The solenoid is a wire wound evenly in the form of a spiral onto a common cylindrical frame (see Fig. 12.14). The product (IN) of the number of turns of a single-layer winding of a solenoid and the current flowing around the turns is called the number ampere-turns.
Solenoids are designed to create a fairly strong magnetic field in a small volume of space. When the turns are tightly wound, the solenoid field is equivalent to the field of a system of circular parallel currents with a common axis. If the diameter d of the solenoid turns is many times smaller than its length (d l), then the solenoid is considered infinitely long (or thin). The magnetic field of such a solenoid is almost entirely concentrated inside, and the magnetic induction vector 
inside it is directed along the axis of the solenoid and is connected with the direction of the current by the rule of the right screw.
R 
is. 12.15
Consider an imaginary closed loop 
inside the solenoid (Fig. 12.15). This circuit does not cover currents, therefore, according to the circulation theorem
Let us divide this circular integral into four integrals (along the sides of the contour) and take into account that on the segments (1-2) and (3-4) the vector 
perpendicular 
, so the scalar product ( 
,
) here vanishes. The field induction at all points of the segment (2-3) is the same and equal to 
23, and on the segment (4-1) 
41, with l 23 = l 41 = l.
Thus, going around the contour clockwise, we get
Because l 0, then IN 23 = IN 41 = IN inside.
Since the circuit inside the solenoid was chosen arbitrarily, the result obtained is valid for any internal points of the solenoid, that is, the field inside the solenoid is uniform:
inside = const.
To find the induction value of this field, consider the circuit L 2 (a –b –c –d –a), covering N turns with current (Fig. 12.15). According to the circulation theorem (and based on previous arguments), we obtain the relation
The field outside an infinitely long solenoid is very weak ( 
outside =0), it can be neglected, therefore,
(12.35)
Where n=N/l- number of turns per unit
solenoid length.
Thus, the magnetic field induction inside an infinitely long solenoid is the same in magnitude and direction and is proportional to the number of ampere-turns per unit length of the solenoid.
Symmetrically located turns make the same contribution to the magnetic induction on the axis of the solenoid, therefore, at the end of a semi-infinite solenoid on its axis, the magnetic induction is equal to half the value given by formula (12.35), i.e.
(12.36)
Practically, if ( l d), then formula (12.35) is valid for points in the middle part of the solenoid, and formula (12.36) is valid for points on the axis near its ends.
Applying the Biot-Savart-Laplace law, one can find the magnetic induction of the field of a solenoid of finite length (Fig. 12.16) at an arbitrary point A on its axis:
(12.37)
G 
de 
- angles between the axis of the solenoid and the radius vector drawn from the point in question to the ends of the solenoid.
The field of such a solenoid is non-uniform, the magnitude of the induction depends on the position of the point A and solenoid length. For an infinitely long solenoid 
,
, and formula (12.37) goes into formula (12.35).
Let's find the magnetic field induction inside a solenoid - a coil whose diameter is significantly greater than its length l. We will consider the field inside the coil to be uniform, and far from the coil to be negligible. Let's select a bypass circuit L in the form of a rectangle 1-2-3-4 (see figure). Let us first find the circulation of the vector IN. Let us write the circulation integral into the expression . Let's split the integral along the contour L into four integrals: 1-2, 2-3, 3-4, 4-1.
Circuit 12341 covers N coil turns in each of which the current I. Thus, it follows from the theorem that B×l = m o NI. From here we will find IN.
Topic 9. Question 8.
Magnetic induction vector flux (magnetic flux)
Let's imagine some closed surface in a magnetic field. Magnetic induction lines are always closed, they have no beginning and end. Therefore, the number of lines entering the surface will be equal to the number of lines leaving it. The magnetic flux is proportional to the number of induction lines, therefore the flux will be zero. The equality to zero of the magnetic flux through any closed surface indicates that the magnetic field has no sources of this field (magnetic charges do not exist). Thus, the magnetic field is vortex, i.e. having no sources of its formation.
Topic 10. Question 1.
Topic 10. Question 2.
Magnetic forces.
Using the expression for the Ampere force, we find the force of interaction between two infinitely long straight conductors with currents I 1 And I 2.
We considered the action of a conductor carrying current I 1 to a current-carrying conductor I 2. In accordance with Newton's III law, the second conductor acts on the first with the same force.
Topic 10. Question 3.
Obtaining an expression for the torque acting on a current-carrying circuit in a magnetic field.
Taking into account the vector nature of these quantities, we can write the general expression:
Topic 10. Question 4.
Circuit with current in a magnetic field.
Homogeneous field.
Thus, in external homogeneous magnetic field under the influence of magnetic forces:
1) a freely oriented circuit with current will rotate until the plane of the circuit is perpendicular to the induction lines, i.e. until the magnetic moment becomes parallel to the induction lines and
2) tensile forces will act on the contour.
Inhomogeneous field.
In a non-uniform magnetic field, in addition to the above forces that rotate and stretch the circuit, a force component appears that tends to move the circuit. If the circuit turns out to be oriented with its magnetic moment along the field (as in the figure), then the force component F 1 will stretch the contour, and the component F 2 will pull the circuit into an area of stronger field. If the circuit is in a field in such a way that its magnetic moment is directed against the field, this position of the circuit will be unstable. The circuit will unfold along the field and will be drawn into the area of a stronger field.
Let us give an expression for the force acting on a circuit with a current in a non-uniform magnetic field, the induction of which varies only along one coordinate X.
Topic 10. Question 5.
Let us calculate, using the circulation theorem, the magnetic field induction inside solenoid. Consider a solenoid with length l having N turns through which current flows (Fig. 175). We consider the length of the solenoid to be many times greater than the diameter of its turns, i.e. the solenoid in question is infinitely long. Experimental study of the magnetic field of the solenoid (see Fig. 162, b) shows that inside the solenoid the field is uniform, outside the solenoid it is inhomogeneous and very weak.
In Fig. 175 shows the magnetic induction lines inside and outside the solenoid. The longer the solenoid, the less magnetic induction outside it. Therefore, we can approximately assume that the field of an infinitely long solenoid is concentrated entirely inside it, and the field outside the solenoid can be neglected.
To find magnetic induction IN select a closed rectangular contour ABCDA as shown in fig. 175. Vector circulation IN in a closed loop ABCDA covering everything N turns, according to (118.1), is equal to
Integral over ABCDA can be represented in the form of four integrals: according AB, BC, CD And D.A. At the sites AB And CD the circuit is perpendicular to the lines of magnetic induction and B l = 0. In the area outside the solenoid B=0. On the site D.A. vector circulation IN equal to Bl(the circuit coincides with the magnetic induction line); hence,
(119.1)
From (119.1) we arrive at the expression for the magnetic induction of the field inside the solenoid (in vacuum):
We found that the field inside the solenoid homogeneously(edge effects in areas adjacent to the ends of the solenoid are neglected in calculations). However, we note that the derivation of this formula is not entirely correct (the magnetic induction lines are closed, and the integral over the external portion of the magnetic field is not strictly equal to zero). The field inside the solenoid can be correctly calculated by applying the Biot-Savart-Laplace law; the result is the same formula (119.2).
The magnetic field is also important for practice. toroid- a ring coil, the turns of which are wound on a torus-shaped core (Fig. 176). The magnetic field, as experience shows, is concentrated inside the toroid; there is no field outside it.
The lines of magnetic induction in this case, as follows from symmetry considerations, are circles whose centers are located along the axis of the toroid. As a contour, we choose one such circle of radius r. Then, according to the circulation theorem (118.1), B× 2p r =m 0 NI whence it follows that magnetic induction inside the toroid (in vacuum)
Where N- number of toroid turns.
If the circuit passes outside the toroid, then it does not cover currents and B× 2p r = 0. This means that there is no field outside the toroid (as experience also shows).
The magnetic field of a solenoid is a superposition of individual fields that are created by each individual turn. The same current flows through all turns. The axes of all turns lie on the same line. The solenoid is an inductor coil having a cylindrical shape. This coil is wound from conductive wire. In this case, the turns are laid tightly to each other and have the same direction. In this case, it is believed that the length of the coil significantly exceeds the diameter of the turns.
Let's look at the magnetic induction created by each turn. It can be seen that the induction inside each turn is directed in the same direction. If you look at the center of the coil, then the induction from its edges will add up. In this case, the magnetic field induction between two adjacent turns is directed in the opposite direction. Since it is created by the same current, it is compensated.
Figure 1 — Field created by individual turns of the solenoid
If the turns of the solenoid are wound tightly enough, then between all the turns the counter-field will be compensated, and inside the turns the individual fields will be added into one common one. The lines of this field will pass inside the solenoid and cover it outside.
If you examine the magnetic field inside the solenoid by any means, for example, using iron filings, you can conclude that it is homogeneous. The magnetic field lines in this region are parallel straight lines. Not only are they parallel to themselves, but they are also parallel to the axis of the solenoid. Going beyond the aisles of the solenoid, they bend and close outside the coil.
Figure 2 - Field created by the solenoid
It can be seen from the figure that the field created by the solenoid is similar to the field created by a permanent bar magnet. At one end, the power lines exit the solenoid and this end is similar to the north pole of a permanent magnet. And they enter the other, and this end corresponds to the south pole. The difference is that the field is also present inside the solenoid. And if you conduct an experiment with iron filings, they will be drawn into the space between the turns.
But if a wooden core or a core made of any other non-magnetic material is inserted inside the solenoid, then when conducting an experiment with iron filings, the field pattern of the permanent magnet and the solenoid will be identical. Since the wooden core will not distort the power lines, but will not allow sawdust to penetrate inside the coil.
Figure 3 — Picture of the field of a permanent bar magnet
Several methods can be used to determine the solenoid poles. For example, the simplest one is to use a magnetic needle. It will be attracted to the opposite pole of the magnet. If the direction of the current in the coil is known, the poles can be determined using the right-hand screw rule. If you rotate the head of the right screw in the direction of the current, the translational movement will indicate the direction of the field in the solenoid. And knowing that the field is directed from the north pole to the south, you can determine which pole is located.
Magnetic field of electric current
A magnetic field is created not only by natural or artificial ones, but also by a conductor if an electric current passes through it. Therefore, there is a connection between magnetic and electrical phenomena.
It is not difficult to verify that a magnetic field is formed around a conductor through which current flows. Place a straight conductor above the moving magnetic needle, parallel to it, and pass an electric current through it. The arrow will take a position perpendicular to the conductor.
What forces could make the magnetic needle turn? Obviously, the strength of the magnetic field that arises around the conductor. Turn off the current and the magnetic needle will return to its normal position. This suggests that when the current was turned off, the magnetic field of the conductor also disappeared.
Thus, an electric current passing through a conductor creates a magnetic field. To find out in which direction the magnetic needle will deviate, use the right-hand rule. If you place your right hand over the conductor, palm down, so that the direction of the current coincides with the direction of the fingers, then the bent thumb will show the direction of deflection of the north pole of the magnetic needle placed under the conductor. Using this rule and knowing the polarity of the arrow, you can also determine the direction of the current in the conductor.
Magnetic field of a straight conductor has the shape of concentric circles. If you place your right hand over the conductor, palm down, so that the current seems to come out of the fingers, then the bent thumb will point to the north pole of the magnetic needle.Such a field is called a circular magnetic field.
The direction of the circular field lines of force depends on the conductor and is determined by the so-called gimlet rule. If you mentally screw the gimlet in the direction of the current, then the direction of rotation of its handle will coincide with the direction of the magnetic field lines. By applying this rule, you can find out the direction of the current in a conductor if you know the direction of the field lines created by this current.
Returning to the experiment with the magnetic needle, we can be convinced that it is always located with its northern end in the direction of the magnetic field lines.
So, A magnetic field arises around a straight conductor through which electric current passes. It has the shape of concentric circles and is called a circular magnetic field.
Pickles d. Magnetic field of the solenoid
A magnetic field arises around any conductor, regardless of its shape, provided that an electric current passes through the conductor.
In electrical engineering we deal with ones consisting of a number of turns. To study the magnetic field of the coil that interests us, let us first consider what shape the magnetic field of one turn has.
Let's imagine a coil of thick wire piercing a sheet of cardboard and connected to a current source. When an electric current passes through a coil, a circular magnetic field is formed around each individual part of the coil. According to the “gimlet” rule, it is not difficult to determine that the magnetic lines of force inside the coil have the same direction (towards us or away from us, depending on the direction of the current in the coil), and they exit from one side of the coil and enter the other side. A series of such turns, having the shape of a spiral, is the so-called solenoid (coil).
Around the solenoid, when current passes through it, a magnetic field is formed. It is obtained as a result of the addition of the magnetic fields of each turn and is shaped like the magnetic field of a rectilinear magnet. The lines of force of the magnetic field of the solenoid, as in a rectilinear magnet, leave one end of the solenoid and return to the other. Inside the solenoid they have the same direction. Thus, the ends of the solenoid have polarity. The end from which the lines of force emerge is north pole solenoid, and the end into which the power lines enter is its south pole.
Solenoid Poles can be determined by right hand rule, but for this you need to know the direction of the current in its turns. If you place your right hand on the solenoid, palm down, so that the current seems to come out of the fingers, then the bent thumb will point to the north pole of the solenoid. From this rule it follows that the polarity of the solenoid depends on the direction of the current in it. It is not difficult to verify this practically by bringing a magnetic needle to one of the poles of the solenoid and then changing the direction of the current in the solenoid. The arrow will instantly rotate 180°, i.e. it will indicate that the poles of the solenoid have changed.
The solenoid has the property of drawing in light iron objects. If a steel bar is placed inside the solenoid, then after some time, under the influence of the magnetic field of the solenoid, the bar will become magnetized. This method is used in manufacturing.
Electromagnets
It is a coil (solenoid) with an iron core placed inside it. The shapes and sizes of electromagnets are varied, but the general structure of all of them is the same.
The electromagnet coil is a frame, most often made of pressed wood or fiber and has various shapes depending on the purpose of the electromagnet. An insulated copper wire is wound on the frame in several layers - the winding of an electromagnet. It has a different number of turns and is made of wire of different diameters, depending on the purpose of the electromagnet.
To protect the winding insulation from mechanical damage, the winding is covered with one or several layers of paper or some other insulating material. The beginning and end of the winding are brought out and connected to output terminals mounted on the frame, or to flexible conductors with lugs at the ends.
The electromagnet coil is mounted on a core made of soft, annealed iron or alloys of iron with silicon, nickel, etc. Such iron has the smallest residual. Cores are most often made of composite thin sheets insulated from each other. The shapes of the cores can be different, depending on the purpose of the electromagnet.
If an electric current is passed through the winding of an electromagnet, a magnetic field is formed around the winding, which magnetizes the core. Since the core is made of soft iron, it will be magnetized instantly. If you then turn off the current, the magnetic properties of the core will also quickly disappear, and it will cease to be a magnet. The poles of an electromagnet, like a solenoid, are determined by the right-hand rule. If you change in the winding of an electromagnet, then the polarity of the electromagnet will change in accordance with this.
The action of an electromagnet is similar to the action of a permanent magnet. However, there is a big difference between them. A permanent magnet always has magnetic properties, and an electromagnet only when an electric current passes through its winding.
In addition, the attractive force of a permanent magnet is constant, since the magnetic flux of a permanent magnet is constant. The force of attraction of an electromagnet is not a constant value. The same electromagnet can have different attractive forces. The attractive force of any magnet depends on the magnitude of its magnetic flux.
The force of attraction, and therefore its magnetic flux, depends on the magnitude of the current passing through the winding of this electromagnet. The greater the current, the greater the attractive force of the electromagnet, and, conversely, the less current in the winding of the electromagnet, the less force it attracts magnetic bodies to itself.
But for electromagnets that are different in structure and size, their strength of attraction depends not only on the magnitude of the current in the winding. If, for example, we take two electromagnets of the same design and size, but one with a small number of winding turns, and the other with a much larger number, then it is easy to see that at the same current the attractive force of the latter will be much greater. Indeed, the greater the number of turns of a winding, the greater the magnetic field created around this winding at a given current, since it is composed of the magnetic fields of each turn. This means that the magnetic flux of the electromagnet, and therefore the force of its attraction, will be greater, the greater the number of turns the winding has.
There is another reason that affects the magnitude of the magnetic flux of an electromagnet. This is the quality of its magnetic circuit. A magnetic circuit is the path along which the magnetic flux is closed. The magnetic circuit has a certain magnetic resistance. Magnetic reluctance depends on the magnetic permeability of the medium through which the magnetic flux passes. The greater the magnetic permeability of this medium, the lower its magnetic resistance.
Since m The magnetic permeability of ferromagnetic bodies (iron, steel) is many times greater than the magnetic permeability of air, so it is more profitable to make electromagnets so that their magnetic circuit does not contain air sections. The product of the current strength and the number of turns of the electromagnet winding is called magnetomotive force. Magnetomotive force is measured in the number of ampere-turns.
For example, a current of 50 mA passes through the winding of an electromagnet with 1200 turns. M magnetomotive force such an electromagnet equals 0.05 x 1200 = 60 ampere-turns.
The action of magnetomotive force is similar to the action of electromotive force in an electrical circuit. Just as EMF causes electric current, magnetomotive force creates magnetic flux in an electromagnet. Just as in an electrical circuit, as the emf increases, the current value increases, so in a magnetic circuit, as the magnetomotive force increases, the magnetic flux increases.
Action magnetic resistance similar to the action of electrical resistance in a circuit. Just as the current decreases as the resistance of an electrical circuit increases, so does the current in a magnetic circuit. An increase in magnetic resistance causes a decrease in magnetic flux.
The dependence of the magnetic flux of an electromagnet on the magnetomotive force and its magnetic resistance can be expressed by a formula similar to the formula of Ohm’s law: magnetomotive force = (magnetic flux / magnetic resistance)
Magnetic flux is equal to magnetomotive force divided by magnetic reluctance.
The number of turns of the winding and the magnetic resistance for each electromagnet is a constant value. Therefore, the magnetic flux of a given electromagnet changes only with a change in the current passing through the winding. Since the force of attraction of an electromagnet is determined by its magnetic flux, in order to increase (or decrease) the force of attraction of the electromagnet, it is necessary to correspondingly increase (or decrease) the current in its winding.
Polarized electromagnet
A polarized electromagnet is a connection between a permanent magnet and an electromagnet. It's designed this way. So-called soft iron pole extensions are attached to the poles of a permanent magnet. Each pole extension serves as the core of an electromagnet; a coil with a winding is mounted on it. Both windings are connected to each other in series.
Since the pole extensions are directly connected to the poles of a permanent magnet, they have magnetic properties even in the absence of current in the windings; At the same time, their force of attraction is constant and is determined by the magnetic flux of a permanent magnet.
The action of a polarized electromagnet is that when current passes through its windings, the attractive force of its poles increases or decreases depending on the magnitude and direction of the current in the windings. The action of other electromagnets is based on this property of a polarized electromagnet. electrical devices.
The effect of a magnetic field on a current-carrying conductor
If you place a conductor in a magnetic field so that it is located perpendicular to the field lines, and pass an electric current through this conductor, the conductor will begin to move and will be pushed out of the magnetic field.
As a result of the interaction of a magnetic field with an electric current, the conductor begins to move, i.e., electrical energy is converted into mechanical energy.
The force with which a conductor is pushed out of a magnetic field depends on the magnitude of the magnetic flux of the magnet, the strength of the current in the conductor, and the length of the part of the conductor that the field lines intersect. The direction of action of this force, i.e. the direction of movement of the conductor, depends on the direction of the current in the conductor and is determined by left hand rule.
If you hold the palm of your left hand so that the magnetic field lines enter it, and the extended four fingers face the direction of the current in the conductor, then the bent thumb will indicate the direction of movement of the conductor. When applying this rule, we must remember that the field lines come out from the north pole of the magnet.
