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
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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
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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
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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
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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
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At long distances ( h >> l) we have from here
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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
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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 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.
The current element I dl excites a magnetic field dB perpendicular to the radius vector r. Let us decompose this field into two components: the axial component dB z and the radical component dB r. When integrated along a circular current contour, the radial components cancel each other out. The resulting field will be directed along the Z axis, and only the axial component needs to be integrated
The angle is the same for all points of the circular current. Integration is reduced to simple multiplication by the contour length 2πa. Thus,
4) Induction magic. Fields on the solenoid axis.
Therefore, the magnetic induction on the solenoid axis can be obtained by integrating the inductions from individual circular currents, according to calculations:
n is the number of turns per unit length of the solenoid.
The direction of vector B along the axis of the solenoid according to the gimlet rule.
33. Ampere's law. Interaction of parallel currents.
On any frame with current placed in a magician. field, a couple of forces act. It can be assumed that this pair of forces is created by the forces acting on each element of the current circuit located in the magic. field.
The magnetic field has an orienting effect on the current-carrying frame. Consequently, the torque experienced by the frame is the result of the action of forces on its individual elements. Ampere established that the force d F, with which the magnetic field acts on a conductor element dl with current located in a magnetic field, is equal to
where d l-vector coinciding in direction with the current, IN- vector of magnetic induction.
Direction of vector d F determined left hand rule: if the palm of the left hand is positioned so that the vector enters it IN, and place four extended fingers in the direction of the current in the conductor, then the bent thumb will show the direction of the force acting on the current.
Ampere force modulus is calculated by the formula
Where a-angle between vectors d l And IN.
Ampere's law is used to determine the strength of interaction between two currents. Consider two infinite rectilinear parallel currents I 1 and I 2, the distance between them is R. Each of the conductors creates a magnetic field, which acts according to Ampere's law on the other conductor with current. It can be shown that two parallel currents of the same direction attract each other with a force
If currents have opposite directions, then, using the left-hand rule, we can show that between them there is repulsive force, defined by the formula.
34. Magnetic constant. Units of magnetic induction and magnetic field strength. Magnetic field of a moving charge.
Magnetic constant. Units of magnetic induction and magnetic field strength
If two parallel conductors carrying current are in a vacuum ( m= 1), then the interaction force per unit length of the conductor is equal to
To find a numeric value m 0 we will use the definition of ampere, according to
which =2×10 –7 N/m at I 1 = I 2 = 1 A and R= 1 m. Substituting this value into the formula, we get
Where Henry(H) - unit of inductance.
Ampere's law allows us to determine the unit of magnetic induction IN. Let us assume that the conductor element d l with current I perpendicular to the direction of the magnetic field. Then Ampere's law will be written in the form dF=IB d l, where
Unit of magnetic induction - tesla(T): 1 T is the magnetic induction of such a uniform magnetic field that acts with a force of 1 N per meter of length of a straight conductor located perpendicular to the direction of the field, if a current of 1 A passes through this conductor:
Because m 0 = 4p×10 –7 N/A 2, and in the case of vacuum ( m= 1), according to (109.3), B=m 0 H, then for this case
Unit of magnetic field strength - ampere per meter(A/m): 1 A/m - the strength of such a field, the magnetic induction of which in a vacuum is equal to 4p × 10 –7 T.
Magnetic field of a moving charge
Each conductor carrying current creates a magnetic field in the surrounding space. Electric current is the ordered movement of electric charges. Therefore, we can say that any charge moving in a vacuum or medium creates a magnetic field around itself. Summarizing the general data: The law of a point charge q freely moving with a non-relativistic speed v. Under free charge movement refers to its movement at a constant speed. This law is expressed by the formula
Where r- radius vector drawn from the charge Q to the observation point M. Vector IN directed perpendicular to the plane in which the vectors are located v And r, namely: its direction coincides with the direction of translational movement of the right screw when it rotates from v To r.
The magnetic induction module is calculated by the formula
Where a- angle between vectors v And r.
The given patterns (1) and (2) are valid only at low speeds ( v<<с) движущихся зарядов, когда электрическое поле свободно движущегося заряда можно считать электростатическим, т. е. создаваемым неподвижным зарядом, находящимся в той точке, где в данный момент времени расположен движущийся заряд.
Formula (1) determines the magnetic induction of a positive charge moving at speed v. If a negative charge moves, then Q should be replaced with -Q. Speed v- relative speed, i.e. speed relative to the observer. Vector IN in the reference frame under consideration depends on both time and the position of the point M observations. Therefore, it is necessary to emphasize the relative nature of the magnetic field of a moving charge.
36. Hall effect. Vector circulation IN for a magnetic field in vacuum.
The Hall effect* (1879) is the occurrence in a metal (or semiconductor) with a current density j placed in a magnetic field IN, electric field in a direction perpendicular to IN And j.
Let us place a metal plate with a current density j into a magnetic field IN, perpendicular j. With this direction j the speed of current carriers in the metal - electrons - is directed from right to left. The electrons experience the Lorentz force, which in this case is directed upward. Thus, at the upper edge of the plate there will be an increased concentration of electrons (it will be negatively charged), and at the lower edge there will be a lack of electrons (it will be charged positively). As a result, an additional transverse electric field will appear between the edges of the plate, directed from bottom to top. When tension E B This transverse field reaches such a value that its action on the charges will balance the Lorentz force, then a stationary distribution of charges in the transverse direction will be established. Then
Where A - record width, Dj - transverse (Hall) potential difference.
Considering that the current strength I=jS=nevS(S- cross-sectional area of the plate thickness d, p - electron concentration, v- average speed of ordered movement of electrons), we obtain
i.e., the Hall transverse potential difference is directly proportional to the magnetic induction IN, current strength I and is inversely proportional to the thickness of the plate d. In formula (1) R= 1/ (en) - Hall constant, depending on the substance. Using the measured value of the Hall constant, it is possible to: 1) determine the concentration of current carriers in the conductor (with the known nature of conductivity and charge of carriers); 2) judge the nature of the conductivity of semiconductors (see § 242, 243), since the sign of the Hall constant coincides with the sign of the charge e current carriers. The Hall effect is therefore the most effective method for studying the energy spectrum of current carriers in metals and semiconductors.
§ 118. Circulation of vector B of magnetic field in vacuum
Circulation of vector B over a given closed contour is called the integral
where d l- vector of the elementary length of the contour, directed along the traversal of the contour, B l =B cos a- vector component IN in the direction tangent to the contour (taking into account the selected traversal direction), a- angle between vectors IN and d l.
The law of total current for a magnetic field in vacuum (theorem on the circulation of vector B):
vector circulation IN along an arbitrary closed contour is equal to the product of the magnetic constant m 0 by the algebraic sum of the currents covered by this circuit: (2)
Where n- number of conductors with currents covered by the circuit L free form. Each current is counted as many times as the number of times it is covered by the circuit. A current is considered positive if its direction forms a right-handed system with the direction of traversal along the contour; current in the opposite direction is considered negative. For example, for the system of currents shown in Fig.,
Expression (2) is valid only for a field in vacuum, since, as will be shown below, for a field in a substance it is necessary to take into account molecular currents.
Let's imagine a closed contour in the form of a circle of radius r. At each point of this contour the vector IN is identical in magnitude and directed tangentially to the circle (it is also a line of magnetic induction). Consequently, the circulation of the vector IN equal to
According to expression (2), we get B× 2p r=m 0 I(in a vacuum), from where
Comparing expressions (3) and (4) for the circulation of vectors E And IN, we see that between them there is fundamental difference. Vector circulation E electrostatic field is always zero, i.e. the electrostatic field is potential. Vector circulation IN magnetic field is not zero. This field is called vortex.
37. Magnetic field of a solenoid and toroid.
Consider a solenoid with length l having N turns through which current flows. 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. The magnetic field inside the solenoid is uniform, but outside the solenoid it is inhomogeneous and very weak.
In Fig. the lines of magnetic induction inside and outside the solenoid are presented. 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. Vector circulation IN in a closed loop ABCDA, covering everything N turns, 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. Location on D.A. vector circulation IN equal to Bl(the circuit coincides with the magnetic induction line); hence,
From (1) we arrive at the expression for the magnetic field induction inside the solenoid (in vacuum): (2)
We found that the field inside the solenoid homogeneously. The field inside the solenoid can be correctly calculated by applying the Biot-Savart-Laplace law; the result is the same formula (2).
The magnetic field is also important for practice. toroid- a ring coil, the turns of which are wound on a torus-shaped core. 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 are circles, the centers of which 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, 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.
38. Magnetic induction vector flux. Gauss's theorem for the magnetic field, including in differential form.
Magnetic induction vector flux (magnetic flux) through the platform dS called scalar physical quantity equal to
Where Bn=IN cos a- vector projection IN to the direction of the normal to the site dS(a- angle between vectors n And IN), d S=d Sn- a vector whose modulus is d S, and its direction coincides with the direction of the normal n to the site. Flow vector IN can be either positive or negative depending on the sign of cos a(determined by choosing the positive direction of the normal n). Flow vector IN connected to the circuit through which current flows. In this case, the positive direction of the normal to the contour is associated with the current by the rule of the right screw. Thus, the magnetic flux created by the circuit through the surface limited by itself is always positive.
Magnetic induction vector flux F B through an arbitrary surface S equal to (1)
For a uniform field and a flat surface located perpendicular to the vector IN, B n =B=const And
From this formula the unit of magnetic flux is determined weber(Wb): 1 Wb is a magnetic flux passing through a flat surface with an area of 1 m 2 located perpendicular to a uniform magnetic field, the induction of which is 1 T (1 Wb = 1 T × m 2).
Gauss's theorem for the field: the flux of the magnetic induction vector through any closed surface is zero:
Let V be the volume bounding the surface under consideration. Then, when contracting the closing surface to a point, we obtain
Thus, at any point in space =0 (and in electrostatics, and only in those places where there are no space charges ρ=0,).
By virtue of equality (2), in the area of magic. phenomena there is no analogue to electric charges.
Gauss's theorem for mag. fields reflects the fact of the absence of magic. charges, as a result of which the lines of mag. inductions have neither beginning nor end - they are closed.
Weber magnetic flux:
39, Magnetic moments of electrons and atoms.
Experience shows that all substances placed in a magnetic field are magnetized. Let us consider the cause of this phenomenon from the point of view of the structure of atoms and molecules, basing it on Ampere’s hypothesis, according to which in any body there are microscopic currents caused by the movement of electrons in atoms and molecules.
For a qualitative explanation of magnetic phenomena, with a sufficient approximation, we can assume that the electron moves in an atom in circular orbits. An electron moving in one of these orbits is equivalent to a circular current, so it has orbital magnetic moment p m = ISn, whose module (1)
Where I=en - current strength, n- frequency of electron rotation in orbit, S- orbital area. If the electron moves clockwise, then the current is directed counterclockwise and the vector R m (in accordance with the right-hand screw rule) is directed perpendicular to the electron orbital plane, as indicated in the figure.
On the other hand, an electron moving in orbit has a mechanical angular momentum Le, whose modulus (2)
Where v = 2pn, pr 2 = S. Vector Le(its direction is also determined by the right screw rule) is called orbital mechanical momentum of the electron.
From Fig. it follows that the directions R m and Le, are opposite, therefore, taking into account expressions (1) and (2), we obtain
where quantity (3)
called gyromagnetic ratio of orbital moments. This ratio, determined by the universal constants, is the same for any orbit, although for different orbits the values v And r are different. Formula (3) was derived for a circular orbit and is also valid for elliptical orbits.
The experimental determination of the gyromagnetic ratio was carried out in experiments by Einstein and de Haas, who observed the rotation of an iron rod freely suspended on a thin quartz thread when it was magnetized in an external magnetic field (alternating current was passed through the solenoid winding with a frequency equal to the frequency of torsional oscillations of the rod). When studying forced torsional vibrations of the rod, the gyromagnetic ratio was determined, which turned out to be equal to – (e/m). Thus, the sign of the carriers responsible for molecular currents coincided with the sign of the electron charge, and the gyromagnetic ratio turned out to be twice as large as the previously introduced value g(3). To explain this result, which was of great importance for the further development of physics, it was assumed and subsequently proven that, in addition to orbital moments (1) and (2), the electron has own mechanical angular momentum Les, called spin. It has now been established that spin is an integral property of the electron, like its charge and mass. Spin the electron Les, corresponds own (cellular) magnetic moment pms, proportional Les and directed in the opposite direction:
Magnitude g s called gyromagnetic ratio of spin moments.
Projection of the intrinsic magnetic moment onto the direction of the vector IN can only take one of the following two values:
Where ħ=h/(2p) (h- Planck's constant), m b- Bohr magneton, which is a unit of the magnetic moment of an electron.
Total magnetic moment of an atom (molecule) p a is equal to the vector sum of the magnetic moments (orbital and spin) of the electrons entering the atom (molecule):
40. Diamagnets and paramagnets
Substances that can affect magic. field – magnetic. Under the influence of an electrostatic field, the dielectric comes into a special state - polarization. That is, at the boundaries of the dielectric and in areas where it is inhomogeneous, electrically bound charges arise. They create their electro-stat. a field that adds up to the original el-stat field. Then the total strength of the el-stat field:
E 0 – initial el-stat. field
E - field resulting from the dielectric field.
In the same way, every magnet located in a magician. the field flowing through the wires is magnetized.
B is the vector of magic induction, the characteristic magic field created by all macro and micro flows.
N – vector of tension, char-th magic field of macrocurrents.
=> magician ate in a thing consists of two fields: ext. the field created by the current and the field created by the magnetization of things. Then the vector is magic. induction of the resulting magic. field is equal to the vector sum of the magnetic inductions external. fields B 0 and microcurrent fields B
Things for which c is in the same direction are called paramagnetic (platinum, aluminum, rare earth elements). Things for which are opposite to c are called diamagnetic (bismuth, silver, gold, copper).
That is, paramagnetic materials are magnetized along the magnetic field. fields, as a result of which they are attracted to the external source. fields. Diamagnets are magnetized against the field and repelled from the external source. fields.
For all diamagnetic bodies and most paramagnetic ones, it is quite small compared to . However, there is a group of bodies for which it can be large compared to . Such bodies are classified into a special group of fugromagnetic bodies (iron, nickel, kobol, etc.). These things are 10 3 - 10 4 more strongly attracted to the external source. fields, i.e. they are strongly magnetized along the field.
According to Ampere's hypothesis, in the molecules of paramagnetic substances there are circular currents called molecular currents.
When there is no external magic field, the axes of these currents are located randomly and the magic field they create is on average 0. Under the influence of magic. fields, these circular currents are oriented, and in doing so they will create a magic field, giving on average an induction that is different from zero; the induction will be added to the initial magic induction of the field. Thus, the increase in the total magnetic induction in a substance is explained. That is, the magnetization of a paramagnet is reduced to a certain orientation of its molecular currents.
Circular currents arise only when external excitation occurs. magic field. The direction of these induced currents is such that the magic field they create is directed against the outside. magician fields. This explains the decrease in field induction in a diamagnetic medium.
41. Magnetic field in matter. Magnetic permeability. The law of total current for the magnetic field in matter, the theorem on the circulation of the vector N.
Magnetization. Magnetic field in matter
Just as polarization was introduced for a quantitative description of the polarization of dielectrics (see § 88), for a quantitative description of the magnetization of magnets, a vector quantity is introduced - magnetization, determined by the magnetic moment of a unit volume of the magnet:
where is the magnetic moment of the magnet, which is the vector sum of the magnetic moments of individual molecules (see (131.6)).
Considering the characteristics of the magnetic field (see § 109), we introduced the magnetic induction vector IN, characterizing the resulting magnetic field created by all macro- and microcurrents, and the intensity vector N, characterizing the magnetic field of macrocurrents. Consequently, the magnetic field in a substance consists of two fields: the external field created by the current and the field created by the magnetized substance. Then we can write that the vector of magnetic induction of the resulting magnetic field in the magnet is equal to the vector sum of the magnetic induction of the external field IN 0 (field created by magnetizing current in a vacuum) and microcurrent fields IN" (field created by molecular currents): (133.1)
Where IN 0 =m 0 N(see (109.3)).
To describe the field created by molecular currents, consider a magnet in the form of a circular cylinder with a cross-section S and length l, introduced into a homogeneous external magnetic hearth with induction IN 0 . The magnetic field of molecular currents arising in a magnet will be directed opposite to the external field for diamagnetic materials and coincide with it in direction for paramagnetic materials. The planes of all molecular currents will be located perpendicular to the vector IN 0, since the vectors of their magnetic moments p m are antiparallel to the vector IN 0 (for diamagnetic materials) and parallel IN 0 (for paramagnetic materials). If we consider any section of the cylinder perpendicular to its axis, then in the internal sections of the cross section of the magnet the molecular currents of neighboring atoms are directed towards each other and are mutually compensated (Fig. 189). Only molecular currents exiting the side surface of the cylinder will be uncompensated.
The current flowing along the side surface of the cylinder is similar to the current in the solenoid and creates a field inside it, magnetic induction IN" which can be calculated taking into account formula (119.2) for N= 1 (single turn solenoid): (133.2)
Where I"- molecular current strength, l is the length of the cylinder under consideration, and the magnetic permeability m taken equal to one.
On the other side, I"/l - magnetic susceptibility of the substance. For diamagnets, c is negative (the field of molecular currents is opposite to the external one), for paramagnets it is positive (the field of molecular currents coincides with the external one).
Using formula (133.6), expression (133.4) can be written as (133.7)
Dimensionless quantity (133.8)
represents the magnetic permeability of a substance. Substituting (133.8) into (133.7), we arrive at relation (109.3) IN=m 0 mN, which was previously postulated.
Since the absolute value of magnetic susceptibility for dia- and paramagnets is very small (about 10 –4 -10 –6), then for them m differs slightly from unity. This is easy to understand, since the magnetic field of molecular currents is much weaker than the magnetizing field. Thus, for diamagnetic materials c<0 и m<1, для парамагнетиков c>0 and m>1.
The law of total current for the magnetic field in matter (the theorem on the circulation of vector B) is a generalization of the law (118.1):
Where I And I"- respectively, algebraic sums of macrocurrents (conduction currents) and microcurrents (molecular currents) covered by an arbitrary closed loop L. Thus, the circulation of the magnetic induction vector IN along an arbitrary closed contour is equal to the algebraic sum of the conduction currents and molecular currents covered by this contour, multiplied by the magnetic constant. Vector IN, thus, characterizes the resulting field created by both macroscopic currents in conductors (conduction currents) and microscopic currents in magnets, therefore the lines of the magnetic induction vector IN have no sources and are closed.
It is known from theory that the circulation of magnetization J along an arbitrary closed contour L equal to algebraic sum molecular currents, covered by this contour:
Then the law of total current for the magnetic field in matter can also be written in the form (133.9)
Where I, let's emphasize this more times, there is an algebraic sum of conduction currents.
The expression in parentheses in (133.9), according to (133.5), is nothing more than the previously introduced vector H magnetic field strength. So, vector circulation N along an arbitrary closed contour L equal to the algebraic sum of the conduction currents covered by this circuit: (133.10)
Expression (133.10) is theorem on the circulation of the vector H.
First, let's solve the more general problem of finding magnetic induction on the axis of a coil with current. To do this, let's make Figure 3.8, in which we depict the current element and the magnetic induction vector that it creates on the axis of the circular contour at some point.
Rice. 3.8 Determination of magnetic induction
on the axis of a circular coil with current
The magnetic induction vector created by an infinitesimal circuit element can be determined using the Biot-Savart-Laplace law (3.10).
As follows from the rules of the vector product, the magnetic induction will be perpendicular to the plane in which the vectors and lie, therefore the magnitude of the vector will be equal
.
To find the total magnetic induction from the entire circuit, it is necessary to add vectorially from all elements of the circuit, i.e., actually calculate the integral along the length of the ring
This integral can be simplified if represented as a sum of two components and
In this case, due to symmetry, therefore, the resulting magnetic induction vector will lie on the axis. Therefore, to find the modulus of a vector, you need to add up the projections of all vectors, each of which is equal to
.
Taking into account that and , we obtain the following expression for the integral
It is easy to see that calculating the resulting integral will give the length of the contour, i.e. As a result, the total magnetic induction created by a circular contour on the axis at point is equal to
. (3.19)
Using the magnetic moment of the circuit, formula (3.19) can be rewritten as follows
.
Now we note that the solution (3.19) obtained in general form allows us to analyze the limiting case when the point is placed at the center of the coil. In this case, the solution for the magnetic field induction in the center of the ring with current will take the form
The resulting magnetic induction vector (3.19) is directed along the current axis, and its direction is related to the direction of the current by the rule of the right screw (Fig. 3.9).
Rice. 3.9 Determination of magnetic induction
in the center of a circular coil with current
Magnetic field induction at the center of a circular arc
This problem can be solved as a special case of the problem considered in the previous paragraph. In this case, the integral in formula (3.18) should not be taken over the entire length of the circle, but only along its arc l. And also take into account that induction is sought at the center of the arc, therefore . As a result we get
, (3.21)
where is the length of the arc; – arc radius.
5 Vector of magnetic field induction of a point charge moving in vacuum(without formula output)
,
where is the electric charge; – constant non-relativistic speed; – radius vector drawn from the charge to the observation point.
Ampere and Lorentz forces
Experiments on deflecting a current-carrying frame in a magnetic field show that any current-carrying conductor placed in a magnetic field is acted upon by a mechanical force called Ampere force.
Ampere's law determines the force acting on a current-carrying conductor placed in a magnetic field:
; 
, (3.22)
where is the current strength; – element of the wire length (the vector coincides in direction with the current); – length of the conductor. The Ampere force is perpendicular to the direction of the current and the direction of the magnetic induction vector.
If a straight conductor of length is in a uniform field, then the ampere force modulus is determined by the expression (Fig. 3.10):
The Ampere force is always directed perpendicular to the plane containing the vectors and , and its direction as a result of the vector product is determined by the right-hand screw rule: if you look along the vector, then the rotation from to along the shortest path should occur clockwise .
Rice. 3.10 Left hand rule and gimlet rule for Ampere force
On the other hand, to determine the direction of the Ampere force, you can also apply the mnemonic rule of the left hand (Fig. 3.10): you need to place your palm so that the lines of magnetic induction enter it, the extended fingers show the direction of the current, then the bent thumb will indicate the direction of the Ampere force.
Based on formula (3.22), we find an expression for the force of interaction between two infinitely long, straight, parallel conductors through which currents flow I 1 and I 2 (Fig. 3.11) (Ampere’s experiment). The distance between the wires is a.
Let's determine the Ampere force d F 21, acting from the magnetic field of the first current I 1 per element l 2d l second current.
The magnitude of the magnetic induction of this field B 1 at the location of the element of the second conductor with current is equal to
Rice. 3.11 Ampere’s experiment to determine the force of interaction
two straight currents
Then, taking (3.22) into account, we obtain
. (3.24)
Reasoning in the same way, it can be shown that the Ampere force acting from the magnetic field created by the second conductor with current on an element of the first conductor I 1 d l, is equal
,
i.e. d F 12 = d F 21 . Thus, we derived formula (3.1), which was obtained experimentally by Ampere.
In Fig. Figure 3.11 shows the direction of the Ampere forces. In the case when the currents are directed in the same direction, then these are attractive forces, and in the case of currents of different directions, these are repulsive forces.
From formula (3.24), we can obtain the Ampere force acting per unit length of the conductor
. (3.25)
Thus, the force of interaction between two parallel straight conductors with currents is directly proportional to the product of the magnitudes of the currents and inversely proportional to the distance between them.
Ampere's law states that a current-carrying element placed in a magnetic field experiences a force. But every current is the movement of charged particles. It is natural to assume that the forces acting on a current-carrying conductor in a magnetic field are due to forces acting on individual moving charges. This conclusion is confirmed by a number of experiments (for example, an electron beam in a magnetic field is deflected).
Let's find an expression for the force acting on a charge moving in a magnetic field based on Ampere's law. To do this, in the formula that determines the elementary Ampere force
let's substitute the expression for the electric current strength
,
Where I– the strength of the current flowing through the conductor; Q– the amount of total charge flowing during the time t; q– the magnitude of the charge of one particle; N– the total number of charged particles passing through a conductor of volume V, length l and section S; n– number of particles per unit volume (concentration); v– particle speed.
As a result we get:
. (3.26)
The direction of the vector coincides with the direction of the speed v, so they can be swapped.
. (3.27)
This force acts on all moving charges in a conductor of length and cross-section S, the number of such charges:
Therefore, the force acting on one charge will be equal to:
. (3.28)
Formula (3.28) determines Lorentz force, the value of which
where a is the angle between the particle velocity and magnetic induction vectors.
In experimental physics, a situation often occurs when a charged particle moves simultaneously in a magnetic and electric field. In this case, consider the complete Lorenz silt as
,
where is the electric charge; – electric field strength; – particle speed; – magnetic field induction.
Only in a magnetic field on a moving charged particle the magnetic component of the Lorentz force acts (Fig. 3.12)
Rice. 3.12 Lorentz force
The magnetic component of the Lorentz force is perpendicular to the velocity vector and the magnetic induction vector. It does not change the magnitude of the speed, but only changes its direction, therefore, it does no work.
The mutual orientation of the three vectors ‑, and , included in (3.30), is shown in Fig. 313 for a positively charged particle.
Rice. 3.13 Lorentz force acting on a positive charge
As can be seen from Fig. 3.13, if a particle flies into a magnetic field at an angle to the lines of force, then it moves uniformly in the magnetic field in a circle with radius and period of revolution:
where is the particle mass.
Ratio of magnetic moment to mechanical moment L(angular momentum) of a charged particle moving in a circular orbit,
where is the charge of the particle; T - particle mass.
Let us consider the general case of the motion of a charged particle in a uniform magnetic field, when its speed is directed at an arbitrary angle a to the magnetic induction vector (Fig. 3.14). If a charged particle flies into a uniform magnetic field at an angle , then it moves along a helical line.
Let's decompose the velocity vector into components v|| (parallel to the vector) and v^ (perpendicular to the vector):
Availability v^ leads to the fact that the Lorentz force will act on the particle and it will move in a circle with a radius R in a plane perpendicular to the vector:
.
The period of such motion (the time of one revolution of a particle around a circle) is equal to
.
Rice. 3.14 Movement along a helix of a charged particle
in a magnetic field
Due to availability v|| the particle will move uniformly along , since on v|| the magnetic field has no effect.
Thus, the particle participates in two movements simultaneously. The resulting trajectory of movement is a helical line, the axis of which coincides with the direction of the magnetic field induction. Distance h between adjacent turns is called helix pitch and equals:
.
The effect of a magnetic field on a moving charge finds great practical application, in particular, in the operation of a cathode ray tube, where the phenomenon of deflection of charged particles by electric and magnetic fields is used, as well as in the operation of mass spectrographs, which make it possible to determine the specific charge of particles ( q/m) and charged particle accelerators (cyclotrons).
Let's consider one such example, called a “magnetic bottle” (Fig. 3.15). Let a non-uniform magnetic field be created by two turns with currents flowing in the same direction. Condensation of induction lines in any spatial region means a greater value of magnetic induction in this region. The magnetic field induction near current-carrying turns is greater than in the space between them. For this reason, the radius of the helical line of the particle trajectory, inversely proportional to the induction modulus, is smaller near the turns than in the space between them. After the particle, moving to the right along the helical line, passes the midpoint, the Lorentz force acting on the particle acquires a component that slows down its movement to the right. At a certain moment, this force component stops the movement of the particle in this direction and pushes it to the left towards coil 1. When a charged particle approaches coil 1, it also slows down and begins to circulate between the coils, finding itself in a magnetic trap, or between “magnetic mirrors”. Magnetic traps are used to contain high-temperature plasma (K) in a certain region of space during controlled thermonuclear fusion.
Rice. 3.15 Magnetic “bottle”
The patterns of motion of charged particles in a magnetic field can explain the peculiarities of the motion of cosmic rays near the Earth. Cosmic rays are streams of high-energy charged particles. When approaching the Earth's surface, these particles begin to experience the action of the Earth's magnetic field. Those that are directed towards the magnetic poles will move almost along the lines of the earth's magnetic field and wind around them. Charged particles approaching the Earth near the equator are directed almost perpendicular to the magnetic field lines, their trajectory will be curved. and only the fastest of them will reach the surface of the Earth (Fig. 3.16).
Rice. 3.16 Formation of the Aurora
Therefore, the intensity of cosmic rays reaching the Earth near the equator is noticeably less than near the poles. Related to this is the fact that the aurora is observed mainly in the circumpolar regions of the Earth.
Hall effect
In 1880 American physicist Hall conducted the following experiment: he passed a direct electric current I through a gold plate and measured the potential difference between opposite points A and C on the upper and lower faces (Fig. 3.17).
The gimlet rule. A clear idea of the nature of the magnetic field arising around any conductor through which an electric current flows is given by pictures of magnetic field lines obtained as described in § 122.
In Fig. 214 and 217 depict such line patterns obtained using iron filings for the field of a long straight conductor and for the field of a circular coil with current. Looking carefully at these drawings, we first of all pay attention to the fact that the magnetic field lines have the appearance of closed lines. This property is common and very important. Whatever the shape of the conductors through which the current flows, the lines of the magnetic field it creates are always closed on themselves, that is, they have neither beginning nor end. This is a significant difference between a magnetic field and an electric field, the lines of which, as we saw in § 18, always begin on some charges and end on others. We saw, for example, that the electric field lines end on the surface of a metal body, which turns out to be charged, and the electric field does not penetrate into the metal. Observation of the magnetic field shows, on the contrary, that its 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 on 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, 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 magnetic field lines 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 electric field lines go from charge to charge, while the magnetic field has neither beginning nor end, and its lines are closed.
If in experiments that give pictures of the magnetic field of a current, the filings are replaced with small magnetic arrows, then their northern ends will indicate the direction of the field lines, i.e., 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 mutual 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 magnetic field lines created by this current: a) the current is directed from top to bottom; b) the current is directed from bottom to top
Rice. 219. To the gimlet rule
If you screw in the gimlet (the 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 the gimlet in so that it goes in the direction of the field (along the field lines), 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 they can be equally applied to determining the relationship between the direction of the current and the direction of the magnetic induction of the field for any shape of conductors.
124.1. Indicate which pole of the magnetic needle in Fig. 73 northern and which southern.
124.2. Wires from the current source are connected to the tops of the wire parallelogram (Fig. 220). What is the magnetic field induction 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. For 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 are equal and have the direction indicated in the figure. Find graphically the direction of the vector at point . Indicate in which plane this vector lies. What angle does it make with the plane passing through and?
Rice. 221. For exercise 124.3
124.4. Carry out the same construction as in problem 124.3, reversing: a) the direction of the current in the conductor; b) direction of 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 in the figure by arrows. 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 coil, then in the horizontal coil, and finally in both.
Rice. 222. For exercise 124.5
Measurements of magnetic induction at different points in the field around a conductor through which current flows show that 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 depends extremely complexly on the size and shape of the conductor through which the current passes. We will limit ourselves to one important case when these dependencies are relatively simple. This is the magnetic field inside the solenoid.
The movement of an electric charge means the movement of the electric force field inherent in the charge, which leads to the appearance of a vortex magnetic field. Like an electric field, a magnetic field is also characterized by intensity, however, the definition of this concept is no longer associated with charge, as was the case with a potential electric field, but with current, i.e., with the movement of electric charges.
The directed translational movement of charges and the vortex magnetic field, which reflects 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 1 theoretically generalized the results of these measurements, ultimately obtaining the formula (for the magnetic field in a vacuum):
(1)
where J is the current strength; - vector coinciding with the elementary section of the current and directed along the current (Fig. 3); - vector drawn from the current element to the point at which it is determined
R is the modulus of this vector.
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1 Bio Jean Baptiste (1774-1862) - French physicist. The works are devoted to optics, electromagnetism, acoustics, and the history of science.
Savard Felix (1791 - 1841) - French physicist. The works relate to optics, electromagnetism, acoustics, and fluid mechanics.
Laplace Pierre Simon (1749 - 1827) - French mathematician, physicist and astronomer. Physical research relates to molecular physics, acoustics, electricity, optics.
As can be seen from expression (1), the vector is directed perpendicular to the plane passing through and the point at which the field is calculated, its direction is determined by the rotation of the head of the right screw, the translational movement of which coincides with the direction. For the dH module, you can write the following expression:
(2)
where a is the angle between the vectors and .
Consider the field created by a current flowing through a thin wire shaped like a circle with radius R (circular current). Let us determine the magnetic field strength at the center
circular current (Fig. 4). Each current element creates a tension in the center, directed along the positive normal to the contour. Therefore, the vector addition of elements is reduced to the addition of their modules. According to formula (2)
Let's calculate dH for the case a=p/2:
Let's integrate this expression over the entire contour:
(3)
If the circuit consists of n turns, then the magnetic field strength at the center will be equal to:
Description of equipment and measurement method
The purpose of this work is to determine the value. A device called tangent galvanometer, which consists of a ring-shaped conductor or flat coil of large radius. The plane of the coil is located vertically and by rotating about the vertical axis it can be given any position. In the center of the coil there is a compass with a magnetic needle. Rice. 5 gives a cross section of the device with a horizontal plane passing through the center of the coil, N.S.- direction of the magnetic meridian, A and D - coil cross-sections, N.S.- magnetic compass needle.
The dial scale is divided into degrees.
If there is no current in the coil, the arrow N.S. Only the Earth's magnetic field acts and the arrow is set in the direction of the magnetic meridian NS.
By turning about the vertical axis, the plane of the coil is aligned with the plane of the magnetic meridian.
If, after installing the coil in this way, a current is passed through it, the arrow will deviate by an angle a. Now the magnetic needle is under the influence of two fields: the Earth's magnetic field () and the magnetic field created by the current (). Provided that the plane of the turn is aligned with the plane of the meridian, the vectors and are mutually perpendicular, then (see Fig. 5)
; = (5)
Since the length of the magnetic needle is small compared to the radius of the coil, it can be considered a constant value within the needle (the field is uniform) and equal to its value at the center of the coil, determined by formula (4).
Solving equations (4) and (5) together, we obtain
where m is the number of turns of the coil.
Formula (6) can be used to determine H 0 in this work
The order of work and processing of measurement results
1. Assemble the installation according to the diagram (Fig. 6) and, without turning on the current, rotate the stand of the tangent galvanometer so that the turns of its coil are in the plane of the magnetic meridian (see above).
2. Turn on the installation and set the current J with a rheostat, selecting a certain angle of deflection of the arrow (within 35 0 -55 0). After waiting for the arrow to reach the equilibrium position, count the angle of its deviation from the plane of the frame a 1. These values of J and a 1 are entered in the table. 1.
3. Without changing the current in magnitude, change its direction with switch P, measure and write down the value of angle a 2 in the table.
4. Check the zero setting of the device and repeat the measurements at the same current again.
Calculate the arithmetic mean value of angle a for a given current J (from four measurements):
5. Perform several more similar experiments (3 - 5) at different currents, choosing the angles of deflection of the arrow within the same limits (35 0 -55 0); enter the results into the table.
6. For each experiment, use formula (6) to calculate H i, (take a= ), and calculate the average value, which is entered into the table (n is the number of experiments at different currents)
7. Assess the measurement errors H. To do this, it is necessary to determine the standard deviation using the formula
s av = 
.
D / = DJ/J +DR/R+D(tga)/tga
The last term of this expression shows that the relative error is a function of the angle, which has the smallest value at a = 45 0 (therefore, the deviation angle a should be taken within the range of 35 0 -55 0). Hence