Magnetic interaction of currents is the ampere law. §16.Magnetic field

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.

Let's apply Ampere's law to calculate the force of interaction between two parallel infinitely long forward currents located in a vacuum (Fig. 12.7).

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

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

, 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. Such a field in vector analysis is called a vortex field.

Let us consider as an example the magnetic field of a closed loop L of arbitrary shape, covering an infinitely long straight conductor with current l, located in a vacuum (Fig. 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 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 - attraction or repulsion 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 force of interaction of current elements that is measured, but the force 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 third 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 area cross section. 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.

Ampere power

A current-carrying conductor in a magnetic field experiences a force equal to

F = I·L·B·sina

I is the current strength in the conductor;

B - module of the magnetic field induction vector;

L is the length of the conductor located in the magnetic field;

a is the angle between the magnetic field vector and the direction of the current in the conductor.

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

The maximum ampere force is:

It corresponds to a = 900.

The direction of the Ampere force is determined by the left hand rule: if the left hand is positioned so that the perpendicular component of the magnetic induction vector B enters the palm, and four extended fingers are directed in the direction of the current, then the thumb bent 90 degrees will show the direction of the force acting on the segment conductor with current, that is, Ampere force.

During the experiment, we observed a force that cannot be explained within the framework of electrostatics. When two parallel conductors carry current in only one direction, there is an attractive force between them. When currents flow in opposite directions, the wires repel each other.

The actual value of this force acting between parallel currents, and its dependence on the distance between the wires, can be measured using a simple device in the form of a scale. In view of the absence of such, we will take on faith the experimental results that show that this force is inversely proportional to the distance between the axes of the wires: F (1/r).

Since this force must be due to some influence spreading from one wire to another, such a cylindrical geometry will create a force that depends inversely on the first power of the distance. Let us remember that the electrostatic field propagates from a charged wire also with a distance dependence of the form 1/r.

Based on experiments, it is also clear that the strength of interaction between the wires depends on the product of the currents flowing through them. From symmetry we can conclude that if this force is proportional to I1, it must be proportional to I2. That this force is directly proportional to each of the currents is simply an experimental fact.

By adding a proportionality coefficient, we can now write down the formula for the force of interaction between two parallel wires: F (l/r, F (I1 I2); therefore,

The proportionality factor will contain the factor 2( associated with it, not the constant itself.

The interaction between two parallel wires is expressed as force per unit length. The longer the wires, the greater the power:

The distance r between the axes of the wires F/l is measured in meters. Force per 1 meter of length is measured in newtons per meter, and currents I1 I2 - in amperes.

In a school physics course, the coulomb is first defined in terms of ampere, without giving the definition of ampere, and then the value of the constant appearing in Coulomb's law is taken on faith.

Only now is it possible to move on to consider the definition of ampere.

When the equation for F/l is assumed to determine the ampere. The constant is called the magnetic constant. It is similar to the constant 0 - the electrical constant. However, there is an operational difference in assigning values ​​to these two constants. We can choose any arbitrary value for any one of them. But then the second constant must be determined experimentally, since the coulomb and ampere are related.

Based now on the formula described above, the value of an ampere can be expressed in words: if the interaction per 1 m of length of two long parallel wires located at a distance of 1 m from each other is equal to 2 * 10-7 N, then the current in each wire is 1A.

In the case where the interacting wires are perpendicular to each other, there is only a very small area of ​​influence where the wires pass close to each other, and therefore the interaction force between the wires can be expected to be small. In fact, this force is zero. Since the force can be considered positive when the currents are parallel, and negative when the currents are antiparallel, it is plausible that this force should be zero when the wires are perpendicular, for this zero value lies midway between the positive and negative values.

The SI unit is 1 Ampere (A) = 1 Coulomb/second.

To measure current, a special device is used - an ammeter (for devices designed to measure small currents, the names milliammeter, microammeter, galvanometer are also used). It is included in the open circuit in the place where the current strength needs to be measured. The main methods for measuring current strength are: magnetoelectric, electromagnetic and indirect (by measuring voltage at a known resistance with a voltmeter).

The force of interaction between current elements, proportional to the currents and the length of the elements, inversely proportional to the square of the distance between them and depending on their relative position

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 sides, 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 “measurement 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:

From here it is not difficult to obtain an expression for the magnetic field induction of each of the straight conductors. The magnetic field of a straight conductor carrying current must have axial symmetry and, therefore, closed lines of magnetic induction can only be concentric circles located in planes perpendicular to the conductor. This means that vectors B1 and B2 of magnetic induction of parallel currents I 1 and I 2 lie in a plane perpendicular to both currents. Therefore, when calculating the Ampere forces acting on current-carrying conductors, in Ampere’s law one must put sin α = 1. From the law of magnetic interaction of parallel currents it follows that the induction modulus B magnetic field of a straight conductor carrying current I on distance R from it is expressed by the relation

In order for parallel currents to attract and antiparallel currents to repel during magnetic interaction, the magnetic induction field lines of a straight conductor must be directed clockwise when viewed along the conductor in the direction of the current. To determine the direction of vector B of the magnetic field of a straight conductor, you can also use the gimlet rule: the direction of rotation of the gimlet handle coincides with the direction of vector B if, during rotation, the gimlet moves in the direction of the current. Magnetic interaction of parallel conductors with current is used in the International System of Units (SI) to determine the unit of force current - ampere:

Magnetic induction vector- this is the main force characteristic of the magnetic field (denoted B).

Lorentz force- the force acting on one charged particle is equal to

F L = q υ B sin α.

Under the influence of the Lorentz force, electric charges in a magnetic field move along curvilinear trajectories. Let us consider the most typical cases of the motion of charged particles in a uniform magnetic field.
a) If a charged particle enters a magnetic field at an angle α = 0°, i.e. flies along the field induction lines, then F l= qvBsma = 0. Such a particle will continue its movement as if there were no magnetic field. The particle trajectory will be a straight line.
b) Particle with charge q enters a magnetic field so that the direction of its velocity v is perpendicular to the induction ^B magnetic field (Figure - 3.34). In this case, the Lorentz force provides centripetal acceleration a = v 2 /R and particle moves in a circle with radius R in a plane perpendicular to the magnetic field induction lines. under the influence of the Lorentz force : F n = qvB sinα, Taking into account that α = 90°, we write the equation of motion of such a particle: t v 2 /R= qvB. Here m- particle mass, R– radius of the circle along which the particle moves. Where can you find the relationship? e/m- called specific charge, which shows the charge per unit mass of the particle.
c) If a charged particle flies in at a speed v 0 into a magnetic field at any angle α, then this movement can be represented as complex and decomposed into two components. The trajectory of movement is a helical line, the axis of which coincides with the direction IN. The direction in which the trajectory twists depends on the sign of the particle's charge. If the charge is positive, the trajectory spins counterclockwise. The trajectory along which a negatively charged particle moves spins clockwise (it is assumed that we are looking at the trajectory along the direction IN; the particle flies away from us.