Magnetic induction of a circular conductor with current. Magnetic field at the center of a circular current-carrying conductor. The Biot-Savart-Laplace law and its application to the calculation of the induction of the magnetic field of the circular current

All elements of a circular conductor with current create magnetic fields in the center of the same direction - along the normal from the coil. therefore, all elements of the coil are perpendicular to the radius vector, then ; since the distances from all elements of the conductor to the center of the coil are the same and equal to the radius of the coil. That's why:

Direct conductor field.

As the integration constant, we choose the angle α (the angle between the vectors dB And r ), and express all other quantities in terms of it. It follows from the figure that:

We substitute these expressions into the formula of the Biot-Savart-Laplace law:

And - the angles at which the ends of the conductor are visible from the point at which the magnetic induction is measured. Substitute in the formula:

In the case of an infinitely long conductor ( and ) we have:

Application of Ampère's law.

Interaction of parallel currents

Consider two infinite rectilinear parallel currents directed in the same direction I 1 And I 2, the distance between which is R. Each of the conductors creates a magnetic field that acts according to Ampère's law on the other current-carrying conductor. Current I 1 creates a magnetic field around itself, the lines of magnetic induction of which are concentric circles. vector direction IN , is determined by the rule of the right screw, its modulus is equal to:

Force direction d F 1 , with which the field B1 operates on the site dl the second current is determined by the rule of the left hand. The modulus of force, taking into account the fact that the angle α between the current elements I 2 and vector B1 straight, equal

Substituting value B1 . we get:

Arguing similarly, one can prove that

It follows that, that is, two parallel currents are attracted to each other with the same force. If the currents have the opposite direction, then using the left hand rule, it can be shown that a repulsive force acts between them.

Interaction force per unit length:

Behavior of a circuit with current in a magnetic field.

We introduce a square frame with side l with current I into a magnetic field B, the circuit will be affected by the torque of a pair of Ampère forces:

The magnetic moment of the circuit,

Magnetic induction at the field point where the circuit is located

A circuit with a current tends to settle in a magnetic field so that the flow through it is maximum and the moment is minimum.

The magnetic induction at a given point of the field is numerically equal to the maximum torque acting at a given point of the field on a circuit with a unit magnetic moment.

Full current law.

Let us find the circulation of the vector B along a closed contour. As a source of the field, we take a long conductor with current I, as a circuit, a force line of radius r.

Let us extend this conclusion to a circuit of any shape, covering any number of currents. Full current law:

The circulation of the magnetic induction vector in a closed loop is proportional to the algebraic sum of the currents covered by this loop.

Applying the total current law to calculate fields

Field inside an infinitely long solenoid:

where τ is the linear density of winding turns, ls is the length of the solenoid, N- number of turns.

Let the closed contour be a rectangle of length X, which braids the turns, then the induction IN along this circuit:

Find the inductance of this solenoid:

Toroid field(wire wound on a frame in the form of a torus).

R is the average radius of the torus, N is the number of turns, where is the linear density of the winding of the turns.

As a contour, we take a line of force with a radius R.

hall effect

Consider a metal plate placed in a magnetic field. Passes across the plate electricity. There is a potential difference. Since the magnetic field acts on moving electric charges (electrons), the Lorentz force will act on them, moving the electrons to the upper edge of the plate, and, consequently, an excess of positive charge will form at the lower edge of the plate. Thus, a potential difference is created between the upper and lower edges. The process of moving electrons will continue until the force acting from the electric field is balanced by the Lorentz force.

Where d- the length of the plate, A is the width of the plate, is the Hall potential difference.

The law of electromagnetic induction.

magnetic flux

where α is the angle between IN and the outer perpendicular to the area of ​​the contour.

With any change in the magnetic flux over time. Thus, the induction emf occurs both when the circuit area changes and when the angle α changes. EMF of induction - the first derivative of the magnetic flux with respect to time:

If the circuit is closed, then an electric current begins to flow through it, called the induction current:

Where R- loop resistance. Current arises due to a change in magnetic flux.

Lenz's rule.

The induction current always has such a direction that the magnetic flux created by this current prevented the change in the magnetic flux that caused this current. The current is directed in such a way as to obstruct the cause that caused it.

Rotation of the frame in a magnetic field.

Let us assume that the frame rotates in a magnetic field with an angular velocity ω, so that the angle α is equal to . in this case the magnetic flux is:

Therefore, a frame rotating in a magnetic field is a source of alternating current.

Eddy currents (Foucault currents).

Eddy currents or Foucault currents occur in the thickness of conductors that are in an alternating magnetic field that creates an alternating magnetic flux. Foucault currents lead to heating of conductors and, consequently, to electrical losses.

The phenomenon of self-induction.

With any change in the magnetic flux, an EMF of induction occurs. Suppose there is an inductor through which an electric current flows. According to the formula, in this case, a magnetic flux is created in the coil. With any change in the current in the coil, the magnetic flux changes and, therefore, an EMF occurs, called the EMF of self-induction ():

Maxwell's system of equations.

The electric field is a set of mutually connected and mutually changing magnetic fields. Maxwell established a quantitative relationship between the quantities characterizing the electric and magnetic fields.

Maxwell's first equation.

From Faraday's law of electromagnetic induction, it follows that with any change in the magnetic flux, an EMF appears. Maxwell suggested that the appearance of an EMF in the surrounding space is associated with the appearance in the surrounding space vortex electromagnetic field. The conducting circuit plays the role of a device that detects the appearance of this electric field in the surrounding space.

The physical meaning of Maxwell's first equation: any change in time of the magnetic field leads to the appearance of a vortex electric field in the surrounding space.

Maxwell's second equation. bias current.

Capacitor included in the circuit direct current. Suppose a circuit containing a capacitor is connected to a DC voltage source. The capacitor is charged and the current in the circuit stops. If a capacitor is included in the circuit AC voltage, then the current in the circuit does not stop. This is due to the process of continuous recharging of the capacitor, as a result of which a time-varying electric field arises between the capacitor plates. Maxwell suggested that a displacement current arises in the space between the capacitor plates, the density of which is determined by the rate of change of the electric field in time. Of all the properties inherent in the electric current, Maxwell attributed only one property to the displacement current: the ability to create a magnetic field in the surrounding space. Maxwell suggested that the conduction current lines do not stop on the capacitor plates, but continuously pass into the displacement current lines. Thus:

Thus, the current density:

where is the conduction current density, is the displacement current density.

According to the total current law:

The physical meaning of Maxwell's second equation: the source of the magnetic field is both the conduction currents and the time-varying electric field.

Maxwell's third equation (Gauss's theorem).

The flow of the electrostatic field strength vector through a closed surface is equal to the charge enclosed inside this surface:

Physical meaning of Maxwell's fourth equation: lines electrostatic fields begin and end on free electric charges. That is, the source of the electrostatic field are electric charges.

Maxwell's Fourth Equation (Magnetic Flux Continuity Principle)

The physical meaning of Maxwell's fourth equation: the lines of the magnetic induction vector do not begin or end anywhere, they are continuous and closed on themselves.

Magnetic properties of substances.

Magnetic field strength.

The main characteristic of the magnetic field is the magnetic induction vector, which determines the force effect of the magnetic field on moving charges and currents, the magnetic induction vector depends on the properties of the medium where the magnetic field is created. Therefore, a characteristic is introduced that depends only on the currents associated with the field, but does not depend on the properties of the medium where the field exists. This characteristic is called the magnetic field strength and is denoted by the letter H.

If we consider a magnetic field in vacuum, then the intensity

where is the vacuum magnetic constant. The unit of tension is ampere/meter.

Magnetic field in matter.

If the entire space surrounding the currents is filled with a homogeneous substance, then the magnetic field induction will change, but the distributed field will not change, that is, the magnetic field induction in the substance is proportional to the magnetic induction in vacuum. - magnetic permeability of the medium. Magnetic permeability shows how many times the magnetic field in a substance differs from the magnetic field in vacuum. The value can be either less or greater than unity, that is, the magnetic field in matter can be either less or greater than the magnetic field in vacuum.

Magnetization vector. Any substance is a magnet, that is, it is capable of acquiring a magnetic moment under the influence of an external magnetic field - to be magnetized. The electrons of atoms under the action of a mutual magnetic field perform precessional motion - such a motion in which the angle between the magnetic moment and the direction of the magnetic field remains constant. In this case, the magnetic moment rotates around the magnetic field with a constant angular velocity ω. Precessional motion is equivalent to circular current. Since the microcurrent is induced by an external magnetic field, then, according to the Lenz rule, the atom has a magnetic field component directed opposite to the external field. The induced component of the magnetic fields adds up and forms its own magnetic field in the substance, directed opposite to the external magnetic field, and, therefore, weakening this field. This effect is called the diamagnetic effect, and substances in which the diamagnetic effect occurs are called diamagnetic substances or diamagnets. In the absence of an external magnetic field, a diamagnet is non-magnetic, since the magnetic moments of the electrons cancel each other out and the total magnetic moment of the atom is zero. Since the diamagnetic effect is due to the action of an external magnetic field on the electrons of the atoms of a substance, diamagnetism is characteristic of ALL SUBSTANCES.

Paramagnets are substances in which, even in the absence of an external magnetic field, atoms and molecules have their own magnetic moment. However, in the absence of an external magnetic field, the magnetic moments of different atoms and molecules are randomly oriented. In this case, the magnetic moment of any macroscopic volume of matter is equal to zero. When a paramagnet is introduced into an external magnetic field, the magnetic moments are oriented along the direction of the external magnetic field, and a magnetic moment arises along the direction of the magnetic field. However, the total magnetic field arising in the paramagnet significantly overlaps the diamagnetic effect.

The magnetization of a substance is the magnetic moment per unit volume of a substance.

where is the magnetic moment of the entire magnet, equal to the vector sum of the magnetic moments of individual atoms and molecules.

The magnetic field in a substance consists of two fields: an external field and a field created by a magnetized substance:

(read "hee") is the magnetic susceptibility of the substance.

Let us substitute formulas (2), (3), (4) into formula (1):

The coefficient is a dimensionless quantity.

For diamagnets (this means that the field of molecular currents is opposite to the external field).

For paramagnets (this means that the field of molecular currents coincides with the external field).

Hence, diamagnets, and for paramagnets. And H .

Hysteresis loop.

Magnetization dependence J from the strength of the external magnetic field H forms the so-called "hysteresis loop". At the beginning (section 0-1) the ferromagnet is magnetized, and the magnetization does not occur linearly, and saturation is reached at point 1, that is, with a further increase in the magnetic field strength, the current growth stops. If we begin to increase the strength of the magnetizing field, then the decrease in magnetization follows the curve 1-2 above the curve 0-1 . When observed residual magnetization (). The presence of residual magnetization is associated with the existence of permanent magnets. The magnetization vanishes at point 3, with a negative value of the magnetic field, which is called the coercive force. With a further increase in the opposite field, the ferromagnet is remagnetized (curve 3-4). Then the ferromagnet can be demagnetized again (curve 4-5-6) and re-magnetize to saturation (curve 6-1). Ferromagnets with a small coercive force (with small values ​​of ) are called soft ferromagnets, and they have a narrow hysteresis loop. ferromagnets that have great importance coercive force are called hard ferromagnets. For each ferromagnet, there is a certain temperature, called the Curie point, at which the ferromagnet loses its ferromagnetic properties.

The nature of ferromagnetism.

According to Weiss. ferromagnets at temperatures below the Curie point have a domain structure, namely, ferromagnets consist of macroscopic regions called domains, each of which has its own magnetic moment, which is the sum of magnetic moments a large number atoms of matter oriented in the same direction. In the absence of an external magnetic field, the domains are randomly oriented and the resulting magnetic moment of the ferromagnet is generally equal to zero. When an external magnetic field is applied, the magnetic moments of the domains begin to orient in the direction of the field. In this case, the magnetization of the substance increases. At a certain value of the strength of the external magnetic field, all domains are oriented along the direction of the field. In this case, the increase in magnetization stops. As the strength of the external magnetic field decreases, the magnetization begins to decrease again, however, not all domains are disoriented simultaneously, so the decrease in magnetization proceeds more slowly, and when the strength of the magnetic field is equal to zero, a sufficiently strong orienting bond remains between some domains, which leads to the presence of residual magnetization coinciding with the direction of the magnetic field that existed before.

To break this bond, it is necessary to apply a magnetic field in the opposite direction. At temperatures above the Curie point, the intensity of thermal motion increases. Chaotic thermal motion breaks the bonds within the domains, that is, the primary orientation of the domains themselves is lost. Thus, a ferromagnet loses its ferromagnetic properties.

Exam questions:

1) Electric charge. The law of conservation of electric charge. Coulomb's law.

2) Electric field strength. The physical meaning of tension. Field strength of a point charge. Force lines of the electric field.

3) Two definitions of potentials. The work of moving the charge in electric field. Relationship between tension and potential. Work on a closed trajectory. The circulation theorem.

4) Electricity. Capacitors. Consistent and parallel connection capacitors. Capacitance of a flat capacitor.

5) Electric current. Conditions for the existence of an electric current. Current strength, current density. Units of current strength.

6) Ohm's law for a homogeneous section of the chain. Electrical resistance. The dependence of the resistance on the length of the cross section of the conductor material. Dependence of resistance on temperature. Series and parallel connection of conductors.

7) Outside forces. EMF. Potential difference and voltage. Ohm's law for an inhomogeneous section of a chain. Ohm's law for a closed circuit.

8) Heating conductors with electric current. Joule-Lenz law. Electric current power.

9) Magnetic field. Ampere power. Left hand rule.

10) Movement of a charged particle in a magnetic field. Lorentz force.

11) Magnetic flux. Faraday's law of electromagnetic induction. Lenz's rule. The phenomenon of self-induction. EMF of self-induction.

dl

RdB, B

It is easy to understand that all elements of the current create a magnetic field of the same direction in the center of the circular current. Since all elements of the conductor are perpendicular to the radius vector, due to which sinα = 1, and are located at the same distance from the center R, then from equation 3.3.6 we obtain the following expression

B = μ 0 μI/2R. (3.3.7)

2. Direct current magnetic field infinite length. Let the current flow from top to bottom. We select several elements with current on it and find their contributions to the total magnetic induction at a point separated from the conductor at a distance R. Each element will give its own vector dB , directed perpendicular to the plane of the sheet "towards us", will also be the direction and the total vector IN . When moving from one element to another, which are located at different heights of the conductor, the angle will change α ranging from 0 to π. Integration will give the following equation

B = (μ 0 μ/4π)2I/R. (3.3.8)

As we said, the magnetic field orients the loop with current in a certain way. This is because the field exerts a force on each element of the frame. And since the currents on opposite sides of the frame, parallel to its axis, flow in opposite directions, the forces acting on them turn out to be multidirectional, as a result of which a torque arises. Ampere established that the force dF , which acts from the side of the field on the conductor element dl , is directly proportional to the current I in the explorer and the vector product of the element of length dl for magnetic induction IN :

dF = I[dl , B ]. (3.3.9)

Expression 3.3.9 is called Ampère's law. The direction of the force vector, which is called by the power of Ampere, are determined according to the rule of the left hand: if the palm of the hand is positioned so that it includes the vector IN , and direct four outstretched fingers along the current in the conductor, then the bent thumb will indicate the direction of the force vector. Ampere's force modulus is calculated by the formula

dF = IBdlsinα, (3.3.10)

Where α is the angle between the vectors d l And B .

Using Ampere's law, you can determine the strength of the interaction of two currents. Imagine two infinite rectilinear currents I 1 And I 2, flowing perpendicular to the plane of Fig. 3.3.4 towards the observer, the distance between which is R. It is clear that each conductor creates a magnetic field in the space around it, which, according to Ampère's law, acts on another conductor located in this field. We choose on the second conductor with current I 2 element d l and calculate the force d F 1 , with which the magnetic field of the conductor with current I 1 affects this element. Lines of magnetic induction of the field that creates a current-carrying conductor I 1, are concentric circles (Fig. 3.3.4).

IN 1

d F 2d F 1

B2

Vector IN 1 lies in the plane of the figure and is directed upwards (this is determined by the rule of the right screw), and its modulus

B1 = (μ 0 μ/4π)2I 1 /R. (3.3.11)

Force d F1 , with which the field of the first current acts on the element of the second current, is determined by the rule of the left hand, it is directed towards the first current. Since the angle between the current element I 2 and vector IN 1 straight line, for the modulus of force, taking into account 3.3.11, we obtain

dF 1= I 2 B 1 dl= (μ 0 μ/4π)2I 1 I 2 dl/R. (3.3.12)

It is easy to show, by reasoning in a similar way, that the force dF2, with which the magnetic field of the second current acts on the same element of the first current

Consider the field created by a current flowing through a thin wire having the shape of a circle of radius R (circular current). Let's determine the magnetic induction at the center of the circular current (Fig. 47.1).

Each current element creates an induction in the center, directed along the positive normal to the circuit. Therefore, vector addition reduces to the addition of their moduli. According to the formula (42.4)

We integrate this expression over the entire contour:

The expression in brackets is equal to the modulus of the dipole magnetic moment (see (46.5)).

Therefore, the magnetic induction at the center of the circular current has the value

From fig. 47.1 it can be seen that the direction of the vector B coincides with the direction of the positive normal to the contour, i.e. with the direction of the vector. Therefore, formula (47.1) can be written in vector form:

Now let's find B on the axis of the circular current at a distance from the center of the circuit (Fig. 47.2). The vectors are perpendicular to the planes passing through the corresponding element and the point at which we are looking for the field. Therefore, they form a symmetrical conical fan (Fig. 47.2, b). From symmetry considerations, we can conclude that the resulting vector B is directed along the contour axis. Each of the constituent vectors contributes to the resulting vector equal in absolute value to the Angle a between and b of the line, therefore

Integrating over the entire contour and replacing with , we get

This formula determines the magnitude of the magnetic induction on the axis of the circular current. Taking into account that the vectors B and have the same direction, we can write formula (47.3) in vector form:

This expression does not depend on the sign of r. Therefore, at the points of the axis that are symmetrical about the center of the current, B has the same magnitude and direction.

When formula (47.4) passes, as it should be, into formula (47.2) for magnetic induction at the center of the circular current.

At large distances from the contour, the denominator can be neglected in comparison with Then formula (47.4) takes the form

similar to expression (9.9) for the electric field strength on the dipole axis.

A calculation beyond the scope of this book gives that any system of currents or moving charges localized in a limited part of space can be assigned a magnetic dipole moment (compare with the electric dipole moment of a system of charges). The magnetic field of such a system at distances large in comparison with its dimensions is determined by using the same formulas by which the field of a system of charges at large distances is determined through the electric dipole moment (see § 10). In particular, the field of a flat contour of any shape at large distances has the form

where is the distance from the contour to the given point, is the angle between the direction of the vector and the direction from the contour to the given point of the field (compare with formula (9.7)). When formula (47.6) gives the module of the vector B the same value as formula (47.5).

On fig. 47.3 shows the lines of magnetic induction of the circular current field. Only lines are shown that lie in one of the planes passing through the current axis. A similar picture takes place in any of these planes.

From everything said in the previous and in this paragraph, it follows that the dipole magnetic moment is very important characteristic circuits with current. This characteristic determines both the field generated by the circuit and the behavior of the circuit in an external magnetic field.

Goal of the work : to study the properties of a magnetic field, to get acquainted with the concept of magnetic induction. Determine the induction of the magnetic field on the axis of the circular current.

Theoretical introduction. A magnetic field. The existence of a magnetic field in nature is manifested in numerous phenomena, the simplest of which are the interaction of moving charges (currents), current and a permanent magnet, two permanent magnets. A magnetic field vector . This means that for its quantitative description at each point in space, it is necessary to set the vector of magnetic induction. Sometimes this quantity is simply called magnetic induction . The direction of the vector of magnetic induction coincides with the direction of the magnetic needle located at the considered point in space and free from other influences.

Since the magnetic field is a force field, it is depicted using lines of magnetic induction - lines, the tangents to which at each point coincide with the direction of the magnetic induction vector at these points of the field. It is customary to draw a number of lines of magnetic induction through a single area perpendicular to , equal to the value of magnetic induction. Thus, the line density corresponds to the value IN . Experiments show that there are no magnetic charges in nature. The consequence of this is that the lines of magnetic induction are closed. The magnetic field is called homogeneous if the induction vectors at all points of this field are the same, that is, they are equal in absolute value and have the same directions.

For a magnetic field, superposition principle: the magnetic induction of the resulting field created by several currents or moving charges is vector sum magnetic induction fields created by each current or moving charge.

In a uniform magnetic field, a straight conductor is acted upon ampere power:

where is a vector equal in absolute value to the length of the conductor l and coinciding with the direction of current I in this conductor.

The direction of the Ampère force is determined right screw rule(vectors , and form a right-hand screw system): if a screw with a right-hand thread is placed perpendicular to the plane formed by the vectors and , and rotate it from to along the smallest angle, then the translational movement of the screw will indicate the direction of the force. In scalar form, relation (1) can be written as follows way:

F = I× l× B× sin a or 2).

From the last relation follows physical meaning of magnetic induction : the magnetic induction of a uniform field is numerically equal to the force acting on a conductor with a current of 1 A, 1 m long, located perpendicular to the direction of the field.

The SI unit for magnetic induction is Tesla (Tl): .

Magnetic field of circular current. An electric current not only interacts with a magnetic field, but also creates it. Experience shows that in a vacuum a current element creates a magnetic field with induction at a point in space

(3) ,

where is the coefficient of proportionality, m 0 \u003d 4p × 10 -7 H / m is the magnetic constant; is a vector numerically equal to the length of the conductor element and coinciding in direction with the elementary current; r is the modulus of the radius vector. Relation (3) was experimentally established by Biot and Savart, analyzed by Laplace, and therefore is called Biot-Savart-Laplace law. According to the right screw rule, the magnetic induction vector at the considered point turns out to be perpendicular to the current element and the radius vector.

Based on the Biot-Savart-Laplace law and the principle of superposition, the calculation of the magnetic fields of electric currents flowing in conductors of arbitrary configuration is carried out by integrating over the entire length of the conductor. For example, the magnetic induction of the magnetic field in the center of a circular coil with a radius R through which current flows I , is equal to:

The lines of magnetic induction of circular and direct currents are shown in Figure 1. On the axis of the circular current, the line of magnetic induction is straight. The direction of magnetic induction is related to the direction of current in the circuit right screw rule. As applied to circular current, it can be formulated as follows: if a screw with a right-hand thread is rotated in the direction of circular current, then the translational movement of the screw will indicate the direction of the magnetic induction lines, the tangents to which at each point coincide with the magnetic induction vector.

First, we will solve a more general problem of finding the magnetic induction on the axis of a coil with current. To do this, let's make a figure 3.8, in which we depict the current element and the magnetic induction vector that it creates on the axis of the circular contour at some point.

Rice. 3.8 Determination of magnetic induction

on the axis of a circular coil with current

The magnetic induction vector created by an infinitesimal circuit element can be determined using the Biot-Savart-Laplace law (3.10).

As follows from the rules of the cross product, the magnetic induction will be perpendicular to the plane in which the vectors and lie, so the vector modulus will be equal to

.

To find the total magnetic induction from the entire circuit, it is necessary to add vectorially from all elements of the circuit, i.e., in fact, calculate the integral over the length of the ring

This integral can be simplified if it is represented as the sum of two components and

In this case, due to symmetry, therefore, the resulting vector of magnetic induction will lie on the axis. Therefore, to find the modulus of the vector, you need to add the projections of all vectors, each of which is equal to

.

Taking into account that and , we obtain the following expression for the integral

It is easy to see that the calculation of the resulting integral will give the length of the contour, i.e. . As a result, the total magnetic induction created by a circular circuit on the axis at the point is equal to

. (3.19)

Using the magnetic moment of the contour, formula (3.19) can be rewritten as follows

.

We now note that the obtained general view solution (3.19) allows us to analyze the limiting case when the point is placed at the center of the coil. In this case, the solution for the magnetic field induction at the center of the ring with current will take the form

The resulting magnetic induction vector (3.19) is directed along the current axis, and its direction is related to the direction of the current by the rule of the right screw (Fig. 3.9).

Rice. 3.9 Determination of magnetic induction

in the center of a circular loop with current

Magnetic field induction at the center of a circular arc

This problem can be solved as a special case of the problem considered in the previous paragraph. In this case, the integral in formula (3.18) should not be taken over the entire circumference, but only over its arc l. And also take into account the fact that the induction is sought in the center of the arc, therefore . As a result, we get

, (3.21)

where is the arc length; is the radius of the arc.

5 Magnetic field induction vector of a point charge moving in vacuum(no formula derivation)

,

where is the electric charge; is the constant nonrelativistic velocity; is the radius vector drawn from the charge to the observation point.

Ampere and Lorentz forces

Experiments on the deflection of a current-carrying frame in a magnetic field show that any current-carrying conductor placed in a magnetic field is subjected to a mechanical force called by the power of Ampere.

Ampère's law determines the force acting on a current-carrying conductor placed in a magnetic field:

; , (3.22)

where is the current strength; - element of the wire length (the vector coincides in direction with the current); - the length of the conductor. The Ampere force is perpendicular to the direction of the current and the direction of the magnetic induction vector.

If a straight conductor with a length is in a uniform field, then the Ampère force modulus is determined by the expression (Fig. 3.10):

The Ampère force is always directed perpendicular to the plane containing the vectors and , and its direction as a result of the cross product is determined by the right screw rule: if you look along the vector , then the rotation from to along the shortest path must be clockwise .

Rice. 3.10 Left hand rule and gimlet rule for Ampère force

On the other hand, to determine the direction of the Ampère force, you can also apply the mnemonic rule of the left hand (Fig. 3.10): you need to place the palm so that the lines of force of magnetic induction enter it, the outstretched fingers show the direction of the current, then the bent thumb will indicate the direction of the Ampère force.

Based on formula (3.22), we find an expression for the interaction force of two infinitely long, straight, parallel conductors, through which currents flow I 1 and I 2 (Fig. 3.11) (Ampère's experiment). The distance between the wires is a.

Let's define the Ampere force d F 21 acting from the side of the magnetic field of the first current I 1 per element l 2d l second current.

The magnitude of the magnetic induction of this field B 1 at the point of location of the element of the second conductor with current is equal to

Rice. 3.11 Ampère's experience in determining the force of interaction

two rectilinear currents

Then, taking into account (3.22), we obtain

. (3.24)

Arguing in exactly the same way, it can be shown that the Ampère force acting from the side of the magnetic field created by the second conductor with current on the element of the first conductor I 1d l, is equal to

,

i.e. d F 12 = d F 21 . Thus, we have derived formula (3.1), which was obtained experimentally by Ampère.

On fig. 3.11 shows the direction of Ampere's forces. In the case when the currents are directed in the same direction, then these are attractive forces, and in the case of currents of different directions, they are repulsive forces.

From formula (3.24), you can get the Ampère force acting per unit length of the conductor

. (3.25)

Thus, the force of interaction of two parallel straight conductors with currents is directly proportional to the product of the magnitudes of the currents and inversely proportional to the distance between them.

Ampère's law states that a force acts on an element with a current placed in a magnetic field. But any current is the movement of charged particles. It is natural to assume that the forces acting on a current-carrying conductor in a magnetic field are due to the forces acting on individual moving charges. This conclusion is confirmed by a number of experiments (for example, an electron beam is deflected in a magnetic field).

Let's find an expression for the force acting on a charge moving in a magnetic field, based on Ampère's law. To do this, in the formula that determines the elementary force of Ampère

we substitute the expression for the strength of the electric current

,

Where I- the strength of the current flowing through the conductor; Q- the value of the total charge that has flowed over time t; q is the charge of one particle; N is the total number of charged particles that have passed through a conductor with a volume V, length l and section S; n is the number of particles per unit volume (concentration); v is the speed of the particle.

As a result, we get:

. (3.26)

The direction of the vector is the same as the direction of the velocity v so they can be swapped.

. (3.27)

This force acts on all moving charges in a conductor with a length and cross section S, the number of such charges:

Therefore, the force acting on one charge will be equal to:

. (3.28)

Formula (3.28) defines Lorentz force, the value of which

where a is the angle between the velocity vectors of the particle and the magnetic induction.

In experimental physics, a situation often occurs when a charged particle moves simultaneously in a magnetic and electric field. In this case, consider the full Lorentz silt as

,

where is the electric charge; is the electric field strength; is the speed of the particle; – magnetic field induction.

Only in a magnetic field on a moving charged particle the magnetic component of the Lorentz force acts (Fig. 3.12)

Rice. 3.12 Lorentz force

The magnetic component of the Lorentz force is perpendicular to the velocity vector and the magnetic induction vector. It does not change the magnitude of the speed, but only changes its direction, therefore, it does no work.

The mutual orientation of the three vectors - , and included in (3.30) is shown in fig. 313 for a positively charged particle.

Rice. 3.13 Lorentz force acting on a positive charge

As can be seen from fig. 3.13, if a particle flies into a magnetic field at an angle to the lines of force, then it moves uniformly in a magnetic field along a circle with a radius and a period of revolution:

where is the particle mass.

Ratio of magnetic moment to mechanical L(momentum) of a charged particle moving in a circular orbit,

where is the particle charge; T - particle mass.

Consider general case the motion of a charged particle in a uniform magnetic field, when its velocity is directed at an arbitrary angle a to the magnetic induction vector (Fig. 3.14). If a charged particle flies into a uniform magnetic field at an angle , then it moves along a helix.

We decompose the velocity vector into components v|| (parallel to the vector ) and v^ (perpendicular to vector ):

Availability v^ leads to the fact that the Lorentz force will act on the particle and it will move along a circle with a radius R in a plane perpendicular to the vector:

.

The period of such motion (the time of one revolution of the particle around the circumference) is equal to

.

Rice. 3.14 Motion along a helix of a charged particle

in a magnetic field

Due to the presence v|| the particle will move uniformly along v|| the magnetic field does not work.

Thus, the particle participates simultaneously in two motions. The resulting trajectory of motion is a helix, the axis of which coincides with the direction of the magnetic field. Distance h between adjacent turns is called helix pitch and equal to:

.

The action of a magnetic field on a moving charge finds great practical application, in particular, in the operation of a cathode ray tube, where the phenomenon of deflection of charged particles by electric and magnetic fields is used, as well as in the operation of mass spectrographs, which make it possible to determine the specific charge of particles ( q/m) and particle accelerators (cyclotrons).

Consider one such example, called the "magnetic bottle" (Figure 3.15). Let an inhomogeneous magnetic field be created by two turns with currents flowing in the same direction. The thickening of the lines of induction in any spatial region means a greater value of the magnitude of the magnetic induction in this region. The magnetic field induction near the coils with current is greater than in the space between them. For this reason, the radius of the helix of the particle trajectory, which is inversely proportional to the modulus of induction, is smaller near the turns than in the space between them. After the particle, moving to the right along the helical line, passes the midpoint, the Lorentz force acting on the particle acquires the component , which slows down its movement to the right. At a certain moment, this component of the force stops the movement of the particle in this direction and pushes it to the left towards coil 1. When a charged particle approaches coil 1, it also slows down and begins to circulate between the coils, being in a magnetic trap, or between “magnetic mirrors”. Magnetic traps are used to hold high-temperature plasma (K) in a certain area of ​​space during controlled thermonuclear fusion.

Rice. 3.15 Magnetic "bottle"

The laws of motion of charged particles in a magnetic field can explain the features of the motion of cosmic rays near the Earth. Cosmic rays are streams of charged particles great energy. When approaching the Earth's surface, these particles begin to experience the action of the Earth's magnetic field. Those of them that are heading towards the magnetic poles will move almost along the lines of the earth's magnetic field and wind around them. Charged particles approaching the Earth near the equator are directed almost perpendicular to the magnetic field lines, their trajectory will be curved. and only the fastest of them will reach the surface of the Earth (Fig. 3.16).

Rice. 3.16 Formation of the Aurora

Therefore, the intensity of cosmic rays reaching the Earth near the equator is noticeably less than near the poles. Related to this is the fact that the aurora is observed mainly in the circumpolar regions of the Earth.

hall effect

In 1880 American physicist Hall conducted the following experiment: he passed a direct electric current I through a plate of gold and measured the potential difference between opposite points A and C on the upper and lower faces (Fig. 3.17).