Kikoin A.K. Magnetic moment of the current // Kvant. - 1986. - No. 3. - S. 22-23.
By special agreement with the editorial board and the editors of the journal "Quantum"
From the course of physics of the ninth grade ("Physics 9", § 88) it is known that on a rectilinear conductor of length l with current I, if it is placed in a uniform magnetic field with induction \(~\vec B\), a force \(~\vec F\) acts, equal in modulus
\(~F = BIl \sin \alpha\) ,
Where α - the angle between the direction of the current and the vector of magnetic induction. This force is directed perpendicular to both the field and the current (according to the rule of the left hand).
A straight conductor is only part of an electrical circuit, since the electric current is always closed. And how does a magnetic field act on a closed current, more precisely, on a closed circuit with current?
Figure 1 shows as an example a contour in the form of a rectangular frame with sides a And b, through which the current flows in the direction indicated by the arrows I.
The frame is placed in a uniform magnetic field with induction \(~\vec B\) so that at the initial moment the vector \(~\vec B\) lies in the plane of the frame and is parallel to two of its sides. Considering each of the sides of the frame separately, we find that on the sides (of length A) there are forces equal in modulus F = BIa and directed in opposite directions. Forces do not act on the other two sides (for them, sin α = 0). Each of the forces F relative to the axis passing through the midpoints of the upper and lower sides of the frame, creates a moment of force (torque) equal to \(~\frac(BIab)(2)\) (\(~\frac(b)(2)\) - arm strength). The signs of the moments are the same (both forces turn the frame in the same direction), so the total torque M equals BIab, or, since the product ab equal to area S framework,
\(~M = BIab = BIS\) .
Under the action of this moment, the frame will begin to rotate (if viewed from above, then clockwise) and will rotate until it becomes its plane perpendicular to the induction vector \(~\vec B\) (Fig. 2).
In this position, the sum of forces and the sum of moments of forces are equal to zero, and the frame is in a state of stable equilibrium. (In fact, the frame will not stop immediately - for some time it will oscillate around its equilibrium position.)
It is easy to show (do it yourself) that at any intermediate position, when the normal to the plane of the contour makes an arbitrary angle β with induction magnetic field, the torque is
\(~M = BIS \sin \beta\) .
It can be seen from this expression that for a given value of the field induction and for a certain position of the circuit with current, the torque depends only on the product of the area of the circuit S for current strength I in him. the value IS and is called the magnetic moment of the circuit with current. More precisely, IS is the modulus of the magnetic moment vector. And this vector is directed perpendicular to the plane of the circuit and, moreover, so that if you mentally rotate the gimlet in the direction of the current in the circuit, then the direction of the translational movement of the gimlet will indicate the direction of the magnetic moment. For example, the magnetic moment of the circuit shown in Figures 1 and 2 is directed away from us beyond the plane of the page. The magnetic moment is measured in A m 2.
Now we can say that a circuit with a current in a uniform magnetic field is set up so that its magnetic moment "looks" in the direction of the field that caused it to turn.
It is known that not only current-carrying circuits have the ability to create their own magnetic field and rotate in an external field. The same properties are observed in a magnetized rod, for example, in a compass needle.
Back in 1820, the remarkable French physicist Ampere expressed the idea that the similarity of the behavior of a magnet and a circuit with a current is due to the fact that there are closed currents in the particles of a magnet. It is now known that atoms and molecules do indeed have the smallest electric currents associated with the movement of electrons in their orbits around nuclei. Because of this, the atoms and molecules of many substances, such as paramagnets, have magnetic moments. The rotation of these moments in an external magnetic field leads to the magnetization of paramagnetic substances.
It also turned out to be something else. All particles that make up the atom also have magnetic moments that are not at all associated with any movement of charges, that is, with currents. For them, the magnetic moment is the same "innate" quality as charge, mass, etc. Even a particle that does not have an electric charge, the neutron, an integral part of atomic nuclei, has a magnetic moment. Therefore, atomic nuclei also have a magnetic moment.
Thus, the magnetic moment is one of the most important concepts in physics.
MAGNETIC TORQUE- physical. quantity characterizing the magnetic. charge system properties. particles (or individual particles) and determining, along with other multipole moments (electric dipole moment, quadrupole moment, etc., see Multipoli) the interaction of the system with the external. el-magn. fields and other similar systems.
According to the ideas of the classical electrodynamics, magnet. the field is created by moving electric. charges. Although modern theory does not reject (and even predicts) the existence of particles with magnetic. charge ( magnetic monopoles), such particles have not yet been experimentally observed and are absent in ordinary matter. Therefore, the elementary characteristic of the magnet. properties turns out to be exactly the M. m. A system that has a M. m. (axial vector) creates a magnetic field at large distances from the system. field
(- radius vector of the observation point). A similar view has an electric. dipole field, consisting of two closely spaced electric. charges of opposite sign. However, unlike electrical dipole moment. M. m. is created not by a system of point "magnetic charges", but by electric. currents flowing within the system. If a closed electric density current flows in a limited volume V, then the M. m. created by him is determined by the f-loy
In the simplest case of a closed circular current I, flowing along a flat coil of area s, , and the vector of the M. m. is directed along the right normal to the coil.
If the current is created by the stationary movement of point electric. charges with masses having velocities , then the resulting M. m., as follows from f-ly (1), has the form
where is meant microscopic averaging. values over time. Since the vector product on the right side is proportional to the momentum vector of the particle's momentum 
(it is assumed that the speeds ), then the contributions of the dep. particles in M. m. and at the moment of the number of movements are proportional:
Proportionality factor e/2ts called gyromagnetic ratio; this value characterizes the universal connection between the magnetic. and mechanical charge properties. particles in the classical electrodynamics. However, the movement of elementary charge carriers in matter (electrons) obeys the laws of quantum mechanics, which makes adjustments to the classical. picture. In addition to the orbital mechanical moment of motion L The electron has an internal mechanical moment - back. The total magnetic field of an electron is equal to the sum of the orbital magnetic field (2) and the spin magnetic field.
As can be seen from this formula (following from the relativistic Dirac equations for electron), gyromagnet. the ratio for the spin turns out to be exactly twice that for the orbital momentum. A feature of the quantum concept of magnet. and mechanical moments is also the fact that vectors cannot have a definite direction in space due to the non-commutativity of the projection operators of these vectors on the coordinate axes.
Spin M. m. charge. particles defined f-loy (3), called. normal, for an electron it is magneton Bora. Experience shows, however, that the M. m. of an electron 
differs from (3) by an order of magnitude ( is the fine structure constant). A similar supplement called abnormal magnetic moment, arises due to the interaction of an electron with photons, it is described in the framework of quantum electrodynamics. Other elementary particles also have anomalous magnetic properties; they are especially large for hadrons, to-rye, according to modern. representations, have vnutr. structure. Thus, the anomalous M. m. of the proton is 2.79 times greater than the "normal" one - the nuclear magneton, ( M- the mass of the proton), and the M. m. of the neutron is equal to -1.91, i.e., it is significantly different from zero, although the neutron does not have electric power. charge. Such large anomalous M. m. hadrons due to internal. the movement of their constituent charges. quarks.
Lit .: Landau L. D., Lifshits E. M., Field Theory, 7th ed., M., 1988; Huang K., Quarks, leptons and gauge fields, transl. from English, M., 1985. D. V. Giltsov.
The magnetic moment of a coil with current is physical quantity, like any other magnetic moment, characterizes the magnetic properties of a given system. In our case, the system is represented by a circular loop with current. This current creates a magnetic field that interacts with an external magnetic field. It can be either the field of the earth, or the field of a constant or electromagnet.
Drawing— 1 circular turn with current
A circular coil with current can be represented as a short magnet. Moreover, this magnet will be directed perpendicular to the plane of the coil. The location of the poles of such a magnet is determined using the gimlet rule. According to which the north plus will be behind the plane of the coil if the current in it moves clockwise.
Drawing— 2 Imaginary bar magnet on the axis of the coil
This magnet, that is, our circular coil with current, like any other magnet, will be affected by an external magnetic field. If this field is uniform, then a torque will arise that will tend to turn the coil. The field will rotate the coil so that its axis is located along the field. In this case, the lines of force of the coil itself, like a small magnet, must coincide in direction with the external field.
If the external field is not uniform, then translational motion will be added to the torque. This movement will arise due to the fact that areas of the field with a higher induction will attract our magnet in the form of a coil more than areas with a lower induction. And the coil will begin to move towards the field with greater induction.
The magnitude of the magnetic moment of a circular coil with current can be determined by the formula.
Formula - 1 Magnetic moment of the coil
Where, I current flowing through the coil
S area of the coil with current
n normal to the plane in which the coil is located
Thus, it can be seen from the formula that the magnetic moment of the coil is a vector quantity. That is, in addition to the magnitude of the force, that is, its module, it also has a direction. This property received a magnetic moment due to the fact that it includes the normal vector to the plane of the coil.
To consolidate the material, you can conduct a simple experiment. To do this, we need a circular coil, made of copper wire, connected to a battery. In this case, the lead wires should be thin enough and preferably twisted together. This will reduce their impact on the experience.
Drawing—
Now let's hang a turn on the lead wires in a uniform magnetic field created, say, by permanent magnets. The coil is still de-energized, and its plane is parallel to the field lines of force. In this case, its axis and poles of an imaginary magnet will be perpendicular to the lines of the external field.
Drawing—
When current is applied to the coil, its plane will turn perpendicular to the lines of force of the permanent magnet, and the axis will become parallel to them. Moreover, the direction of rotation of the coil will be determined by the gimlet rule. And strictly speaking, the direction in which the current flows through the coil.
Experiments by Stern and Gerlach
In $1921$, O. Stern put forward the idea of an experiment in measuring the magnetic moment of an atom. He carried out this experiment in co-authorship with W. Gerlach in $1922$. The method of Stern and Gerlach uses the fact that a beam of atoms (molecules) is able to deviate in an inhomogeneous magnetic field. An atom that has a magnetic moment can be represented as an elementary magnet with small but finite dimensions. If such a magnet is placed in a uniform magnetic field, then it does not experience force. The field will act on the north and south poles of such a magnet with forces that are equal in magnitude and opposite in direction. As a result, the center of inertia of the atom will either be at rest or move in a straight line. (In this case, the axis of the magnet can oscillate or precess). That is, in a uniform magnetic field there are no forces that act on an atom and impart acceleration to it. A uniform magnetic field does not change the angle between the directions of the magnetic field induction and the magnetic moment of the atom.
The situation is different if the external field is inhomogeneous. In this case, the forces that act on the north and south poles of the magnet are not equal. The resulting force acting on the magnet is non-zero, and it imparts an acceleration to the atom, along the field or against it. As a result, when moving in an inhomogeneous field, the magnet under consideration will deviate from the original direction of movement. In this case, the size of the deviation depends on the degree of field inhomogeneity. In order to obtain significant deviations, the field must change sharply already within the length of the magnet (the linear dimensions of the atom are $\approx (10)^(-8)cm$). Experimenters achieved such heterogeneity with the help of the design of a magnet that created a field. One magnet in the experiment looked like a blade, the other was flat or had a notch. The magnetic lines thickened at the "blade", so that the intensity in this area was significantly greater than at the flat pole. A thin beam of atoms flew between these magnets. Individual atoms were deflected in the generated field. Traces of individual particles were observed on the screen.
According to the concepts of classical physics, magnetic moments in an atomic beam have different directions with respect to some axis $Z$. What does it mean: the projection of the magnetic moment ($p_(mz)$) on this axis takes all the values of the interval from $\left|p_m\right|$ to -$\left|p_m\right|$ (where $\left|p_( mz)\right|-$ magnetic moment modulus). On the screen, the beam should appear expanded. However, in quantum physics, if quantization is taken into account, then not all orientations of the magnetic moment become possible, but only a finite number of them. Thus, on the screen, the trace of a beam of atoms was split into a certain number of individual traces.
The experiments performed showed that, for example, a beam of lithium atoms split into $24$ beams. This is justified, since the main term $Li - 2S$ is a term (one valence electron with spin $\frac(1)(2)\ $ in the s-orbit, $l=0).$ it is possible to draw a conclusion about the magnitude of the magnetic moment. This is how Gerlach proved that the spin magnetic moment is equal to the Bohr magneton. Studies of various elements showed complete agreement with theory.
Stern and Rabi measured the magnetic moments of nuclei using this approach.
So, if the projection $p_(mz)$ is quantized, the average force that acts on the atom from the magnetic field is quantized along with it. The experiments of Stern and Gerlach proved the quantization of the projection of the magnetic quantum number onto the $Z$ axis. It turned out that the magnetic moments of the atoms are directed parallel to the $Z$ axis, they cannot be directed at an angle to this axis, so we had to accept that the orientation of the magnetic moments relative to the magnetic field changes discretely. This phenomenon has been called spatial quantization. The discreteness of not only the states of atoms, but also the orientations of the magnetic moments of an atom in an external field is a fundamentally new property of the movement of atoms.
The experiments were fully explained after the discovery of the electron spin, when it was found that the magnetic moment of the atom is caused not by the orbital moment of the electron, but by the internal magnetic moment of the particle, which is associated with its internal mechanical moment (spin).
Calculation of the motion of the magnetic moment in an inhomogeneous field
Let an atom move in an inhomogeneous magnetic field, its magnetic moment is equal to $(\overrightarrow(p))_m$. The force acting on it is:
In general, an atom is an electrically neutral particle, so other forces do not act on it in a magnetic field. By studying the motion of an atom in an inhomogeneous field, one can measure its magnetic moment. Let us assume that the atom moves along the $X$ axis, the field inhomogeneity is created in the direction of the $Z$ axis (Fig. 1):
Picture 1.
\frac()()\frac()()
Using conditions (2), we transform expression (1) into the form:
The magnetic field is symmetrical with respect to the y=0 plane. It can be assumed that the atom moves in this plane, which means that $B_x=0.$ The equality $B_y=0$ is violated only in small areas near the edges of the magnet (we neglect this violation). From the above it follows that:
In this case, expressions (3) have the form:
The precession of atoms in a magnetic field does not affect $p_(mz)$. We write the equation of motion of an atom in the space between the magnets in the form:
where $m$ is the mass of the atom. If an atom passes the path $a$ between the magnets, then it deviates from the X axis by a distance equal to:
where $v$ is the speed of the atom along the $X$ axis. Leaving the space between the magnets, the atom continues to move at a constant angle with respect to the $X$ axis along a straight line. In formula (7) the quantities $\frac(\partial B_z)(\partial z)$, $a$, $v\ and\ m$ are known, by measuring z one can calculate $p_(mz)$.
Example 1
Exercise: How many components, when conducting an experiment similar to the experiment of Stern and Gerlach, will the beam of atoms split if they are in the state $()^3(D_1)$?
Solution:
A term splits into $N=2J+1$ sublevels if the Lande multiplier is $g\ne 0$, where
To find the number of components into which the beam of atoms will split, we should determine the total internal quantum number $(J)$, the multiplicity $(S)$, the orbital quantum number, compare the Lande multiplier with zero, and if it is nonzero, then calculate the number sublevels.
1) To do this, consider the structure of the symbolic record of the state of the atom ($3D_1$). Our term is deciphered as follows: the symbol $D$ corresponds to the orbital quantum number $l=2$, $J=1$, the multiplicity of $(S)$ is equal to $2S+1=3\to S=1$.
We calculate $g,$ by applying formula (1.1):
The number of components into which the beam of atoms is split is equal to:
Answer:$N=3.$
Example 2
Exercise: Why was a beam of hydrogen atoms, which were in the $1s$ state, used in the experiment of Stern and Gerlach to detect the spin of an electron?
Solution:
In the $s-$ state, the angular momentum of the electron $(L)$ is equal to zero, since $l=0$:
The magnetic moment of an atom, which is associated with the movement of an electron in orbit, is proportional to the mechanical moment:
\[(\overrightarrow(p))_m=-\frac(q_e)(2m)\overrightarrow(L)(2.2)\]
hence it is equal to zero. This means that the magnetic field should not affect the movement of hydrogen atoms in the ground state, that is, split the flow of particles. But when using spectral instruments, it was shown that the lines of the hydrogen spectrum show the presence of a fine structure (doublets) even if there is no magnetic field. In order to explain the presence of a fine structure, the idea of an intrinsic mechanical angular momentum of an electron in space (spin) was put forward.
Various media, when considering their magnetic properties, are called magnets .
All substances in one way or another interact with a magnetic field. Some materials retain their magnetic properties even in the absence of an external magnetic field. The magnetization of materials occurs due to the currents circulating inside the atoms - the rotation of electrons and their movement in the atom. Therefore, the magnetization of a substance should be described using real atomic currents, called Ampere currents.
In the absence of an external magnetic field, the magnetic moments of the atoms of a substance are usually randomly oriented, so that the magnetic fields they create cancel each other out. When an external magnetic field is applied, the atoms tend to orient their magnetic moments in the direction of the external magnetic field, and then the compensation of magnetic moments is violated, the body acquires magnetic properties - it becomes magnetized. Most bodies are magnetized very weakly and the magnitude of the magnetic field induction B in such substances differs little from the magnitude of the magnetic field induction in vacuum. If the magnetic field is weakly amplified in a substance, then such a substance is called paramagnetic :
( , , , , , , Li, Na);
if it weakens, then it diamagnetic :
(Bi, Cu, Ag, Au, etc.) .
But there are substances that have strong magnetic properties. Such substances are called ferromagnets :
(Fe, Co, Ni, etc.).
These substances are able to retain magnetic properties even in the absence of an external magnetic field, representing permanent magnets.
All bodies when they are introduced into an external magnetic field are magnetized to one degree or another, i.e. create their own magnetic field, which is superimposed on an external magnetic field.
Magnetic properties of matter are determined by the magnetic properties of electrons and atoms.
Magnetics consist of atoms, which, in turn, consist of positive nuclei and, relatively speaking, electrons revolving around them.
An electron moving in an orbit in an atom is equivalent to a closed circuit with orbital current :
Where e is the electron charge, ν is the frequency of its orbital rotation:
The orbital current corresponds to orbital magnetic moment electron
 ,
|
(6.1.1) |
Where S is the area of the orbit, is the unit normal vector to S, is the electron velocity. Figure 6.1 shows the direction of the orbital magnetic moment of an electron.
An electron moving in an orbit has orbital angular momentum , which is directed opposite to and is related to it by the relation
Where m is the mass of the electron.
In addition, the electron has own angular momentum, which is called electron spin
| , | (6.1.4) |
Where 
, 
is Planck's constant
The spin of an electron corresponds to spin magnetic moment electron directed in the opposite direction:
| , | (6.1.5) |
The value is called gyromagnetic ratio of spin moments