Magnetic field strength on the axis of a circular current (Fig. 6.17-1) created by a conductor element IDl, is equal

because in this case

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

When integrated over a turn, the vector will describe a cone, so that as a result only the field component along the axis will “survive” 0z. Therefore, it is enough to sum up the value

Integration

is carried out taking into account the fact that the integrand does not depend on the variable l, A

Accordingly, complete magnetic induction on the coil axis equal to

In particular, in the center of the turn ( h= 0) field is equal

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 magnitude of the magnetic moment of a turn Р m, equal to the product I per area of ​​the turn. The magnetic field forms a right-handed system with the circular current, so (6.13) can be written in vector form

For comparison, let's calculate the field of an electric dipole (Fig. 6.17-2). The electric fields from positive and negative charges are equal, respectively,

so the resulting field will be

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

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

The analogy with (6.14) is obvious.

Power lines circular magnetic field with current are shown in Fig. 6.18. and 6.19

Rice. 6.18. Magnetic field lines of a circular coil with current at short distances from the wire

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

In Fig. 6.20 presents an experiment in 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 turning on a direct current of 25 A and tapping on the plate, the sawdust forms chains that repeat the shape of the magnetic field lines.

The magnetic lines of force for a coil whose axis lies in the plane of the plate are concentrated inside the coil. Near the wires they have a ring shape, and far 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= 53 pm (this value is called the Bohr radius after one of the creators of quantum mechanics, who was the first to calculate the orbital radius theoretically) (Fig. 6.21). Find the strength of the equivalent circular current and magnetic induction IN 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 at 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 a ring-shaped loop with a radius of 10 cm (Fig. 6.22). Find the magnetic induction at the center of the loop.

Rice. 6.22. Magnetic field of a long conductor with a circular loop

Solution. The magnetic field at the center of the loop is created by an infinitely long straight wire and a ring coil. The field from a straight wire is directed orthogonally to the plane of the drawing “at us”, its value is equal to (see (6.9))

The field created by the ring-shaped part of the conductor has the same direction and is equal to (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.

Magnetism

Characteristics of the magnetic field (strength, induction). Lines of force, tension and magnetic induction of direct current, in the center of circular current.

MAGNETIC FIELD INDUCTION

Magnetic induction- vector quantity: at each point in the field, the magnetic induction vector is directed tangentially to the magnetic lines of force.

The presence of a magnetic field is detected by the force exerted on current-carrying conductors or permanent magnets introduced into it. The name “magnetic field” is associated with the orientation of the magnetic needle under the influence of the field created by the current. This phenomenon was first discovered by the Danish physicist H. Oersted (1777-1851).

When studying the magnetic field, two facts were established:

1. The magnetic field acts only on moving charges;

2. Moving charges, in turn, create a magnetic field.

Thus, we see that the magnetic field differs significantly from the electrostatic field, which acts on both moving and stationary charges.

A magnetic field – a force field acting on moving electric charges and on bodies with a magnetic moment.

Any magnetic field has energy that manifests itself when interacting with other bodies. Under the influence of magnetic forces, moving particles change the direction of their flow. A magnetic field appears only around those electric charges that are in motion. Any change in the electric field entails the appearance of magnetic fields.

The opposite statement is also true: a change in the magnetic field is a prerequisite for the emergence of an electric field. Such close interaction led to the creation of the theory of electromagnetic forces, with the help of which various physical phenomena are successfully explained today.

Magnetic field strength- vector physical quantity equal to the difference in the magnetic induction vector B and magnetization vector M . Usually indicated by the symbol N .

Magnetic field of direct and circular currents.

Magnetic field of direct current, i.e. current flowing through a straight wire of infinite length

Magnetic field of the current element ,dl – wire length element

Having integrated the last expression within these limits, we obtain a magnetic field equal to:

Direct current magnetic field

from all current elements a cone of vectors will be formed, the resulting vector is directed upward along the Z axis. Let's add the projections of the vectors onto the Z axis, then each projection has the form:

Angle between and radius vector r equal to .

Integrating over dl and taking into account , we get

- magnetic field on the axis of the circular coil

Magnetic field lines

Magnetic field lines are circles. Magnetic field lines are lines drawn so that the tangents to them at each point indicate the direction of the field at that point. The field lines are drawn so that their density, that is, the number of lines passing through a unit area, gives the module of the magnetic induction of the magnetic field. Thus, we will receive “magnetic maps”, the method of construction and use of which is similar to “electric maps”. The main difference between the magnetic field is that its lines are always closed. constructing magnetic field lines

Magnetic field in the center of a circular conductor carrying current.

dl

RdB,B

It is easy to understand that all current elements 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. Let us select several elements with current on it and find their contributions to the total magnetic induction at a point located at a distance from the conductor R. Each element will give its own vector dB , directed perpendicular to the plane of the sheet “towards us”, the total vector will also be in the same direction 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 current-carrying frame in a certain way. This happens 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 in different directions, as a result of which a torque arises. Ampere established that the force dF , which acts from the field side on the conductor element dl , is directly proportional to the current strength I in the conductor and the cross product of an element of length dl for magnetic induction IN :

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

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

dF = IBdlsinα, (3.3.10)

Where α – angle between vectors d l And B .

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

IN 1

d F 2d F 1

B 2

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

B 1 = (μ 0 μ/4π)2I 1 /R. (3.3.11)

Force d F 1 , with which the field of the first current acts on the element of the second current, is determined by the left-hand rule, it is directed towards the first current. Since the angle between the current element I 2 and vector IN 1 direct, 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 similar reasoning, that the force dF 2, with which the magnetic field of the second current acts on the same element of the first current

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

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

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

The direction of a vector is determined by the cross product. The vector has two components: and . When we start summing these vectors, all the perpendicular components add up to zero. . And now we write: , = , a . , and finally1), .

We got the following result:

And now, as a check, the field in the center of the turn is equal to: .

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

Let's consider a piece of conductor carrying current that can move freely along two guides in an external magnetic field (Fig. 9.5). We will consider the magnetic field to be uniform and directed at an angle α in relation to the normal to the plane of movement of the conductor.

Fig.9.5. A section of conductor carrying current 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 movement of the conductor. In absolute value this force is equal to:

,

Where I– current strength in the conductor; l– length of the conductor; B– magnetic field induction.

The work of this force on the elementary path of movement ds There is:

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

dA=IdФ.

Considering a section of a conductor with current as part of a closed loop and integrating this relationship, we find the work done when moving a loop with current in a magnetic field:

A = I(Ф 2 – Ф 1)

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

Movement of charged particles

Uniform magnetic field

Let's consider a 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. We will consider this field 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 speed of the particle. This means that the work done by the Lorentz force is always zero; therefore, the absolute value of the particle’s speed, and therefore the energy of the particle, remains constant during movement. Since the particle velocity u does not change, the magnitude of the Lorentz force

remains constant. This force, being perpendicular to the direction of movement, is a centripetal force. But motion under the influence of a constant centripetal force 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 an important feature: the time of complete revolution of a particle in a circle (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 formula (3.6), we have:

(3.7)

The frequency turns out to be equal

For a given type of particle, both period and 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 motion will have if the initial velocity of the particle makes a certain angle with the direction of the field.
In this case, it is convenient to decompose the speed 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 in a circle lying in a plane perpendicular to the field. The component Ut does not cause the appearance of additional force, since the Lorentz force when moving parallel to the field is 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 movements, the particle will move along a cylindrical spiral.

The pitch of the screw of this spiral is equal to

substituting its expression (3.7) for T, we have:

The Hall effect is the phenomenon of the appearance of a transverse potential difference (also called the Hall voltage) when a conductor with direct current is placed in a magnetic field. Discovered by Edwin Hall in 1879 in thin plates of gold. 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 influence of tension. The magnetic field will deflect charge carriers (electrons to be specific) from their movement along or against the electric field to one of the faces of the beam. In this case, the criterion for smallness will be the condition that 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 side 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 proportionality coefficient between and is called coefficient(or constant) Hall. In this approximation, the sign of the Hall constant depends on the sign of charge carriers, which makes it possible to determine their type for a large number of metals. For some metals (for example, lead, zinc, iron, cobalt, tungsten), a positive sign is observed in strong fields, which is explained in the semiclassical and quantum theories of solids.

Electromagnetic induction- the phenomenon of the occurrence of 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 discovered that the electromotive force arising in a closed conducting circuit is proportional to the rate of change of the magnetic flux through the surface bounded by this circuit. The magnitude of the electromotive force (EMF) does not depend on what is causing the flux change - a change in the magnetic field itself or the movement of the circuit (or part of it) in the magnetic field. The electric current caused by this emf is called induced current.

Goal of the work : study the properties of the magnetic field, become familiar with the concept of magnetic induction. Determine the magnetic field induction 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 magnetic induction vector. Sometimes this quantity is simply called magnetic induction . The direction of the magnetic induction vector coincides with the direction of the magnetic needle located at the point in space under consideration and free from other influences.

Since the magnetic field is a force field, it is depicted using magnetic induction lines – 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 through a single area perpendicular to , a number of magnetic induction lines equal to the magnitude of the magnetic induction. Thus, the density of the lines corresponds to the value IN . Experiments show that there are no magnetic charges in nature. The consequence of this is that the magnetic induction lines are closed. The magnetic field is called homogeneous, if the induction vectors at all points of this field are the same, that is, equal in magnitude and have the same directions.

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

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

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

The direction of the Ampere force is determined right screw rule(vectors , and form a right-handed screw system): if a screw with a right-hand thread is placed perpendicular to the plane formed by the vectors and , and rotated from to at 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 it follows physical meaning of magnetic induction : 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 of magnetic induction is Tesla (T): .

Magnetic field of circular current. 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 proportionality coefficient, m 0 =4p×10 -7 H/m– magnetic constant, – vector numerically equal to the length of the conductor element and coinciding in direction with the elementary current, – radius vector drawn from the conductor element to the field point under consideration, r – modulus of the radius vector. Relationship (3) was established experimentally by Biot and Savart, analyzed by Laplace and is therefore called Biot-Savart-Laplace law. According to the rule of the right screw, the magnetic induction vector at the point under consideration 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 magnetic fields of electric currents flowing in conductors of arbitrary configuration are calculated by integrating over the entire length of the conductor. For example, the magnetic induction of a magnetic field at the center of a circular coil with a radius R , through which current flows I , is equal to:

The magnetic induction lines of circular and forward currents are shown in Figure 1. On the axis of the circular current, the magnetic induction line is straight. The direction of magnetic induction is related to the direction of current in the circuit right screw rule. When applied to circular current, it can be formulated as follows: if a screw with a right-hand thread is rotated in the direction of the 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.