Force acting from outside magnetic field on a moving electrically charged particle.
where q is the charge of the particle;
V - charge speed;
a is the angle between the charge velocity vector and the magnetic induction vector.
The direction of the Lorentz force is determined according to the left hand rule:
If you put left hand so that the component of the induction vector perpendicular to the speed enters the palm, and the four fingers are located in the direction of the speed of movement of the positive charge (or against the direction of the speed negative charge), then bent thumb will indicate the direction of the Lorentz force:
. 
Since the Lorentz force is always perpendicular to the charge speed, it does not do work (i.e. does not change the value of the charge speed and its kinetic energy).
If a charged particle moves parallel to the magnetic field lines, then Fl = 0, and the charge in the magnetic field moves uniformly and rectilinearly.
If a charged particle moves perpendicular to the magnetic field lines, then the Lorentz force is centripetal:
and creates centripetal acceleration equal:
In this case, the particle moves in a circle.
. 
According to Newton's second law: the Lorentz force is equal to the product of the mass of the particle and the centripetal acceleration:
then the radius of the circle:
and the period of charge revolution in a magnetic field:
Since electric current represents the ordered movement of charges, the effect of a magnetic field on a conductor carrying current is the result of its action on individual moving charges. If we introduce a current-carrying conductor into a magnetic field (Fig. 96a), we will see that as a result of the addition of the magnetic fields of the magnet and the conductor, the resulting magnetic field will increase on one side of the conductor (in the drawing above) and the magnetic field will weaken on the other side conductor (in the drawing below). As a result of the action of two magnetic fields, the magnetic lines will bend and, trying to contract, they will push the conductor down (Fig. 96, b).
The direction of the force acting on a current-carrying conductor in a magnetic field can be determined by the “left-hand rule.” If the left hand is placed in a magnetic field so that the magnetic lines coming out of the north pole seem to enter the palm, and the four extended fingers coincide with the direction of the current in the conductor, then the large bent finger of the hand will show the direction of the force. Ampere force acting on an element of the length of the conductor depends on: the magnitude of the magnetic induction B, the magnitude of the current in the conductor I, the element of the length of the conductor and the sine of the angle a between the direction of the element of the length of the conductor and the direction of the magnetic field.
This dependence can be expressed by the formula:
For a straight conductor of finite length, placed perpendicular to the direction of a uniform magnetic field, the force acting on the conductor will be equal to:
From the last formula we determine the dimension of magnetic induction.
Since the dimension of force is:
i.e., the dimension of induction is the same as what we obtained from Biot and Savart’s law.
Tesla (unit of magnetic induction)
Tesla, unit of magnetic induction International System of Units, equal magnetic induction, at which the magnetic flux through a cross section of area 1 m 2 equals 1 Weber. Named after N. Tesla. Designations: Russian tl, international T. 1 tl = 104 gs(gauss).
Magnetic torque, magnetic dipole moment- the main quantity characterizing magnetic properties substances. The magnetic moment is measured in A⋅m 2 or J/T (SI), or erg/Gs (SGS), 1 erg/Gs = 10 -3 J/T. The specific unit of elementary magnetic moment is the Bohr magneton. In the case of a flat circuit with electric current, the magnetic moment is calculated as
Where - current strength in a contour, is the area of the contour, is the unit vector of the normal to the plane of the contour. The direction of the magnetic moment is usually found according to the gimlet rule: if you rotate the handle of the gimlet in the direction of the current, then the direction of the magnetic moment will coincide with the direction of the translational movement of the gimlet.
For an arbitrary closed loop, the magnetic moment is found from:
,
where is the radius vector drawn from the origin to the contour length element
In the general case of arbitrary current distribution in a medium:
,
where is the current density in the volume element.
So, a torque acts on a current-carrying circuit in a magnetic field. The contour is oriented at a given point in the field in only one way. Let's take the positive direction of the normal to be the direction of the magnetic field at a given point. Torque is directly proportional to current I, contour area S and the sine of the angle between the direction of the magnetic field and the normal.
Here M - torque , or moment of power , - magnetic moment circuit (similarly - the electric moment of the dipole).
In an inhomogeneous field (), the formula is valid if the outline size is quite small(then the field can be considered approximately uniform within the contour). Consequently, the circuit with current still tends to turn around so that its magnetic moment is directed along the lines of the vector.
But, in addition, a resultant force acts on the circuit (in the case of a uniform field and . This force acts on a circuit with current or on a permanent magnet with a moment and draws them into a region of a stronger magnetic field.
Work on moving a circuit with current in a magnetic field.
It is easy to prove that the work done to move a current-carrying circuit in a magnetic field is equal to 
, where and are the magnetic fluxes through the contour area in the final and initial positions. This formula is valid if the current in the circuit is constant, i.e. When moving the contour, the phenomenon is not taken into account electromagnetic induction.
The formula is also valid for large circuits in a highly inhomogeneous magnetic field (provided I= const).
Finally, if the circuit with current is not displaced, but the magnetic field is changed, i.e. change the magnetic flux through the surface covered by the circuit from value to then for this you need to do the same work 
. This work is called the work of changing the magnetic flux associated with the circuit. Magnetic induction vector flux (magnetic flux) through the pad dS is called scalar physical quantity, which is equal
where B n =Вcosα is the projection of the vector IN to the direction of the normal to the site dS (α is the angle between the vectors n And IN), d S= dS n- a vector whose module is equal to dS, 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 cosα (set by choosing the positive direction of the normal n). Flow vector IN usually associated with a circuit through which current flows. In this case, we specified the positive direction of the normal to the contour: it is associated with the current by the rule of the right screw. This means that the magnetic flux that is created by the circuit through the surface limited by itself is always positive.
The flux of the magnetic induction vector Ф B through an arbitrary given surface S is equal to
(2)
For a uniform field and a flat surface, which is located perpendicular to the vector IN, B n =B=const and
This formula gives the unit of magnetic flux weber(Wb): 1 Wb is a magnetic flux that passes through a flat surface with an area of 1 m 2, which is located perpendicular to a uniform magnetic field and whose induction is 1 T (1 Wb = 1 T.m 2).
Gauss's theorem for field B: the flux of the magnetic induction vector through any closed surface is zero:
(3)
This theorem is a reflection of the fact that no magnetic charges, as a result of which the lines of magnetic induction have neither beginning nor end and are closed.
Therefore, for streams of vectors IN And E through a closed surface in the vortex and potential fields, different formulas are obtained.
As an example, let's find the vector flow IN through the solenoid. The magnetic induction of a uniform field inside a solenoid with a core with magnetic permeability μ is equal to
The magnetic flux through one turn of the solenoid with area S is equal to
and the total magnetic flux, which is linked to all turns of the solenoid and is called flux linkage,
ABSTRACT
In the subject "Physics"
Topic: “Application of the Lorentz force”
Completed by: Student of group T-10915 Logunova M.V.
Teacher Vorontsov B.S.
Kurgan 2016
Introduction. 3
1. Use of the Lorentz force. 4
.. 4
1. 2 Mass spectrometry. 6
1. 3 MHD generator. 7
1. 4 Cyclotron. 8
Conclusion. eleven
List of used literature... 13
Introduction
Lorentz force- the force with which the electromagnetic field, according to classical (non-quantum) electrodynamics, acts on a point charged particle. Sometimes the Lorentz force is called the force acting on a moving object with speed υ charge q only from the side of the magnetic field, often full force- from the side electromagnetic field in general, in other words, from the electrical side E and magnetic B fields.
In the International System of Units (SI) it is expressed as:
F L = q υ B sin α
It is named after the Dutch physicist Hendrik Lorentz, who derived an expression for this force in 1892. Three years before Lorenz, the correct expression was found by O. Heaviside.
The macroscopic manifestation of the Lorentz force is the Ampere force.
Using the Lorentz force
The effect exerted by a magnetic field on moving charged particles is very widely used in technology.
The main application of the Lorentz force (more precisely, its special case - the Ampere force) is electrical machines (electric motors and generators). The Lorentz force is widely used in electronic devices to influence charged particles (electrons and sometimes ions), for example, in television cathode ray tubes , V mass spectrometry And MHD generators.
Also, in the currently created experimental installations for carrying out a controlled thermonuclear reaction, the action of a magnetic field on the plasma is used to twist it into a cord that does not touch the walls of the working chamber. The circular motion of charged particles in a uniform magnetic field and the independence of the period of such motion from the particle speed are used in cyclic accelerators of charged particles - cyclotrons.
1. 1. Electron beam devices
Electron beam devices (EBDs) are a class of vacuum electronic devices that use a flow of electrons, concentrated in the form of a single beam or beam of beams, which are controlled both in intensity (current) and position in space, and interact with a stationary spatial target (screen) of the device. The main area of application of ELP is the conversion of optical information into electrical signals and the reverse conversion of the electrical signal into an optical signal - for example, into a visible television image.
The class of cathode-ray devices does not include X-ray tubes, photocells, photomultipliers, gas-discharge devices (dekatrons) and receiving and amplifying electron tubes (beam tetrodes, electric vacuum indicators, lamps with secondary emission, etc.) with a beam form of currents.
An electron beam device consists of at least three main parts:
· An electronic spotlight (gun) forms an electron beam (or a beam of rays, for example, three beams in a color picture tube) and controls its intensity (current);
· The deflection system controls the spatial position of the beam (its deviation from the axis of the spotlight);
· The target (screen) of the receiving ELP converts the energy of the beam into the luminous flux of a visible image; the target of the transmitting or storing ELP accumulates a spatial potential relief, read by a scanning electron beam
| Rice. 1 CRT device |
General principles of the device.
A deep vacuum is created in the CRT cylinder. To create an electron beam, a device called an electron gun is used. The cathode, heated by the filament, emits electrons. By changing the voltage on the control electrode (modulator), you can change the intensity of the electron beam and, accordingly, the brightness of the image. After leaving the gun, the electrons are accelerated by the anode. Next, the beam passes through a deflection system, which can change the direction of the beam. Television CRTs use a magnetic deflection system as it provides large deflection angles. Oscillographic CRTs use an electrostatic deflection system as it provides greater performance. The electron beam hits a screen covered with phosphor. Bombarded by electrons, the phosphor glows and a rapidly moving spot of variable brightness creates an image on the screen.
1. 2 Mass spectrometry
| Rice. 2 |
The Lorentz force is also used in instruments called mass spectrographs, which are designed to separate charged particles according to their specific charges.
Mass spectrometry(mass spectroscopy, mass spectrography, mass spectral analysis, mass spectrometric analysis) - a method for studying a substance based on determining the mass-to-charge ratio of ions formed by ionization of the sample components of interest. One of the most powerful ways of qualitative identification of substances, which also allows quantitative determination. We can say that mass spectrometry is the “weighing” of the molecules in a sample.
The diagram of the simplest mass spectrograph is shown in Figure 2.
In chamber 1, from which the air has been evacuated, there is an ion source 3. The chamber is placed in a uniform magnetic field, at each point of which the induction B⃗ B→ is perpendicular to the plane of the drawing and directed towards us (in Figure 1 this field is indicated by circles). An accelerating voltage is applied between electrodes A and B, under the influence of which the ions emitted from the source are accelerated and at a certain speed enter the magnetic field perpendicular to the induction lines. Moving in a magnetic field along a circular arc, the ions fall on photographic plate 2, which makes it possible to determine the radius R of this arc. Knowing the magnetic field induction B and the speed υ of ions, according to the formula
the specific charge of ions can be determined. And if the charge of the ion is known, its mass can be calculated.
The history of mass spectrometry dates back to the seminal experiments of J. J. Thomson at the beginning of the 20th century. The ending “-metry” in the name of the method appeared after the widespread transition from detecting charged particles using photographic plates to electrical measurements ion currents.
Mass spectrometry is especially widely used in the analysis organic matter, since it provides confident identification of both relatively simple and complex molecules. The only general requirement is that the molecule be ionizable. However, by now it has been invented
There are so many ways to ionize sample components that mass spectrometry can be considered an almost all-encompassing method.
1. 3 MHD generator
Magnetohydrodynamic generator, MHD generator is a power plant in which the energy of a working fluid (liquid or gaseous electrically conducting medium) moving in a magnetic field is converted directly into electrical energy.
The operating principle of an MHD generator, like a conventional machine generator, is based on the phenomenon of electromagnetic induction, that is, on the occurrence of a current in a conductor crossing magnetic field lines. Unlike machine generators, the conductor in an MHD generator is the working fluid itself.
The working fluid moves across the magnetic field, and under the influence of the magnetic field, oppositely directed flows of charge carriers of opposite signs arise.
The Lorentz force acts on a charged particle.
The following media can serve as the working fluid of the MHD generator:
· electrolytes;
· liquid metals;
· plasma (ionized gas).
The first MHD generators used electrically conductive liquids (electrolytes) as a working fluid. Currently, plasma is used in which the charge carriers are mainly free electrons and positive ions. Under the influence of a magnetic field, charge carriers deviate from the trajectory along which the gas would move in the absence of the field. In this case, in a strong magnetic field, a Hall field can arise (see Hall effect) - an electric field formed as a result of collisions and displacements of charged particles in a plane perpendicular to the magnetic field.
1. 4 Cyclotron
A cyclotron is a resonant cyclic accelerator of non-relativistic heavy charged particles (protons, ions), in which the particles move in a constant and uniform magnetic field, and a high-frequency electric field of constant frequency is used to accelerate them.
The circuit diagram of the cyclotron is shown in Fig. 3. Heavy charged particles (protons, ions) enter the chamber from an injector near the center of the chamber and are accelerated by an alternating field of a fixed frequency applied to the accelerating electrodes (there are two of them and they are called dees). Particles with charge Ze and mass m move in a constant magnetic field of intensity B, directed perpendicular to the plane of motion of the particles, in an unwinding spiral. The radius R of the trajectory of a particle having a speed v is determined by the formula
where γ = -1/2 is the relativistic factor.
In a cyclotron, for a non-relativistic (γ ≈ 1) particle in a constant and uniform magnetic field, the orbital radius is proportional to the speed (1), and the rotation frequency of the non-relativistic particle (the cyclotron frequency does not depend on the particle energy
| E = mv 2 /2 = (Ze) 2 B 2 R 2 /(2m) (3) |
In the gap between the dees, particles are accelerated by a pulse electric field(there is no electric field inside hollow metal dees). As a result, the energy and radius of the orbit increase. By repeating the acceleration by the electric field at each revolution, the energy and radius of the orbit are brought to the maximum acceptable values. In this case, the particles acquire a speed v = ZeBR/m and the corresponding energy:
At the last turn of the spiral, a deflecting electric field is turned on, leading the beam out. The constancy of the magnetic field and the frequency of the accelerating field make continuous acceleration possible. While some particles are moving along the outer turns of the spiral, others are in the middle of the path, and others are just beginning to move.
The disadvantage of the cyclotron is the limitation by essentially non-relativistic energies of particles, since even not very large relativistic corrections (deviations of γ from unity) disrupt the synchronism of acceleration at different turns and particles with significantly increased energies no longer have time to end up in the gap between the dees in the phase of the electric field required for acceleration . In conventional cyclotrons, protons can be accelerated to 20-25 MeV.
To accelerate heavy particles in an unwinding spiral mode to energies tens of times higher (up to 1000 MeV), a modification of the cyclotron called isochronous(relativistic) cyclotron, as well as a phasotron. In isochronous cyclotrons, relativistic effects are compensated by a radial increase in the magnetic field.
Conclusion
Hidden text
Written conclusion (the most basic for all subparagraphs of the first section - principles of operation, definitions)
List of used literature
1. Wikipedia [Electronic resource]: Lorentz force. URL: https://ru.wikipedia.org/wiki/Lorentz_Force
2. Wikipedia [Electronic resource]: Magnetohydrodynamic generator. URL: https://ru.wikipedia.org/wiki/ Magnetohydrodynamic_generator
3. Wikipedia [Electronic resource]: Electron beam devices. URL: https://ru.wikipedia.org/wiki/ Electron-beam_devices
4. Wikipedia [Electronic resource]: Mass spectrometry. URL: https://ru.wikipedia.org/wiki/Mass spectrometry
5. Nuclear physics on the Internet [Electronic resource]: Cyclotron. URL: http://nuclphys.sinp.msu.ru/experiment/accelerators/ciclotron.htm
6. Electronic textbook of physics [Electronic resource]: T. Applications of the Lorentz force // URL: http://www.physbook.ru/index.php/ T. Applications of the Lorentz force
7. Academician [Electronic resource]: Magnetohydrodynamic generator // URL: http://dic.academic.ru/dic.nsf/enc_physics/MAGNETOHYDRODYNAMIC
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All rights belong to their authors. This site does not claim authorship, but provides free use.
Page creation date: 2017-03-31
Ampere power, acting on a conductor segment of length Δ l with current strength I, located in a magnetic field B,
The expression for the Ampere force can be written as:
This force is called Lorentz force . Angle α in this expression equal to angle between speed and vector of magnetic induction The direction of the Lorentz force acting on a positively charged particle, as well as the direction of the Ampere force, can be found by left hand rule or by gimlet rule. The relative position of the vectors , and for a positively charged particle is shown in Fig. 1.18.1.
|
|
|
Figure 1.18.1. The relative position of the vectors , and The modulus of the Lorentz force is numerically equal to the area of the parallelogram built on the vectors and multiplied by the charge q |
The Lorentz force is directed perpendicular to the vectors and
When a charged particle moves in a magnetic field, the Lorentz force does no work. Therefore, the magnitude of the velocity vector does not change when the particle moves.
If a charged particle moves in a uniform magnetic field under the influence of the Lorentz force, and its speed lies in a plane perpendicular to the vector, then the particle will move in a circle of radius
The period of revolution of a particle in a uniform magnetic field is equal to
called cyclotron frequency . The cyclotron frequency does not depend on the speed (and therefore on the kinetic energy) of the particle. This circumstance is used in cyclotrons – accelerators of heavy particles (protons, ions). The schematic diagram of the cyclotron is shown in Fig. 1.18.3.
A vacuum chamber is placed between the poles of a strong electromagnet, in which there are two electrodes in the form of hollow metal half-cylinders ( dees ). An alternating electrical voltage is applied to the dees, whose frequency is equal to the cyclotron frequency. Charged particles are injected into the center of the vacuum chamber. The particles are accelerated by the electric field in the gap between the dees. Inside the dees, the particles move under the influence of the Lorentz force in semicircles, the radius of which increases as the energy of the particles increases. Every time a particle flies through the gap between the dees, it is accelerated by the electric field. Thus, in a cyclotron, as in all other accelerators, a charged particle is accelerated by an electric field and kept on its trajectory by a magnetic field. Cyclotrons make it possible to accelerate protons to energies of the order of 20 MeV.
Uniform magnetic fields are used in many devices and, in particular, in mass spectrometers – devices with which you can measure the masses of charged particles – ions or nuclei of various atoms. Mass spectrometers are used for separation isotopes, that is, atomic nuclei with the same charge, but different masses(eg 20 Ne and 22 Ne). The simplest mass spectrometer is shown in Fig. 1.18.4. Ions escaping from the source S, pass through several small holes, forming a narrow beam. Then they get into speed selector , in which particles move in crossed homogeneous electric and magnetic fields. An electric field is created between the plates of a flat capacitor, a magnetic field is created in the gap between the poles of an electromagnet. The initial speed of charged particles is directed perpendicular to the vectors and
A particle moving in crossed electric and magnetic fields is acted upon by an electric force and magnetic Lorentz force. Given that E = υ B these forces exactly balance each other. If this condition is met, the particle will move uniformly and rectilinearly and, after flying through the capacitor, will pass through the hole in the screen. For given values of electric and magnetic fields, the selector will select particles moving at speed υ = E / B.
Next, particles with the same speed value enter the mass spectrometer chamber, in which a uniform magnetic field is created. The particles move in the chamber in a plane perpendicular to the magnetic field under the influence of the Lorentz force. Particle trajectories are circles of radii R = mυ / qB". Measuring the radii of trajectories for known values of υ and B" relationship can be determined q / m. In the case of isotopes ( q 1 = q 2) a mass spectrometer allows you to separate particles with different masses.
Modern mass spectrometers make it possible to measure the masses of charged particles with an accuracy higher than 10 –4.
If the velocity of a particle has a component along the direction of the magnetic field, then such a particle will move in a uniform magnetic field in a spiral. In this case, the radius of the spiral R depends on the modulus of the component perpendicular to the magnetic field υ ┴ of the vector and the pitch of the spiral p– from the modulus of the longitudinal component υ || (Fig. 1.18.5).
Thus, the trajectory of a charged particle seems to wind around the magnetic induction line. This phenomenon is used in technology for magnetic thermal insulation of high temperature plasma, that is, a completely ionized gas at a temperature of the order of 10 6 K. A substance in this state is obtained in Tokamak-type installations when studying controlled thermonuclear reactions. The plasma should not come into contact with the walls of the chamber. Thermal insulation is achieved by creating a magnetic field of a special configuration. As an example in Fig. 1.18.6 shows the trajectory of a charged particle in magnetic “bottle”(or trapped ).
A similar phenomenon occurs in the Earth’s magnetic field, which is a protection for all living things from flows of charged particles from outer space. Fast charged particles from space (mainly from the Sun) are “captured” by the Earth’s magnetic field and form so-called radiation belts (Fig. 1.18.7), in which particles, as in magnetic traps, move back and forth along spiral trajectories between the north and south magnetic poles in times of the order of fractions of a second. Only in the polar regions do some particles invade the upper atmosphere, causing auroras. The Earth's radiation belts extend from distances of the order of 500 km to tens of Earth radii. It should be remembered that the south magnetic pole of the Earth is located near the north geographic pole (in northwest Greenland). The nature of terrestrial magnetism has not yet been studied.
Control questions
1.Describe the experiments of Oersted and Ampere.
2.What is the source of the magnetic field?
3. What is Ampere’s hypothesis that explains the existence of the magnetic field of a permanent magnet?
4.What is the fundamental difference between a magnetic field and an electric one?
5. Formulate the definition of the magnetic induction vector.
6. Why is the magnetic field called vortex?
7. Formulate laws:
A) Ampere;
B) Bio-Savart-Laplace.
8. What is the magnitude of the magnetic induction vector of the forward current field?
9. State the definition of the unit of current (ampere) in the International System of Units.
10. Write down the formula expressing the quantity:
A) module of the magnetic induction vector;
B) Ampere forces;
B) Lorentz forces;
D) the period of revolution of a particle in a uniform magnetic field;
D) radius of curvature of a circle when a charged particle moves in a magnetic field;
Self-control test
What was observed in Oersted's experiment?
1) Interaction of two parallel conductors with current.
2) Interaction of two magnetic needles
3) Rotate a magnetic needle near a conductor when current is passed through it.
4) Emergence electric current in a coil when a magnet is pushed into it.
How do two parallel conductors interact if they carry currents in the same direction?
Attracted;
They push off;
The force and moment of forces are zero.
The force is zero, but the moment of force is not zero.
What formula determines the expression for the modulus of the Ampere force?
What formula determines the expression for the modulus of the Lorentz force?
B) 
IN) 
G) 
0.6 N; 2) 1 N; 3) 1.4 N; 4) 2.4 N.
1) 0.5 T; 2) 1 T; 3) 2 T; 4) 0.8 T .
An electron with a speed V flies into a magnetic field with an induction module B perpendicular to the magnetic lines. What expression corresponds to the radius of the electron's orbit?
Answer: 1) 
2)
4)
8. How will the period of revolution of a charged particle in a cyclotron change when its speed is doubled? (V<< c).
1) Increase by 2 times; 2) Increase by 2 times;
3) Increase by 16 times; 4) Will not change.
9. What formula determines the modulus of induction of a magnetic field created at the center of a circular current with a circle radius R?
1)
2)
3)
4)
10. The current strength in the coil is equal to I. Which formula determines the modulus of magnetic field induction in the middle of a coil of length l with the number of turns N?
1)
2)
3)
4)
Laboratory work No.
Determination of the horizontal component of the Earth's magnetic field induction.
Brief theory for laboratory work.
A magnetic field is a material medium that transmits so-called magnetic interactions. The magnetic field is one of the forms of manifestation of the electromagnetic field.
The sources of magnetic fields are moving electric charges, current-carrying conductors and alternating electric fields. Generated by moving charges (currents), the magnetic field, in turn, acts only on moving charges (currents), but has no effect on stationary charges.
The main characteristic of a magnetic field is the magnetic induction vector 
:
|
|
The magnitude of the magnetic induction vector is numerically equal to the maximum force acting from the magnetic field on a conductor of unit length through which a current of unit strength flows. Vector 
forms a right-handed triple with the force vector and current direction. Thus, magnetic induction is a force characteristic of a magnetic field.
The SI unit of magnetic induction is Tesla (T).
Magnetic field lines are imaginary lines, at each point of which the tangents coincide with the direction of the magnetic induction vector. Magnetic lines of force are always closed and never intersect.
Ampere's law determines the force action of a magnetic field on a current-carrying conductor.
If in a magnetic field with induction 
a current-carrying conductor is placed, then each current-directed element 
the conductor is acted upon by the Ampere force, determined by the relation
|
|
.The direction of the Ampere force coincides with the direction of the vector product 
,
those. it is perpendicular to the plane in which the vectors lie 
And 
(Fig. 1).
Rice. 1. To determine the direction of the Ampere force
If 
perpendicular 
,
then the direction of the Ampere force can be determined by the rule of the left hand: direct four extended fingers along the current, place the palm perpendicular to the lines of force, then the thumb will show the direction of the Ampere force. Ampere's law is the basis for the definition of magnetic induction, i.e. relation (1) follows from formula (2), written in scalar form.
The Lorentz force is the force with which an electromagnetic field acts on a charged particle moving in this field. The Lorentz force formula was first obtained by G. Lorentz as a result of generalization of experience and has the form:
|
|
.Where 
– force acting on a charged particle in an electric field with intensity 
;
–
force acting on a charged particle in a magnetic field.
The formula for the magnetic component of the Lorentz force can be obtained from Ampere's law, taking into account that current is the ordered movement of electric charges. If the magnetic field did not act on moving charges, it would not have an effect on the current-carrying conductor. The magnetic component of the Lorentz force is determined by the expression:
|
|
.This force is directed perpendicular to the plane in which the velocity vectors lie 
and magnetic field induction 
; its direction coincides with the direction of the vector product 
For q
> 0 and with direction 
For q>0
(Fig. 2). 
Rice. 2. To determine the direction of the magnetic component of the Lorentz force
If the vector 
perpendicular to the vector 
, then the direction of the magnetic component of the Lorentz force for positively charged particles can be found using the left-hand rule, and for negatively charged particles using the right-hand rule. Since the magnetic component of the Lorentz force is always directed perpendicular to the speed 
, then it does not do any work to move the particle. It can only change the direction of speed 
,
bend the trajectory of a particle, i.e. act as a centripetal force.
The Biot-Savart-Laplace law is used to calculate magnetic fields (definitions 
) created by conductors carrying current.
According to the Biot-Savart-Laplace law, each current-directed element of a conductor 
creates at a point at a distance 
from this element, a magnetic field, the induction of which is determined by the relation:
|
|
.Where 
H/m – magnetic constant; µ
– magnetic permeability of the medium.
Rice. 3. Towards the Biot-Savart-Laplace law
Direction 
coincides with the direction of the vector product 
, i.e. 
perpendicular to the plane in which the vectors lie 
And 
. Simultaneously 
is tangent to the line of force, the direction of which can be determined by the gimlet rule: if the translational movement of the tip of the gimlet is directed along the current, then the direction of rotation of the handle will determine the direction of the magnetic field line (Fig. 3).
To find the magnetic field created by the entire conductor, you need to apply the principle of field superposition:
|
|
.For example, let's calculate the magnetic induction in the center of the circular current (Fig. 4).
Rice. 4. Towards the calculation of the field at the center of the circular current
For circular current 
And 
, therefore relation (5) in scalar form has the form:
The total current law (magnetic induction circulation theorem) is another law for calculating magnetic fields.
The total current law for a magnetic field in a vacuum has the form:
|
|
.Where B l
–
projection 
per conductor element 
, directed along the current.
The circulation of the magnetic induction vector along any closed circuit is equal to the product of the magnetic constant and the algebraic sum of the currents covered by this circuit.
The Ostrogradsky-Gauss theorem for the magnetic field is as follows:
|
|
.Where B n
–
vector projection 
to normal 
to the site dS.
The flux of the magnetic induction vector through an arbitrary closed surface is zero.
The nature of the magnetic field follows from formulas (9), (10).
The condition for the potentiality of the electric field is that the circulation of the intensity vector is equal to zero 
.
A potential electric field is generated by stationary electric charges; The field lines are not closed, they begin on positive charges and end on negative ones.
From formula (9) we see that in a magnetic field the circulation of the magnetic induction vector is different from zero, therefore, the magnetic field is not potential.
From relation (10) it follows that magnetic charges capable of creating potential magnetic fields do not exist. (In electrostatics, a similar theorem smolders in the form 
.
Magnetic lines of force close on themselves. Such a field is called a vortex field. Thus, the magnetic field is a vortex field. The direction of the field lines is determined by the gimlet rule. In a straight, infinitely long conductor carrying current, the lines of force have the form of concentric circles surrounding the conductor (Fig. 3).
Why does history include some scientists on its pages in golden letters, while others are erased without a trace? Everyone who comes to science is obliged to leave their mark on it. It is by the size and depth of this trace that history judges. Thus, Ampere and Lorentz made an invaluable contribution to the development of physics, which made it possible not only to develop scientific theories, but has received significant practical value. How did the telegraph come about? What are electromagnets? Today's lesson will answer all these questions.
For science, the acquired knowledge is of great value, which can subsequently find its practical use. New discoveries not only expand research horizons, but also raise new questions and problems.
Let's highlight the main Ampere's discoveries in the field of electromagnetism.
Firstly, these are the interactions of conductors with current. Two parallel conductors with currents are attracted to each other if the currents in them are in the same direction, and repel if the currents in them are in the opposite direction (Fig. 1).
Rice. 1. Current carrying conductors
Ampere's law reads:
The force of interaction between two parallel conductors is proportional to the product of the currents in the conductors, proportional to the length of these conductors and inversely proportional to the distance between them.
The force of interaction between two parallel conductors,
The magnitude of currents in conductors,
− length of conductors,
Distance between conductors,
Magnetic constant.
The discovery of this law made it possible to introduce into units of measurement a current value that did not exist before that time. So, if we proceed from the definition of current strength as the ratio of the amount of charge transferred through the cross-section of the conductor per unit time, then we obtain a fundamentally unmeasurable quantity, namely the amount of charge transferred through the cross-section of the conductor. Based on this definition, we will not be able to introduce a unit of current. Ampere's law allows us to establish a connection between the magnitudes of current in conductors and quantities that can be measured empirically: mechanical force and distance. Thus, it is possible to introduce into consideration the unit of current - 1 A (1 ampere).
One ampere current - this is a current at which two homogeneous parallel conductors located in a vacuum at a distance of one meter from each other interact with Newton’s force.
Law of interaction of currents - two parallel conductors in a vacuum, the diameters of which are much smaller than the distances between them, interact with a force directly proportional to the product of the currents in these conductors and inversely proportional to the distance between them.
Another discovery of Ampere is the law of the action of a magnetic field on a current-carrying conductor. It is expressed primarily in the action of a magnetic field on a coil or frame with current. Thus, a coil with current in a magnetic field is acted upon by a moment of force, which tends to rotate this coil so that its plane becomes perpendicular to the lines of the magnetic field. The angle of rotation of the coil is directly proportional to the amount of current in the coil. If the external magnetic field in the coil is constant, then the value of the magnetic induction module is also constant. The area of the coil at not very high currents can also be considered constant; therefore, it is true that the current strength is equal to the product of the moment of the forces turning the coil with the current by a certain constant value under constant conditions.
– current strength,
– the moment of forces unwinding the coil with current.
Consequently, it becomes possible to measure the current strength by the angle of rotation of the frame, which is implemented in a measuring device - an ammeter (Fig. 2).
Rice. 2. Ammeter
After discovering the effect of a magnetic field on a current-carrying conductor, Ampere realized that this discovery could be used to make a conductor move in a magnetic field. So, magnetism can be turned into mechanical movement - to create an engine. One of the first to operate on direct current was an electric motor (Fig. 3), created in 1834 by the Russian electrical engineer B.S. Jacobi.
Rice. 3. Engine
Let's consider a simplified model of a motor, which consists of a stationary part with magnets attached to it - the stator. Inside the stator, a frame of conductive material called a rotor can rotate freely. In order for electric current to flow through the frame, it is connected to the terminals using sliding contacts (Fig. 4). If you connect the motor to a direct current source in a circuit with a voltmeter, then when the circuit is closed, the frame with current will begin to rotate.
Rice. 4. Operating principle of the electric motor
In 1269, the French naturalist Pierre de Maricourt wrote a work entitled “Letter on the Magnet.” The main goal of Pierre de Maricourt was to create a perpetual motion machine, in which he was going to use amazing properties magnets. How successful his attempts were is unknown, but what is certain is that Jacobi used his electric motor to propel the boat, and he managed to accelerate it to a speed of 4.5 km/h.
It is necessary to mention one more device that works on the basis of Ampere's laws. Ampere showed that a coil with current behaves like permanent magnet. This means that it is possible to design electromagnet– a device whose power can be adjusted (Fig. 5).
Rice. 5. Electromagnet
It was Ampere who came up with the idea that by combining conductors and magnetic needles, one could create a device that transmits information over a distance.
Rice. 6. Electric telegraph
The idea of the telegraph (Fig. 6) arose in the very first months after the discovery of electromagnetism.
However, the electromagnetic telegraph became widespread after Samuel Morse created a more convenient device and, most importantly, developed a binary alphabet consisting of dots and dashes, which is called Morse code.
From the transmitting telegraph apparatus using a Morse key, which closes electrical circuit, short or long electrical signals corresponding to dots or dashes of Morse code are generated in the communication line. On a receiving telegraph apparatus (writing instrument), while the signal (electric current) is passing, an electromagnet attracts an armature, to which a metal writing wheel or scribe is rigidly connected, which leaves an ink mark on the paper tape (Fig. 7).
Rice. 7. Telegraph operation diagram
The mathematician Gauss, when he became acquainted with Ampere's research, proposed creating an original cannon (Fig. 8), working on the principle of the action of a magnetic field on an iron ball - a projectile.
Rice. 8. Gauss gun
It is necessary to pay attention to what historical era these discoveries were made. In the first half of the 19th century, Europe took leaps and bounds along the path of the industrial revolution - it was a fertile time for scientific research discoveries and their rapid implementation into practice. Ampere undoubtedly made a significant contribution to this process, giving civilization electromagnets, electric motors and the telegraph, which are still in wide use today.
Let us highlight the main discoveries of Lorenz.
Lorentz established that a magnetic field acts on a particle moving in it, causing it to move along a circular arc:
The Lorentz force is a centripetal force perpendicular to the direction of velocity. First of all, the law discovered by Lorentz allows us to determine such an important characteristic as the ratio of charge to mass - specific charge.
The specific charge value is a value unique to each charged particle, which allows them to be identified, be it an electron, a proton or any other particle. Thus, scientists received a powerful research tool. For example, Rutherford was able to analyze radioactive radiation and identified its components, among which there are alpha particles - the nuclei of the helium atom - and beta particles - electrons.
In the twentieth century, accelerators appeared, the operation of which is based on the fact that charged particles are accelerated in a magnetic field. The magnetic field bends the trajectories of particles (Fig. 9). The direction of the bend of the trace allows one to judge the sign of the particle's charge; By measuring the radius of the trajectory, you can determine the speed of the particle if its mass and charge are known.
Rice. 9. Curvature of particle trajectory in a magnetic field
The Large Hadron Collider was developed on this principle (Fig. 10). Thanks to Lorenz's discoveries, science has received a fundamentally new tool for physical research, opening the way to the world of elementary particles.
Rice. 10. Large Hadron Collider
In order to characterize the influence of a scientist on technical progress, let us remember that from the expression for the Lorentz force it follows that we can calculate the radius of curvature of the trajectory of a particle moving in a constant magnetic field. At constant external conditions this radius depends on the mass of the particle, its speed and charge. Thus, we get the opportunity to classify charged particles according to these parameters and, therefore, we can analyze any mixture. If a mixture of substances in a gaseous state is ionized, accelerated and directed into a magnetic field, then the particles will begin to move along circular arcs with different radii - the particles will leave the field at different points, and all that remains is to fix these departure points, which is realized using a screen covered with a phosphor , which glows when charged particles hit it. This is exactly how it works mass analyzer(Fig. 11) . Mass analyzers are widely used in physics and chemistry to analyze the composition of mixtures.
Rice. 11. Mass analyzer
This is not all the technical devices that work on the basis of the developments and discoveries of Ampere and Lorentz, because scientific knowledge sooner or later it ceases to be the exclusive property of scientists and becomes the property of civilization, while it is embodied in various technical devices that make our lives more comfortable.
Bibliography
- Kasyanov V.A., Physics 11th grade: Textbook. for general education institutions. - 4th ed., stereotype. - M.: Bustard, 2004. - 416 p.: ill., 8 p. color on
- Gendenstein L.E., Dick Yu.I., Physics 11. - M.: Mnemosyne.
- Tikhomirova S.A., Yarovsky B.M., Physics 11. - M.: Mnemosyne.
- Internet portal “Chip and Dip” ().
- Internet portal “Kiev City Library” ().
- Internet portal “Institute of Distance Education” ().
Homework
1. Kasyanov V.A., Physics 11th grade: Textbook. for general education institutions. - 4th ed., stereotype. - M.: Bustard, 2004. - 416 p.: ill., 8 p. color on, st. 88, v. 1-5.
2. In a cloud chamber, which is placed in a uniform magnetic field with an induction of 1.5 Tesla, an alpha particle, flying perpendicular to the induction lines, leaves a trace in the form of a circular arc with a radius of 2.7 cm. Determine the momentum and kinetic energy of the particle. The mass of the alpha particle is 6.7∙10 -27 kg, and the charge is 3.2∙10 -19 C.
3. Mass spectrograph. A beam of ions, accelerated by a potential difference of 4 kV, flies into a uniform magnetic field with a magnetic induction of 80 mT perpendicular to the magnetic induction lines. The beam consists of two types of ions with molecular masses of 0.02 kg/mol and 0.022 kg/mol. All ions have a charge of 1.6 ∙ 10 -19 C. The ions fly out of the field in two beams (Fig. 5). Find the distance between the beams of ions that fly out.
4. * Using a DC electric motor, the load is lifted on a cable. If you disconnect the electric motor from the voltage source and short-circuit the rotor, the load will fall from constant speed. Explain this phenomenon. What form does the potential energy of the load go into?
The effect exerted by a magnetic field on moving charged particles is very widely used in technology.
For example, the deflection of an electron beam in TV picture tubes is carried out using a magnetic field, which is created by special coils. A number of electronic devices use a magnetic field to focus beams of charged particles.
In currently created experimental installations for carrying out a controlled thermonuclear reaction, the action of a magnetic field on the plasma is used to twist it into a cord that does not touch the walls of the working chamber. The circular motion of charged particles in a uniform magnetic field and the independence of the period of such motion from the particle speed are used in cyclic accelerators of charged particles - cyclotrons.
The Lorentz force is also used in devices called mass spectrographs, which are designed to separate charged particles according to their specific charges.
The diagram of the simplest mass spectrograph is shown in Figure 1.
In chamber 1, from which air has been pumped out, there is an ion source 3. The chamber is placed in a uniform magnetic field, at each point of which the induction \(~\vec B\) is perpendicular to the plane of the drawing and directed towards us (in Figure 1 this field is indicated by circles) . An accelerating voltage is applied between the electrodes A and B, under the influence of which the ions emitted from the source are accelerated and at a certain speed enter the magnetic field perpendicular to the induction lines. Moving in a magnetic field in a circular arc, the ions fall on photographic plate 2, which makes it possible to determine the radius R this arc. Knowing the magnetic field induction IN and speed υ ions, according to the formula
\(~\frac q m = \frac (v)(RB)\)
the specific charge of ions can be determined. And if the charge of the ion is known, its mass can be calculated.
Literature
Aksenovich L. A. Physics in high school: Theory. Tasks. Tests: Textbook. allowance for institutions providing general education. environment, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsiya i vyakhavanne, 2004. - P. 328.
