but current and then
BecausenS d l – number of charges in volume S d l, then for one charge
or
| , | (2.5.2) |
Lorentz force – force exerted by a magnetic field on a moving positive charge(here is the speed of the ordered motion of positive charge carriers). Lorentz force modulus:
| , | (2.5.3) |
where α is the angle between and .
From (2.5.4) it can be seen that the charge moving along the line is not affected by the force ().
| Lorenz Hendrik Anton(1853–1928) – Dutch theoretical physicist, creator of classical electron theory, member of the Netherlands Academy of Sciences. He derived a formula relating the permittivity to the density of a dielectric, gave an expression for the force acting on a moving charge in an electromagnetic field (Lorentz force), explained the dependence of the electrical conductivity of a substance on thermal conductivity, developed the theory of light dispersion. Developed the electrodynamics of moving bodies. In 1904 he derived formulas relating the coordinates and time of the same event in two different inertial frames of reference (Lorentz transformations). |
The Lorentz force is directed perpendicular to the plane in which the vectors lie and . To a moving positive charge left hand rule applies or« gimlet rule» (Fig. 2.6).
The direction of the force for a negative charge is opposite, therefore, to right hand rule applies to electrons.
Since the Lorentz force is directed perpendicular to the moving charge, i.e. perpendicular ,the work done by this force is always zero . Therefore, acting on a charged particle, the Lorentz force cannot change the kinetic energy of the particle.
Often Lorentz force is the sum of electric and magnetic forces:
 ,
|
(2.5.4) |
here the electric force accelerates the particle, changes its energy.
Every day, we observe the effect of magnetic force on a moving charge on a television screen (Fig. 2.7).
The motion of the electron beam along the plane of the screen is stimulated by the magnetic field of the deflecting coil. If you bring a permanent magnet to the plane of the screen, then it is easy to notice its effect on the electron beam by the distortions that appear in the image.
The action of the Lorentz force in charged particle accelerators is described in detail in Section 4.3.
Definition 1The Ampere force acting on a part of the conductor with a length Δ l with a certain current strength I, located in a magnetic field B, F = I B Δ l sin α can be expressed through the forces acting on specific charge carriers.
Let the charge of the carrier be denoted as q, and n be the value of the concentration of free charge carriers in the conductor. In this case, the product n · q · υ · S, in which S is the cross-sectional area of the conductor, is equivalent to the current flowing in the conductor, and υ is the modulus of the speed of the ordered movement of carriers in the conductor:
I = q · n · υ · S .
Definition 2
Formula Ampere forces can be written in the following form:
F = q n S Δ l υ B sin α .
Due to the fact that the total number N of free charge carriers in a conductor with a cross section S and a length Δ l is equal to the product n S Δ l, the force acting on one charged particle is equal to the expression: F L \u003d q υ B sin α.
The power found is called Lorentz forces. The angle α in the above formula is equivalent to the angle between the magnetic induction vector B → and the speed ν → .
The direction of the Lorentz force, which acts on a particle with a positive charge, in the same way as the direction of the Ampère force, is found by the gimlet rule or by using the left hand rule. The mutual arrangement of the vectors ν → , B → and F L → for a particle carrying a positive charge is illustrated in fig. one . eighteen . one .
Picture 1 . eighteen . one . Mutual arrangement of vectors ν → , B → and F Л → . The Lorentz force modulus F L → is numerically equivalent to the product of the area of the parallelogram built on the vectors ν → and B → and the charge q.
The Lorentz force is directed normally, that is, perpendicular to the vectors ν → and B →.
The Lorentz force does no work when a particle carrying a charge moves in a magnetic field. This fact leads to the fact that the modulus of the velocity vector under the conditions of particle motion also does not change its value.
If a charged particle moves in a uniform magnetic field under the action of the Lorentz force, and its velocity ν → lies in a plane that is directed normally with respect to the vector B →, then the particle will move along a circle of a certain radius, calculated using the following formula:
The Lorentz force in this case is used as a centripetal force (Fig. 1.18.2).
Picture 1 . eighteen . 2 . Circular motion of a charged particle in a uniform magnetic field.
For the period of revolution of a particle in a uniform magnetic field, the following expression will be valid:
T = 2 π R υ = 2 π m q B .
This formula clearly demonstrates the absence of dependence of charged particles of a given mass m on the velocity υ and the radius of the trajectory R .
Definition 3The relation below is the formula for the angular velocity of a charged particle moving along a circular path:
ω = υ R = υ q B m υ = q B m .
It bears the name cyclotron frequency. This physical quantity does not depend on the speed of the particle, from which we can conclude that it does not depend on its kinetic energy either.
Definition 4
This circumstance finds its application in cyclotrons, namely in accelerators of heavy particles (protons, ions).
Figure 1. eighteen . 3 shows a schematic diagram of the cyclotron.
Picture 1 . eighteen . 3 . Movement of charged particles in the vacuum chamber of the cyclotron.
Definition 5
Duant- this is a hollow metal half-cylinder placed in a vacuum chamber between the poles of an electromagnet as one of the two accelerating D-shaped electrodes in the cyclotron.
An alternating electrical voltage is applied to the dees, whose frequency is equivalent to the cyclotron frequency. Particles carrying some charge are injected into the center of the vacuum chamber. In the gap between the dees, they experience acceleration caused by an electric field. Particles inside the dees, in the process of moving along semicircles, experience the action of the Lorentz force. The radius of the semicircles increases with increasing particle energy. As in all other accelerators, in cyclotrons the acceleration of a charged particle is achieved by applying an electric field, and its retention on the trajectory by means of a magnetic field. Cyclotrons make it possible to accelerate protons to energies close to 20 MeV.
Homogeneous magnetic fields are used in many devices for a wide variety of applications. In particular, they have found their application in the so-called mass spectrometers.
Definition 6
Mass spectrometers- These are such devices, the use of which allows us to measure the masses of charged particles, that is, ions or nuclei of various atoms.
These devices are used to separate isotopes (nuclei of atoms with the same charge but different masses, for example, Ne 20 and Ne 22). On fig. one . eighteen . 4 shows the simplest version of the mass spectrometer. The ions emitted from the source S pass through several small holes, which together form a narrow beam. After that, they enter the speed selector, where the particles move in crossed homogeneous electric fields, which are created between the plates of a flat capacitor, and magnetic fields, which appear in the gap between the poles of an electromagnet. The initial velocity υ → of charged particles is directed perpendicular to the vectors E → and B → .
A particle that moves in crossed magnetic and electric fields experiences the effects of the electric force q E → and the Lorentz magnetic force. Under conditions when E = υ B is fulfilled, these forces completely compensate each other. In this case, the particle will move uniformly and rectilinearly and, having flown through the capacitor, will pass through the hole in the screen. For given values of the electric and magnetic fields, the selector will select particles that move at a speed υ = E B .
After these processes, particles with the same velocities enter a uniform magnetic field B → mass spectrometer chambers. Particles under the action of the Lorentz force move in a chamber perpendicular to the magnetic field plane. Their trajectories are circles with radii R = m υ q B ". In the process of measuring the radii of the trajectories with known values of υ and B " , we are able to determine the ratio q m . In the case of isotopes, that is, under the condition q 1 = q 2 , the mass spectrometer can separate particles with different masses.
With the help of modern mass spectrometers, we are able to measure the masses of charged particles with an accuracy exceeding 10 - 4 .
Picture 1 . eighteen . four . Velocity selector and mass spectrometer.
In the case when the particle velocity υ → has a component υ ∥ → along the direction of the magnetic field, such a particle in a uniform magnetic field will make a spiral motion. The radius of such a spiral R depends on the modulus of the component perpendicular to the magnetic field υ ┴ vector υ → , and the pitch of the spiral p depends on the modulus of the longitudinal component υ ∥ (Fig. 1 . 18 . 5).
Picture 1 . eighteen . 5 . The movement of a charged particle in a spiral in a uniform magnetic field.
Based on this, we can say that the trajectory of a charged particle in a sense "winds" on the lines of magnetic induction. This phenomenon is used in technology for magnetic thermal insulation of high-temperature plasma - a fully ionized gas at a temperature of about 10 6 K . When studying controlled thermonuclear reactions, a substance in a similar state is obtained in facilities of the "Tokamak" type. The plasma must not touch the walls of the chamber. Thermal insulation is achieved by creating a magnetic field of a special configuration. Figure 1. eighteen . 6 illustrates as an example the trajectory of a charge-carrying particle in a magnetic "bottle" (or trap).
Picture 1 . eighteen . 6. Magnetic bottle. Charged particles do not go beyond its limits. The required magnetic field can be created using two round current coils.
The same phenomenon occurs in the Earth's magnetic field, which protects all living things from the flow of charge-carrying particles from outer space.
Definition 7
Fast charged particles from space, mostly from the Sun, are "intercepted" by the Earth's magnetic field, resulting in the formation of radiation belts (Fig. 1.18.7), in which particles, as if in magnetic traps, move back and forth along spiral trajectories between the north and south magnetic poles in a fraction of a second.
An exception is the polar regions, in which some of the particles break through into the upper layers of the atmosphere, which can lead to the emergence of phenomena such as "auroras". The radiation belts of the Earth extend from distances of about 500 km to tens of radii of our planet. It is worth remembering that the south magnetic pole of the Earth is located near the north geographic pole in the northwest of Greenland. The nature of terrestrial magnetism has not yet been studied.
Picture 1 . eighteen . 7. Radiation belts of the Earth. Fast charged particles from the Sun, mostly electrons and protons, are trapped in the magnetic traps of the radiation belts.
Their invasion into the upper layers of the atmosphere is possible, which is the cause of the appearance of the "northern lights".
Picture 1 . eighteen . eight . Model of charge motion in a magnetic field.
Picture 1 . eighteen . 9 . Mass spectrometer model.
Picture 1 . eighteen . ten . speed selector model.
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The action 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 kinescopes is carried out using a magnetic field, which is created by special coils. In a number of electronic devices, a magnetic field is used to focus beams of charged particles.
In the currently created experimental facilities for the implementation of 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 movement of charged particles in a circle in a uniform magnetic field and the independence of the period of such movement from the speed of the particle are used in cyclic accelerators of charged particles - cyclotrons.
The action of the Lorentz force is also used in devices called mass spectrographs, which are designed to separate charged particles according to their specific charges.
The scheme of the simplest mass spectrograph is shown in Figure 1.
In chamber 1, from which the air is evacuated, 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 h B, under the influence of which the ions emitted from the source are accelerated and enter the magnetic field at a certain speed perpendicular to the induction lines. Moving in a magnetic field along an arc of a circle, the ions fall on the photographic plate 2, which makes it possible to determine the radius R this arc. Knowing the induction of the magnetic field AT and speed υ ions, according to the formula
\(~\frac q m = \frac (v)(RB)\)
the specific charge of the ions can be determined. And if the charge of an ion is known, its mass can be calculated.
Literature
Aksenovich L. A. Physics in high school: Theory. Tasks. Tests: Proc. allowance for institutions providing general. environments, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsia i vykhavanne, 2004. - C. 328.
« Physics - Grade 11 "
The magnetic field acts with force on moving charged particles, including current-carrying conductors.
What is the force acting on one particle?
1.
The force exerted on a moving charged particle by a magnetic field is called Lorentz force in honor of the great Dutch physicist X. Lorenz, who created the electronic theory of the structure of matter.
The Lorentz force can be found using Ampère's law.
Lorentz force modulus is equal to the ratio of the modulus of force F, acting on a section of the conductor of length Δl, to the number N of charged particles moving in an orderly manner in this section of the conductor:
Since the force (Ampère force) acting on the section of the conductor from the magnetic field
is equal to F=| I | BΔl sin α,
and the current in the conductor is I = qnvS
where
q - particle charge
n is the concentration of particles (i.e. the number of charges per unit volume)
v - speed of particles
S is the cross section of the conductor.
Then we get:
Each moving charge is affected by the magnetic field Lorentz force equal to:
where α is the angle between the velocity vector and the magnetic induction vector.
The Lorentz force is perpendicular to the vectors and .
2.
Direction of the Lorentz force
The direction of the Lorentz force is determined using the same left hand rules, which is the direction of the Ampère force:
If the left hand is positioned so that the component of magnetic induction, perpendicular to the charge velocity, enters the palm, and four outstretched fingers are directed along the movement of the positive charge (against the movement of the negative), then the thumb bent by 90 ° will indicate the direction of the Lorentz force acting on the charge F l
3.
If in the space where the charged particle is moving, there is both an electric field and a magnetic field, then the total force acting on the charge is equal to: = el + l where the force with which the electric field acts on the charge q is equal to F el = q .
4.
The Lorentz force does no work, because it is perpendicular to the velocity vector of the particle.
This means that the Lorentz force does not change the kinetic energy of the particle and, consequently, the modulus of its velocity.
Under the action of the Lorentz force, only the direction of the particle's velocity changes.
5.
Motion of a charged particle in a uniform magnetic field
There is homogeneous magnetic field directed perpendicular to the particle's initial velocity. 
The Lorentz force depends on the moduli of the particle velocity vectors and the magnetic field induction.
The magnetic field does not change the modulus of the velocity of a moving particle, which means that the modulus of the Lorentz force remains unchanged.
The Lorentz force is perpendicular to the velocity and therefore determines the centripetal acceleration of the particle.
The invariance in modulus of the centripetal acceleration of a particle moving with a constant modulo velocity means that
In a uniform magnetic field, a charged particle moves uniformly along a circle of radius r.
According to Newton's second law 
Then the radius of the circle along which the particle moves is equal to:
The time it takes for a particle to make a complete revolution (orbital period) is: 
6.
Using the action of a magnetic field on a moving charge.
The action of a magnetic field on a moving charge is used in television kinescope tubes, in which electrons flying towards the screen are deflected by a magnetic field created by special coils.
The Lorentz force is used in the cyclotron - charged particle accelerator to produce particles with high energies.
The device of mass spectrographs is also based on the action of a magnetic field, which makes it possible to accurately determine the masses of particles.
