Magnetic field of the circular current axis. The magnetic field of a circular coil with current

The gimlet's rule. A clear idea of ​​the nature of the magnetic field that arises around any conductor through which an electric current flows is given by pictures of the lines of the magnetic field, obtained as described in § 122.

On fig. 214 and 217 show such line patterns obtained with iron filings for the field of a long rectilinear conductor and for the field of a circular coil with current. Looking carefully at these figures, we first of all pay attention to the fact that the lines of the magnetic field have the form of closed lines. This property of them is common and very important. Whatever the shape of the conductors through which the current flows, the lines of the magnetic field created by it are always closed on themselves, that is, they have neither beginning nor end. This is the essential difference between the magnetic field and the electric field, whose lines, as we saw in § 18, always begin on some charges and end on others. We have seen, for example, that the lines of the electric field end on the surface of a metallic body, which turns out to be charged, and inside the metal electric field does not penetrate. The observation over magnetic field shows, on the contrary, that his lines never end on any surface. When a magnetic field is created by permanent magnets, it is not so easy to see that in this case the magnetic field does not end at the surface of the magnets, but penetrates into them, because we cannot use iron filings to observe what is happening inside the iron. However, even in these cases, a careful study shows that the magnetic field passes through the iron, and its lines close on themselves, that is, they are closed.

Rice. 217. Picture of the lines of the magnetic field of a circular coil with current

This important difference between electric and magnetic fields is due to the fact that in nature there are electric charges and there are no magnetic ones. Therefore the lines electric field go from charge to charge, but the magnetic field has neither beginning nor end, and its lines have a closed character.

If, in experiments that give pictures of the magnetic field of the current, we replace the filings with small magnetic arrows, then their northern ends will indicate the direction of the field lines, that is, the direction of the field (§ 122). Rice. 218 shows that when the direction of the current changes, the direction of the magnetic field also changes. The relationship between the direction of the current and the direction of the field it creates is easy to remember using the gimlet rule (Fig. 219).

Rice. 218. The relationship between the direction of the current in a straight conductor and the direction of the lines of the magnetic field created by this current: a) the current is directed from top to bottom; b) the current is directed from the bottom up

Rice. 219. To the gimlet rule

If you screw in the gimlet (right screw) so that it goes in the direction of the current, then the direction of rotation of its handle will indicate the direction of the field (the direction of the field lines).

In this form, this rule is especially convenient for establishing the direction of the field around long straight conductors. In the case of a ring conductor, the same rule applies to each section of it. It is even more convenient for ring conductors to formulate the gimlet rule as follows:

If you screw in the gimlet so that it goes in the direction of the field (along the lines of the field), then the direction of rotation of its handle will indicate the direction of the current.

It is easy to see that both formulations of the gimlet rule are completely equivalent and can be equally applied to determining the relationship between the direction of the current and the direction of the magnetic field induction for any form of conductors.

124.1. Indicate which of the poles of the magnetic needle in Fig. 73 north and which south.

124.2. Wires from a current source are connected to the vertices of the wire parallelogram (Fig. 220). What is the magnetic induction of the field at the center of the parallelogram? How will the magnetic induction be directed at the point if the branch of the parallelogram is made of copper wire, and the branch is made of aluminum wire of the same cross section?

Rice. 220. To exercise 124.2

124.3. Two long straight conductors and not lying in the same plane are perpendicular to each other (Fig. 221). The point lies in the middle of the shortest distance between these lines - the segment. The currents in the conductors and are equal and have the direction indicated in the figure. Find graphically the direction of the vector at the point . Indicate in which plane this vector lies. What angle does it form with the plane passing through and?

Rice. 221. To exercise 124.3

124.4. Perform the same construction as in problem 124.3, changing to the opposite: a) the direction of the current in the conductor; b) the direction of the current in the conductor; c) direction of current in both conductors.

124.5. Two circular turns - vertical and horizontal - carry currents of the same strength (Fig. 222). Their directions are indicated by arrows in the figure. Find graphically the direction of the vector in the common center of the turns. At what angle will this vector be inclined to the plane of each of the circular turns? Perform the same construction by reversing the direction of the current first in the vertical turn, then in the horizontal, and finally in both.

Rice. 222. To exercise 124.5

Measurements of magnetic induction in different points the fields around the conductor through which the current flows show that the magnetic induction at each point is always proportional to the strength of the current in the conductor. But for a given current strength, the magnetic induction at different points of the field is different and extremely complex depends on the size and shape of the conductor through which the current passes. We confine ourselves to one important case where these dependencies are relatively simple. This is the magnetic field inside the solenoid.

Let a wire coil of radius R be located in the YZ plane, along which a current of force Á flows. We are interested in the magnetic field that creates the current. The lines of force near the coil are: Polarization of light. Wave optics

The general picture of the lines of force is also visible (Fig. 7.10). Addition harmonic vibrations If the system participates simultaneously in several oscillatory processes, then the addition of oscillations means finding a law that describes the resulting oscillatory process.

In theory, we would be interested in the field , but it is impossible to specify the field of this coil in elementary functions. It can only be found on the axis of symmetry. We are looking for a field at points (x, 0, 0).

The direction of the vector is determined vector product. The vector has two components: and . When we start summing these vectors, then all the perpendicular components add up to zero. . And now we write: , = , and . , and finally 1), .

We got this result:

And now, as a test, the field at the center of the coil is: .

The work done when moving a current-carrying circuit in a magnetic field.

Consider a segment of a current-carrying conductor that can move freely along two guides in an external magnetic field (Fig. 9.5). The magnetic field will be considered uniform and directed at an angle α in relation to the normal to the plane of movement of the conductor.

Fig.9.5. A segment of a current-carrying conductor in a uniform magnetic field.

As can be seen from Fig. 9.5, the vector has two components and , of which only the component creates a force acting in the plane of conductor movement. In absolute value, this force is equal to:

,

where I- current strength in the conductor; l- conductor length; B– magnetic field induction.

The work of this force on the elementary path of displacement ds there is:

Work lds equal to area dS, swept by the conductor during movement, and the value BdScosα equal to the flux of magnetic induction dФ through this area. Therefore, we can write:

dA=IdФ.

Considering a segment of a current-carrying conductor as part of a closed circuit and integrating this relation, we find the work when moving the current-carrying circuit in a magnetic field:

A \u003d I (F 2 - F 1)

where F 1 and F 2 denote the flow of magnetic field induction through the contour area, respectively, in the initial and final positions.

Movement of charged particles

Uniform magnetic field

Consider special case when there is no electric field, but there is a magnetic field. Let us assume that a particle with an initial velocity u0 enters a magnetic field with induction B. This field will be assumed to be uniform and directed perpendicular to the velocity u0.

The main features of motion in this case can be clarified without resorting to a complete solution of the equations of motion. First of all, we note that the Lorentz force acting on a particle is always perpendicular to the velocity of the particle. This means that the work of the Lorentz force is always zero; consequently, the absolute value of the speed of the particle, and hence the energy of the particle remain constant during the motion. Since the speed of the particle u does not change, the value of the Lorentz force

remains constant. This force, being perpendicular to the direction of motion, is a centripetal force. But motion under the action of a centripetal force of constant magnitude is motion in a circle. The radius r of this circle is determined by the condition

If the electron energy is expressed in eV and is equal to U, then

(3.6)

and therefore

The circular motion of charged particles in a magnetic field has important feature: the time of complete revolution of the particle in a circle (the period of motion) does not depend on the energy of the particle. Indeed, the period of revolution is equal to

Substituting here instead of r its expression according to the formula (3.6), we have:

(3.7)

The frequency turns out to be

For a given type of particle, both the period and the frequency depend only on the magnetic field induction.

Above, we assumed that the direction of the initial velocity is perpendicular to the direction of the magnetic field. It is not difficult to imagine what character the movement will have if starting speed particle makes some angle with the direction of the field.
In this case, it is convenient to decompose the velocity into two components, one of which is parallel to the field and the other is perpendicular to the field. The Lorentz force acts on the particle, and the particle moves along a circle lying in a plane perpendicular to the field. The component Ut does not cause the appearance of an additional force, since the Lorentz force when moving parallel to the field is equal to zero. Therefore, in the direction of the field, the particle moves by inertia uniformly, with a speed

As a result of the addition of both motions, the particle will move in a cylindrical spiral.

The screw pitch of this spiral is

substituting its expression (3.7) instead of T, we have:

Hall effect - the phenomenon of the occurrence of a transverse potential difference (also called the Hall voltage) when a conductor with a direct current is placed in a magnetic field. Discovered by Edwin Hall in 1879 in thin gold plates. Properties

In its simplest form, the Hall effect looks like this. Let an electric current flow through a metal bar in a weak magnetic field under the action of tension. The magnetic field will deflect charge carriers (for definiteness, electrons) from their movement along or against the electric field to one of the faces of the bar. In this case, the criterion of smallness will be the condition that in this case the electron does not begin to move along the cycloid.

Thus, the Lorentz force will lead to the accumulation of a negative charge near one face of the bar, and a positive charge near the opposite. The accumulation of charge will continue until the resulting electric field of charges compensates for the magnetic component of the Lorentz force:

The speed of electrons can be expressed in terms of current density:

where is the concentration of charge carriers. Then

The coefficient of proportionality between and is called coefficient(or constant) Hall. In this approximation, the sign of the Hall constant depends on the sign of the charge carriers, which makes it possible to determine their type for a large number of metals. For some metals (for example, such as lead, zinc, iron, cobalt, tungsten), a positive sign is observed in strong fields, which is explained in the semiclassical and quantum theories solid body.

Electromagnetic induction - the phenomenon of the occurrence of an electric current in a closed circuit when the magnetic flux passing through it changes.

Electromagnetic induction was discovered by Michael Faraday on August 29 [ source not specified 111 days] 1831. He found that the electromotive force that occurs in a closed conducting circuit is proportional to the rate of change of the magnetic flux through the surface bounded by this circuit. Value electromotive force(EMF) does not depend on what causes the change in flux - a change in the magnetic field itself or the movement of a circuit (or part of it) in a magnetic field. The electric current caused by this EMF is called the induction current.

The intensity of the magnetic field on the axis of the circular current (Fig. 6.17-1) created by the conductor element idl, is equal to

because in this case

Rice. 6.17. Magnetic field on the circular current axis (left) and electric field on the dipole axis (right)

When integrating over a coil, the vector will describe a cone, so that as a result, only the field component along the axis will “survive”. 0z. Therefore, it suffices to sum the value

Integration

is performed taking into account the fact that the integrand does not depend on the variable l, a

Accordingly, complete magnetic induction on the axis of the coil is equal to

In particular, at the center of the coil ( h= 0) the field is

At a great distance from the coil ( h >> R) we can neglect the unit under the radical in the denominator. As a result, we get

Here we have used the expression for the modulus of the magnetic moment of the coil P m equal to the product I to the area of ​​the coil The magnetic field forms a right-handed system with a circular current, so that (6.13) can be written in vector form

For comparison, we calculate the field of an electric dipole (Fig. 6.17-2). Electric fields from positive and negative charges equal, respectively,

so the resulting field will be

Over long distances ( h >> l) we have from here

Here we have used the concept of the dipole electric moment vector introduced in (3.5). Field E parallel to the dipole moment vector, so that (6.16) can be written in vector form

The analogy with (6.14) is obvious.

lines of force magnetic field of a circular coil with current are shown in fig. 6.18. and 6.19

Rice. 6.18. Lines of force of the magnetic field of a circular coil with current at short distances from the wire

Rice. 6.19. Distribution of lines of force of the magnetic field of a circular coil with current in the plane of its symmetry axis.
The magnetic moment of the coil is directed along this axis

On fig. 6.20 presents the experience of studying the distribution of magnetic field lines around a circular coil with current. A thick copper conductor is passed through holes in a transparent plate, on which iron filings are poured. After switching on direct current with a force of 25 A and tapping on a sawdust plate form chains that repeat the shape of the magnetic field lines.

The magnetic lines of force for the coil, the axis of which lies in the plane of the plate, thicken inside the coil. Near the wires, they have an annular shape, and away from the coil, the field quickly decreases, so that the sawdust is practically not oriented.

Rice. 6.20. Visualization of magnetic field lines around a circular coil with current

Example 1 An electron in a hydrogen atom moves around a proton in a circle of radius a B\u003d 53 pm (this value is called the Bohr radius after one of the creators quantum mechanics, who first calculated the radius of the orbit theoretically) (Fig. 6.21). Find the strength of the equivalent circular current and magnetic induction AT fields in the center of the circle.

Rice. 6.21. Electron in a hydrogen atom and B = 2.18 10 6 m/s. A moving charge creates a magnetic field at the center of the orbit

The same result can be obtained using expression (6.12) for the field in the center of the coil with a current, the strength of which we found above

Example 2 An infinitely long thin conductor with a current of 50 A has an annular loop with a radius of 10 cm (Fig. 6.22). Find the magnetic induction at the center of the loop.

Rice. 6.22. A magnetic field long conductor with circular loop

Solution. The magnetic field in the center of the loop is created by an infinitely long straight wire and an annular coil. The field from a rectilinear wire is directed orthogonally to the plane of the figure "on us", its value is equal to (see (6.9))

The field generated by the ring-shaped part of the conductor has the same direction and is equal (see 6.12)

The total field at the center of the coil will be equal to

Additional Information

http://n-t.ru/nl/fz/bohr.htm - Niels Bohr (1885–1962);

http://www.gumer.info/bibliotek_Buks/Science/broil/06.php - Bohr's theory of the hydrogen atom in Louis de Broglie's book "Revolution in Physics";

http://nobelprize.org/nobel_prizes/physics/laureates/1922/bohr-bio.html - Nobel Prizes. Nobel Prize in Physics 1922 Niels Bohr.

The movement of an electric charge means the movement of the electric force field inherent in the charge. The kinetics of the potential electric field manifests itself in the form of an emerging eddy magnetic field covering the current. To detect a magnetic field, a ferromagnetic rod with freedom of rotation (for example, a magnetic needle) can serve as an indicator.

Like an electric field, a magnetic field is also characterized by strength , however, the definition of this concept is no longer associated with the charge, as it was in the case of a potential electric field, but with the current, i.e. movement of electricity.

The directed translational movement of charges and the vortex magnetic field, which reflect the movement of the electric field of these charges, are two sides of a single electromagnetic process, called electric current.

An experimental study of the magnetic field of currents was carried out in 1820 by French physicists J. Biot and F. Savard, and P. Laplace theoretically generalized the results of these measurements, resulting in a formula (for a magnetic field in vacuum):

, (1)

where 1/4 is the coefficient of proportionality, depending on the choice of units of measurement; I– current strength; is a vector coinciding with the elementary section of the current (Fig. 3); is a vector drawn from the current element to the point at which the

As can be seen from expression (1), the vector
directed perpendicular to the plane passing through and the point at which the field is calculated, and so that the rotation around in the direction
associated with right screw rule (see Fig. 3). For module dH you can write the following expression:

, (2)

where  is the angle between the vectors and .

R

Calculate according to the formula dH for the case   /2:

. (3)

We integrate this expression over the entire contour, taking into account that r  R:

H 
. (4)

If the contour is n turns, then the magnetic field strength in its center will be equal to

H  . (5)

Description of equipment and measurement method

The purpose of this work is to determine the value H 0. For measurement H 0, a device called a tangent galvanometer is used, which consists of a ring-shaped conductor or a very flat coil of large radius. The plane of the coil is located vertically, and by rotation about the vertical axis it can be given any position.

A compass with a very short magnetic needle is fixed in the center of the coil. Rice. 5 gives the cross section of the device by a horizontal plane passing through the center of the coil, where NS is the direction of the magnetic meridian, AD is the section of the coil by the horizontal plane, ab is the magnetic needle of the compass.

In the absence of current in the coil, only the Earth's magnetic field acts on the arrow ab and the arrow is set in the direction of the magnetic meridian NS.

If a current is passed through the coil, then the arrow deviates by an angle . Now the magnetic needle ab is under the action of two fields: the Earth's magnetic field ( ) and the magnetic field created by the current ( ).

Under conditions of alignment of the coil with the plane of the meridian, the vectors and are mutually perpendicular, then (see Fig. 5):

;
. (6)

Since the length of the magnetic needle ab is small compared to the radius of the coil, then within the limits of the arrow H can be considered constant (the field is uniform) and equal to its value at the center of the coil, determined by formula (5).

Solving equations (5) and (6) together, we obtain:

. (7)

This calculation formula is used to determine H 0 in this work.

Consider the field created by the current I, flowing along a thin wire having the shape of a circle of radius R .

We define the magnetic induction on the axis of the conductor with current at a distance X from the plane of the circular current. The vectors are perpendicular to the planes passing through the corresponding and . Therefore, they form a symmetrical conical fan. It can be seen from symmetry considerations that the resulting vector is directed along the axis of the circular current. Each of the vectors contributes equal to , and cancel each other out. But,, and because the angle between and α is right, then we get

,

Substituting into and, integrating over the entire contour , we obtain an expression for finding magnetic induction circular current :

,

For , we get magnetic induction at the center of the circular current :

Note that the numerator is the magnetic moment of the circuit. Then, at a great distance from the contour, at , the magnetic induction can be calculated by the formula:

The lines of force of the circular current magnetic field are clearly visible in the experiment with iron filings.

The magnetic moment of a coil with current is physical quantity, like any other magnetic moment, characterizes magnetic properties this system. In our case, the system is represented by a circular loop with current. This current creates a magnetic field that interacts with an external magnetic field. It can be either the field of the earth, or the field of a constant or electromagnet.

A circular coil with current can be represented as a short magnet. Moreover, this magnet will be directed perpendicular to the plane of the coil. The location of the poles of such a magnet is determined using the gimlet rule. According to which the north plus will be behind the plane of the coil if the current in it moves clockwise.

This magnet, that is, our circular coil with current, like any other magnet, will be affected by an external magnetic field. If this field is uniform, then a torque will arise that will tend to turn the coil. The field will rotate the coil so that its axis is located along the field. In this case, the lines of force of the coil itself, like a small magnet, must coincide in direction with the external field.

If the external field is not uniform, then the torque will be added and forward movement. This movement will arise due to the fact that areas of the field with a higher induction will attract our magnet in the form of a coil more than areas with a lower induction. And the coil will begin to move towards the field with greater induction.

The magnitude of the magnetic moment of a circular coil with current can be determined by the formula.