagnetic wave.

The value of k in formula (39.1) is equal to 1 in the CGSM system and in the CGSE system. It follows that a current of 1 CGSM-unit is equivalent to a current of 3-10 ° CGSE-units:

Multiplying this ratio by 1 s, we get

The strength of the interaction of parallel currents. Ampere's law

If we take two conductors with electric currents, then they will be attracted to each other if the currents in them are directed in the same way and repel if the currents flow in opposite directions. The force of interaction, which is per unit length of the conductor, if they are parallel, can be expressed as:

where $ I_1 (, I) _2 $ are the currents that flow in the conductors, $ b $ is the distance between the conductors, $ in \ system \ SI \ (\ mu) _0 = 4 \ pi \ cdot (10) ^ (- 7) \ frac (Hn) (m) \ (Henry \ per \ meter) $ magnetic constant.

The law of interaction of currents was established in 1820 by Ampere. Based on Ampere's law, the units of current strength are established in the SI and CGSM systems. Since ampere is equal to the strength of a direct current, which, when flowing through two parallel infinitely long straight conductors of infinitely small circular cross-section, located at a distance of 1 m from each other in vacuum, causes the interaction force of these conductors equal to $ 2 \ cdot (10) ^ (- 7) N $ for each meter of length.

Ampere's law for a conductor of arbitrary shape

If a conductor with a current is in a magnetic field, then a force equal to each current carrier acts:

where $ \ overrightarrow (v) $ is the speed of thermal motion of charges, $ \ overrightarrow (u) $ is the speed of their ordered motion. From the charge, this action is transferred to the conductor along which the charge moves. This means that a force acts on a conductor with a current that is in a magnetic field.

Let us choose a conductor element with a current of length $ dl $. Let's find the force ($ \ overrightarrow (dF) $) with which the magnetic field acts on the selected element. Let's average expression (2) over the current carriers that are in the element:

where $ \ overrightarrow (B) $ is the magnetic induction vector at the point of the element $ dl $. If n is the concentration of current carriers per unit volume, S is the cross-sectional area of ​​the wire at a given place, then N is the number of moving charges in the element $ dl $, equal to:

Multiply (3) by the number of current carriers, we get:

Knowing that:

where $ \ overrightarrow (j) $ is the current density vector, and $ Sdl = dV $, we can write:

From (7) it follows that the force acting on a unit volume of the conductor is equal to the density of the force ($ f $):

Formula (7) can be written as:

where $ \ overrightarrow (j) Sd \ overrightarrow (l) = Id \ overrightarrow (l). $

Formula (9) Ampere's law for a conductor of arbitrary shape. Ampere force modulus from (9) is obviously equal to:

where $ \ alpha $ is the angle between the vectors $ \ overrightarrow (dl) $ and $ \ overrightarrow (B) $. The Ampere force is directed perpendicular to the plane in which the vectors $ \ overrightarrow (dl) $ and $ \ overrightarrow (B) $ lie. The force that acts on a wire of finite length can be found from (10) by integrating over the length of the conductor:

The forces that act on conductors with currents are called Ampere forces.

The direction of Ampere's force is determined by the left-hand rule ( Left hand should be positioned so that the field lines enter the palm, four fingers were directed along the current, then bent to 900 thumb will indicate the direction of Ampere's force).

Example 1

Task: A straight conductor of mass m and length l is suspended horizontally on two light threads in a uniform magnetic field, the induction vector of this field has a horizontal direction perpendicular to the conductor (Fig. 1). Find the current strength and direction that will break one of the suspension strands. Field induction B. Each strand will break under N load.

To solve the problem, let us depict the forces that act on the conductor (Fig. 2). We assume that the conductor is homogeneous, then we can assume that the point of application of all forces is the middle of the conductor. In order for the Ampere force to be directed downward, the current must flow in the direction from point A to point B (Fig. 2) (In Fig. 1, the magnetic field is shown directed at us, perpendicular to the plane of the figure).

In this case, the equation of equilibrium of forces applied to a conductor with a current can be written as:

\ [\ overrightarrow (mg) + \ overrightarrow (F_A) +2 \ overrightarrow (N) = 0 \ \ left (1.1 \ right), \]

where $ \ overrightarrow (mg) $ is the force of gravity, $ \ overrightarrow (F_A) $ is the Ampere force, $ \ overrightarrow (N) $ is the thread reaction (there are two of them).

We project (1.1) onto the X-axis, we get:

Ampere force modulus for a straight final conductor with current is:

where $ \ alpha = 0 $ is the angle between the vectors of magnetic induction and the direction of current flow.

Substituting (1.3) into (1.2), we express the current strength, we get:

Answer: $ I = \ frac (2N-mg) (Bl). $ From point A and point B.

Example 2

Task: A constant current of force I flows through a conductor in the form of a half ring of radius R. The conductor is in a uniform magnetic field, the induction of which is B, the field is perpendicular to the plane in which the conductor lies. Find the power of Ampere. Wires that carry current outside the field.

Let the conductor be in the plane of the drawing (Fig. 3), then the field lines are perpendicular to the plane of the drawing (from us). Let us single out on the semiring an infinitesimal current element dl.

The current element is acted upon by an Ampere force equal to:

\\ \ left (2.1 \ right). \]

The direction of force is determined by the left hand rule. Let's choose the coordinate axes (Fig. 3). Then the element of force can be written through its projections ($ (dF) _x, (dF) _y $) as:

where $ \ overrightarrow (i) $ and $ \ overrightarrow (j) $ are unit vectors. Then the force that acts on the conductor is found as an integral over the length of the wire L:

\ [\ overrightarrow (F) = \ int \ limits_L (d \ overrightarrow (F) =) \ overrightarrow (i) \ int \ limits_L (dF_x) + \ overrightarrow (j) \ int \ limits_L ((dF) _y) \ left (2.3 \ right). \]

Due to symmetry, the integral $ \ int \ limits_L (dF_x) = 0. $ Then

\ [\ overrightarrow (F) = \ overrightarrow (j) \ int \ limits_L ((dF) _y) \ left (2.4 \ right). \]

Considering Fig. 3, we write that:

\ [(dF) _y = dFcos \ alpha \ left (2.5 \ right), \]

where, according to Ampere's law for a current element, we write that

By condition $ \ overrightarrow (dl) \ bot \ overrightarrow (B) $. Let us express the length of the arc dl in terms of the radius R and the angle $ \ alpha $, we get:

\ [(dF) _y = IBRd \ alpha cos \ alpha \ \ left (2.8 \ right). \]

Let us integrate (2.4) for $ - \ frac (\ pi) (2) \ le \ alpha \ le \ frac (\ pi) (2) \ $ substituting (2.8), we get:

\ [\ overrightarrow (F) = \ overrightarrow (j) \ int \ limits ^ (\ frac (\ pi) (2)) _ (- \ frac (\ pi) (2)) (IBRcos \ alpha d \ alpha) = \ overrightarrow (j) IBR \ int \ limits ^ (\ frac (\ pi) (2)) _ (- \ frac (\ pi) (2)) (cos \ alpha d \ alpha) = 2IBR \ overrightarrow (j ). \]

Answer: $ \ overrightarrow (F) = 2IBR \ overrightarrow (j). $

Consider a wire that is in a magnetic field and through which current flows (Figure 12.6).

For each current carrier (electron), acts Lorentz force... We define the force acting on an element of a wire of length d l

The last expression is called Ampere's law.

Ampere's modulus is calculated by the formula:

.

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

We apply Ampere's law to calculate the interaction force of two parallel infinitely long direct currents in a vacuum (Figure 12.7).

Distance between conductors - b. Suppose that the conductor I 1 creates a magnetic field by induction

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

considering that (sinα = 1)

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

.

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

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

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

Consider, as an example, the magnetic field of a closed loop L of arbitrary shape, covering an infinitely long straight conductor with a current l in a vacuum (Figure 12.8).

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 with a dotted line). At point A of the contour L, the vector B of the magnetic induction of the field of this current is perpendicular to the radius vector.

The figure shows that

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

Then we get that the circulation of the induction vector

At all points of the line, the magnetic induction vector is

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 the circulation of the vector in it V along the line of magnetic induction is not zero;

2.vector circulation V The magnetic induction of a closed loop covering the field of rectilinear current in 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 the magnetic field is formed by several conductors with current, then the circulation of the resulting field

This expression is called total current theorem.