The formula for the force of interaction of parallel currents. Interaction of parallel currents. Ampere's law for a conductor of arbitrary shape
From here it is easy to obtain an expression for the magnetic induction of each of the rectilinear conductors. The magnetic field of a straight conductor with 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 the vectors B1 and B2 of the magnetic induction parallel currents I 1 and I 2 lie in a plane perpendicular to both currents. Therefore, when calculating the Ampere forces acting on conductors with current, in the Ampere's law it is necessary to set 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 with current I on distance R from it is expressed by the ratio
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In order for parallel currents to be attracted during magnetic interaction and antiparallel currents to repel, the lines of magnetic induction of the field of a rectilinear conductor must be directed clockwise, when viewed along the conductor in the direction of the current. To determine the direction of the vector B of the magnetic field of a rectilinear conductor, you can also use the gimbal rule: the direction of rotation of the grip handle coincides with the direction of the vector B if the gimbal moves in the direction of the current during rotation The magnetic interaction of parallel conductors with current is used in the International System of Units (SI) to determine the unit of force current - ampere:
Vector of magnetic induction is the main power characteristic magnetic field (indicated by B).
Lorentz force- the force acting on one charged particle is
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Under the action of the Lorentz force, electric charges in a magnetic field move along curvilinear trajectories. Let us consider the most typical cases of motion of charged particles in a uniform magnetic field.
a) If a charged particle falls into a magnetic field at an angle α = 0 °, i.e. flies along the lines of field inductions, then F l= qvBsma = 0. Such a particle will continue to move as if there were no magnetic field. The particle trajectory will be a straight line.
b) Particle with charge q falls into 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 the particle moves in a circle with a radius R in a plane perpendicular to the lines of induction of the magnetic field. under the action of the Lorentz force : F n = qvB sinα, taking into account that α = 90 °, we write down the equation of motion of such a particle: m v 2 / R = qvB. Here m- particle mass, R- the radius of the circle along which the particle moves. Where can the attitude be found e / m- called specific charge, which shows the charge per unit mass of a particle.
c) If a charged particle flies in with a speed v 0 into a magnetic field at any angle α, then this motion can be presented as complex and decomposed into two components. The trajectory of movement is a helical line, the axis of which coincides with the direction V... The direction in which the trajectory is twisted depends on the sign of the particle's charge. If the charge is positive, the trajectory spins counterclockwise. The trajectory along which the negatively charged particle moves is twisted clockwise (it is assumed that we are looking at the trajectory along the direction V; the particle then flies away from us.
A magnetic needle located near a conductor with a current is acted upon by forces that tend to turn the arrow. The French physicist A. Ampere observed the force interaction of two conductors with currents and established the law of the interaction of currents. A magnetic field, unlike an electric one, has a forceful effect only on moving charges (currents). The characteristic for describing the magnetic field is the vector of magnetic induction. The magnetic induction vector determines the forces acting on currents or moving charges in a magnetic field. For the positive direction of the vector, the direction from the south pole S to the north pole N of the magnetic needle, which is freely installed in the magnetic field, is taken. Thus, by examining the magnetic field created by a current or a permanent magnet, using a small magnetic arrow, it is possible to determine the direction of the vector at each point in space. The interaction of currents is caused by their magnetic fields: the magnetic field of one current acts by the Ampere force on another current and vice versa. As shown by Ampere's experiments, the force acting on the section of the conductor is proportional to the current I, the length Δl of this section and the sine of the angle α between the directions of the current and the magnetic induction vector: F ~ IΔl sin α
This power is called by Ampere... It reaches the maximum modulus value F max, when the conductor with current is oriented perpendicular to the lines of magnetic induction. The modulus of the vector is determined as follows: the modulus of the magnetic induction vector is equal to the ratio of the maximum value of the Ampere force acting on a straight conductor with current to the current I in the conductor and its length Δl:
In the general case, the Ampere force is expressed by the ratio: F = IBΔl sin α
This ratio is usually called Ampere's law. In the SI system of units, the unit of magnetic induction is the induction of such a magnetic field, in which the maximum Ampere force 1 N acts for each meter of the length of the conductor with a current of 1 A. This unit is called a tesla (T).
Tesla is a very large unit. The Earth's magnetic field is approximately equal to 0.5 · 10 –4 Tesla. A large laboratory electromagnet can create a field of no more than 5 T. The Ampere force is directed perpendicular to the magnetic induction vector and the direction of the current flowing through the conductor. The left hand rule is usually used to determine the direction of Ampere's force. The magnetic interaction of parallel conductors with current is used in the SI system to determine the unit of current strength - ampere: Ampere- the strength of a constant current, which, when passing through two parallel 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 magnetic interaction force between these conductors equal to 2 · 10 -7 N for each meter length. The formula expressing the law of magnetic interaction of parallel currents has the form:
14. Law of Bio-Savar-Laplace. Vector of magnetic induction. The theorem on the circulation of the magnetic induction vector.
Bio Savard Laplace's law determines the magnitude of the modulus of the magnetic induction vector at a point chosen arbitrarily in a magnetic field. In this case, the field is created by direct current in a certain area.
The magnetic field of any current can be calculated as the vector sum (superposition) of the fields created by individual elementary sections of the current:
A current element of length dl creates a field with magnetic induction: or in vector form: 
Here I- current; - a vector that coincides with an elementary section of the current and is directed in the direction where the current flows; - the radius vector drawn from the current element to the point at which we define; r- radius vector module; k
The magnetic induction vector is the main force characteristic of the magnetic field (indicated). The magnetic induction vector is directed perpendicular to the plane passing through the point at which the field is calculated.
Direction is related to direction « gimlet rule »: The direction of rotation of the screw head gives the direction, the translational movement of the screw corresponds to the direction of the current in the element.
Thus, the Biot – Savart – Laplace law establishes the magnitude and direction of the vector at an arbitrary point of the magnetic field created by a conductor with current I.
The modulus of the vector is determined by the ratio: 
where α is the angle between and ; k- coefficient of proportionality, depending on the system of units.
In the international system of units SI, the Bio-Savard-Laplace law for vacuum can be written as follows: 
where 
- magnetic constant.
Vector circulation theorem: the circulation of the magnetic induction vector is equal to the current in the circuit multiplied by the magnetic constant. 
,
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 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 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, Savart and Laplace, a formula was found that describes the strength of the current acting on the 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 - the attraction or repulsion of 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 the elements of the current, 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, the Ampere formula for the force differs from (16.2) by the presence of a complete 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. Considering that in the experiments it is not the force of interaction of the elements of the current that is measured, but the force 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)
Easy to see:
Since the vectors and have an angle not equal to 180 °, it is obvious 
, that is, Newton's third law for the current elements is not fulfilled. 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, that is, for currents III-th Newton's law is fulfilled.
Description of the interaction of currents using a magnetic field.
In full 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)
Relation (16.5), which describes the generation of a magnetic field by a current, is called the Biot-Savard law. 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 is the 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 magnitudes 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 relation (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 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 that passes 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 c).
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
Consequently, 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 in 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
Consider a wire with a magnetic field and current flowing through it (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 force modulus is calculated by the formula:
.
The Ampere force is directed perpendicular to the plane in which the vectors dl and B lie.
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Let's apply Ampere's law to calculate the force of interaction 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 four extended 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 of the field along the closed circuit is not zero and depends on the choice of the circuit. Such a field in vector analysis is called a vortex field.
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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.
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