Magnetic field lines

Magnetic fields, like electric fields, can be represented graphically using lines of force. A magnetic field line, or a magnetic field induction line, is a line, the tangent to which at each point coincides with the direction of the magnetic field induction vector.

a)	b)	in)

Rice. 1.2. Lines of force of the direct current magnetic field (a),

circular current (b), solenoid (c)

Magnetic lines of force, like electric lines, do not intersect. They are drawn with such density that the number of lines crossing a unit surface perpendicular to them is equal to (or proportional to) the magnitude of the magnetic induction of the magnetic field at a given location.

On fig. 1.2 a the field lines of the direct current field are shown, which are concentric circles, the center of which is located on the current axis, and the direction is determined by the rule of the right screw (the current in the conductor is directed to the reader).

Lines of magnetic induction can be "showed" using iron filings that are magnetized in the field under study and behave like small magnetic needles. On fig. 1.2 b shows the lines of force of the magnetic field of the circular current. The magnetic field of the solenoid is shown in fig. 1.2 in.

The lines of force of the magnetic field are closed. Fields with closed lines of force are called vortex fields. Obviously, the magnetic field is a vortex field. This is the essential difference between a magnetic field and an electrostatic one.

In an electrostatic field, the lines of force are always open: they begin and end on electric charges. Magnetic lines of force have neither beginning nor end. This corresponds to the fact that there are no magnetic charges in nature.

1.4. Biot-Savart-Laplace law

French physicists J. Biot and F. Savard conducted in 1820 a study of magnetic fields created by currents flowing through thin wires of various shapes. Laplace analyzed the experimental data obtained by Biot and Savart and established a relationship that was called the Biot–Savart–Laplace law.

According to this law, the induction of a magnetic field of any current can be calculated as a vector sum (superposition) of the inductions of magnetic fields created by individual elementary sections of the current. For the magnetic induction of the field created by a current element with a length, Laplace obtained the formula:

, (1.3)

where is a vector, modulo equal to the length of the conductor element and coinciding in direction with the current (Fig. 1.3); is the radius vector drawn from the element to the point where ; is the modulus of the radius vector .

Without a doubt, the magnetic field lines are now known to everyone. At least, even at school, their manifestation is demonstrated in physics lessons. Remember how a teacher placed a permanent magnet (or even two, combining the orientation of their poles) under a sheet of paper, and on top of it he poured metal filings taken in a labor training classroom? It is quite clear that the metal had to be held on the sheet, but something strange was observed - lines were clearly traced along which sawdust lined up. Notice - not evenly, but in stripes. These are the magnetic field lines. Or rather, their manifestation. What happened then and how can it be explained?

Let's start from afar. Together with us in the visible physical world coexists a special kind of matter - a magnetic field. It ensures the interaction of moving elementary particles or larger bodies that have an electric charge or a natural electric charge and are not only interconnected with each other, but often generate themselves. For example, a wire carrying an electric current creates magnetic field lines around it. The reverse is also true: the action of alternating magnetic fields on a closed conducting circuit creates a movement of charge carriers in it. The latter property is used in generators that supply electrical energy to all consumers. A striking example of electromagnetic fields is light.

The lines of force of the magnetic field around the conductor rotate or, which is also true, are characterized by a directed vector of magnetic induction. The direction of rotation is determined by the gimlet rule. The indicated lines are a convention, since the field spreads evenly in all directions. The thing is that it can be represented as an infinite number of lines, some of which have a more pronounced tension. That is why some “lines” are clearly traced in and sawdust. Interestingly, the lines of force of the magnetic field are never interrupted, so it is impossible to say unequivocally where the beginning is and where the end is.

In the case of a permanent magnet (or an electromagnet similar to it), there are always two poles, conventionally named North and South. The lines mentioned in this case are rings and ovals connecting both poles. Sometimes this is described in terms of interacting monopoles, but then a contradiction arises, according to which the monopoles cannot be separated. That is, any attempt to divide the magnet will result in several bipolar parts.

Of great interest are the properties of lines of force. We have already talked about continuity, but the ability to create an electric current in a conductor is of practical interest. The meaning of this is as follows: if the conducting circuit is crossed by lines (or the conductor itself is moving in a magnetic field), then additional energy is imparted to the electrons in the outer orbits of the atoms of the material, allowing them to begin independent directed movement. It can be said that the magnetic field seems to “knock out” charged particles from the crystal lattice. This phenomenon is called electromagnetic induction and is currently the main way to obtain primary electrical energy. It was discovered experimentally in 1831 by the English physicist Michael Faraday.

The study of magnetic fields began as early as 1269, when P. Peregrine discovered the interaction of a spherical magnet with steel needles. Almost 300 years later, W. G. Colchester suggested that he himself was a huge magnet with two poles. Further, magnetic phenomena were studied by such famous scientists as Lorentz, Maxwell, Ampère, Einstein, etc.

Let's understand together what a magnetic field is. After all, many people live in this field all their lives and do not even think about it. Time to fix it!

A magnetic field

A magnetic field is a special kind of matter. It manifests itself in the action on moving electric charges and bodies that have their own magnetic moment (permanent magnets).

Important: a magnetic field does not act on stationary charges! A magnetic field is also created by moving electric charges, or by a time-varying electric field, or by the magnetic moments of electrons in atoms. That is, any wire through which current flows also becomes a magnet!

A body that has its own magnetic field.

A magnet has poles called north and south. The designations "northern" and "southern" are given only for convenience (as "plus" and "minus" in electricity).

The magnetic field is represented by force magnetic lines. The lines of force are continuous and closed, and their direction always coincides with the direction of the field forces. If metal shavings are scattered around a permanent magnet, the metal particles will show a clear picture of the magnetic field lines emerging from the north and entering the south pole. Graphical characteristic of the magnetic field - lines of force.

Magnetic field characteristics

The main characteristics of the magnetic field are magnetic induction, magnetic flux and magnetic permeability. But let's talk about everything in order.

Immediately, we note that all units of measurement are given in the system SI.

Magnetic induction B - vector physical quantity, which is the main power characteristic of the magnetic field. Denoted by letter B . The unit of measurement of magnetic induction - Tesla (Tl).

Magnetic induction indicates how strong a field is by determining the force with which it acts on a charge. This force is called Lorentz force.

Here q - charge, v - its speed in a magnetic field, B - induction, F is the Lorentz force with which the field acts on the charge.

F- a physical quantity equal to the product of magnetic induction by the area of ​​the contour and the cosine between the induction vector and the normal to the plane of the contour through which the flow passes. Magnetic flux is a scalar characteristic of a magnetic field.

We can say that the magnetic flux characterizes the number of magnetic induction lines penetrating a unit area. The magnetic flux is measured in Weberach (WB).

Magnetic permeability is the coefficient that determines the magnetic properties of the medium. One of the parameters on which the magnetic induction of the field depends is the magnetic permeability.

Our planet has been a huge magnet for several billion years. The induction of the Earth's magnetic field varies depending on the coordinates. At the equator, it is about 3.1 times 10 to the minus fifth power of Tesla. In addition, there are magnetic anomalies, where the value and direction of the field differ significantly from neighboring areas. One of the largest magnetic anomalies on the planet - Kursk and Brazilian magnetic anomaly.

The origin of the Earth's magnetic field is still a mystery to scientists. It is assumed that the source of the field is the liquid metal core of the Earth. The core is moving, which means that the molten iron-nickel alloy is moving, and the movement of charged particles is the electric current that generates the magnetic field. The problem is that this theory geodynamo) does not explain how the field is kept stable.

The earth is a huge magnetic dipole. The magnetic poles do not coincide with the geographic ones, although they are in close proximity. Moreover, the Earth's magnetic poles are moving. Their displacement has been recorded since 1885. For example, over the past hundred years, the magnetic pole in the Southern Hemisphere has shifted by almost 900 kilometers and is now in the Southern Ocean. The pole of the Arctic hemisphere is moving across the Arctic Ocean towards the East Siberian magnetic anomaly, the speed of its movement (according to 2004 data) was about 60 kilometers per year. Now there is an acceleration of the movement of the poles - on average, the speed is growing by 3 kilometers per year.

What is the significance of the Earth's magnetic field for us? First of all, the Earth's magnetic field protects the planet from cosmic rays and the solar wind. Charged particles from deep space do not fall directly to the ground, but are deflected by a giant magnet and move along its lines of force. Thus, all living things are protected from harmful radiation.

During the history of the Earth, there have been several inversions(changes) of magnetic poles. Pole inversion is when they change places. The last time this phenomenon occurred about 800 thousand years ago, and there were more than 400 geomagnetic reversals in the history of the Earth. Some scientists believe that, given the observed acceleration of the movement of the magnetic poles, the next pole reversal should be expected in the next couple of thousand years.

Fortunately, no reversal of poles is expected in our century. So, you can think about the pleasant and enjoy life in the good old constant field of the Earth, having considered the main properties and characteristics of the magnetic field. And so that you can do this, there are our authors, who can be entrusted with some of the educational troubles with confidence in success! and other types of work you can order at the link.

1. The description of the properties of a magnetic field, as well as an electric field, is often greatly facilitated by introducing into consideration the so-called lines of force of this field. By definition, magnetic field lines are lines, the direction of the tangents to which at each point of the field coincides with the direction of the field strength at the same point. The differential equation of these lines will obviously have the form equation (10.3)]

Magnetic lines of force, like electric lines, are usually drawn in such a way that in any section of the field the number of lines crossing the area of ​​\u200b\u200bthe unit surface perpendicular to them is, if possible, proportional to the field strength on this area; however, as we shall see below, this requirement is by no means always feasible.

2 Based on equation (3.6)

we came to the following conclusion in § 10: electric lines of force can begin or end only at those points in the field at which electric charges are located. Applying the Gauss theorem (17) to the magnetic vector flux, we obtain on the basis of equation (47.1)

Thus, in contrast to the flow of an electric vector, the flow of a magnetic vector through an arbitrary closed surface is always equal to zero. This position is a mathematical expression of the fact that there are no magnetic charges similar to electric charges: the magnetic field is excited not by magnetic charges, but by the movement of electric charges (ie, currents). Based on this position and on comparing equation (53.2) with equation (3.6), it is easy to verify, by the reasoning given in § 10, that the magnetic lines of force at any point in the field can neither begin nor end

3. From this circumstance, it is usually concluded that magnetic lines of force, unlike electric lines, must be closed lines or go from infinity to infinity.

Indeed, both of these cases are possible. According to the results of solving problem 25 in § 42, the lines of force in the field of an infinite rectilinear current are circles perpendicular to the current and centered on the current axis. On the other hand (see Problem 26), the direction of the magnetic vector in the field of a circular current at all points lying on the axis of the current coincides with the direction of this axis. Thus, the axis of the circular current coincides with the line of force going from infinity to infinity; the drawing shown in fig. 53, is a section of the circular current by the meridional plane (i.e., the plane

perpendicular to the plane of the current and passing through its center), on which the dashed lines show the lines of force of this current

However, a third case is also possible, to which attention is not always drawn, namely: a line of force may have neither beginning nor end and at the same time not be closed and not go from infinity to infinity. This case takes place if the line of force fills a certain surface and, moreover, using a mathematical term, fills it densely everywhere. The easiest way to explain this is with a specific example.

4. Consider the field of two currents - a circular flat current and an infinite rectilinear current flowing along the current axis (Fig. 54). If there were only one current, then the field lines of the field of this current would lie in meridional planes and would have the form shown in the previous figure. Consider one of these lines shown in Fig. 54 dashed line. The set of all lines similar to it, which can be obtained by rotating the meridional plane around the axis, forms the surface of a certain ring or torus (Fig. 55).

The lines of force of the rectilinear current field are concentric circles. Therefore, at each point of the surface, both and are tangent to this surface; therefore, the intensity vector of the resulting field is also tangent to it. This means that each line of force of the field passing through one point of the surface must lie on this surface with all its points. This line will obviously be a helix on

the surface of the torus The course of this helix will depend on the ratio of the strength of the currents and on the position and shape of the surface. It is obvious that only under certain specific selection of these conditions will this helix be closed; Generally speaking, when the line is continued, new turns of it will lie between the previous turns. When the line is continued indefinitely, it will come as close as it likes to any point it has passed, but it will never return to it a second time. And this means that, while remaining open, this line will densely fill the surface of the torus everywhere.

5. To strictly prove the possibility of the existence of non-closed lines of force, we introduce orthogonal curvilinear coordinates on the surface of the torus y (azimuth of the meridional plane) and (polar angle in the meridional plane with the vertex located at the intersection of this plane with the axis of the ring - Fig. 54).

The field strength on the surface of the torus is a function of only one angle, with the vector directed in the direction of increase (or decrease) of this angle, and the vector in the direction of increase (or decrease) of the angle. Let there be the distance of a given point of the surface from the center line of the torus, its distance from the vertical axis As it is easy to see, the element of the length of the line lying on is expressed by the formula

Accordingly, the differential equation of the lines of forces [cf. equation (53.1)] on the surface takes the form

Taking into account that they are proportional to the strength of the currents and integrating, we obtain

where is some angle function independent of .

For the line to be closed, i.e., for it to return to the starting point, it is necessary that a certain integer number of revolutions of the line around the torus correspond to an integer number of its revolutions around the vertical axis. In other words, it is necessary that it be possible to find two such integers nm, so that an increase in the angle by corresponds to an increase in the angle by

Let us now take into account what the integral of the periodic function of the angle with period is. As is known, the integral

of a periodic function in the general case is the sum of a periodic function and a linear function. Means,

where K is some constant, there is a function with a period Therefore,

Introducing this into the previous equation, we obtain the condition for the closure of the lines of force on the surface of the torus

Here K is a quantity independent of. It is obvious that two integers of heels satisfying this condition can be found only if the value - K is a rational number (integer or fractional); this will take place only for a certain ratio between the forces of the currents. Generally speaking, - K will be an irrational quantity and, therefore, the lines of force on the surface of the torus under consideration will be open. However, in this case, you can always choose an integer so that - arbitrarily little differs from some integer. This means that an open line of force, after a sufficient number of revolutions, will come as close as you like to any point of the field once passed. In a similar way, it can be shown that this line, after a sufficient number of revolutions, will come as close as desired to any predetermined point on the surface, and this means, by definition, that it densely fills this surface everywhere.

6. The existence of non-closed magnetic lines of force densely filling a certain surface everywhere obviously makes it impossible to accurately represent the field graphically with the help of these lines. In particular, it is far from always possible to satisfy the requirement that the number of lines crossing a unit area perpendicular to them be proportional to the field strength on this area. So, for example, in the case just considered, the same open line intersects an infinite number of times any finite area that intersects the surface of the ring

However, with due diligence, the use of the concept of lines of force is, although approximate, but still a convenient and illustrative way of describing a magnetic field.

7. According to equation (47.5), the circulation of the magnetic field vector along the curve that does not cover currents is equal to zero, while the circulation along the curve that covers currents is equal to the sum of the strengths of the covered currents (taken with proper signs). The circulation of the vector along the field line cannot be equal to zero (due to the parallelism of the length element of the field line and the vector, the value is essentially positive). Therefore, each closed magnetic field line must cover at least one of the current-carrying conductors. Moreover, open lines of force that densely fill some surface (unless they go from infinity to infinity) must also wind around currents. Indeed, the vector integral over an almost closed turn of such a line is essentially positive. Therefore, the circulation along the closed contour obtained from this coil by adding an arbitrarily small segment closing it is nonzero. Therefore, this circuit must be pierced by current.