The 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. 1 . 18 . 1 .

Picture 1 . 18 . 1 . 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 . 18 . 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 .

The 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. 18 . 3 shows a schematic diagram of the cyclotron.

Picture 1 . 18 . 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. 1 . 18 . 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 . 18 . 4 . 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 . 18 . 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 engineering for magnetic thermal insulation of high-temperature plasma - a completely 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. 18 . 6 illustrates as an example the trajectory of a charge-carrying particle in a magnetic "bottle" (or trap).

Picture 1 . 18 . 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 . 18 . 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 . 18 . 8 . Model of charge motion in a magnetic field.

Picture 1 . 18 . 9 . Mass spectrometer model.

Picture 1 . 18 . 10 . speed selector model.

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The Lorentz force is the force that acts from the side of the electromagnetic field on a moving electric charge. Quite often, only the magnetic component of this field is called the Lorentz force. Formula for determining:

F = q(E+vB),

Where q is the particle charge;E is the electric field strength;B— magnetic field induction;v is the speed of the particle.

The Lorentz force is very similar in principle to, the difference lies in the fact that the latter acts on the entire conductor, which is generally electrically neutral, and the Lorentz force describes the influence of an electromagnetic field only on a single moving charge.

It is characterized by the fact that it does not change the speed of movement of charges, but only affects the velocity vector, that is, it is able to change the direction of movement of charged particles.

In nature, the Lorentz force allows you to protect the Earth from the effects of cosmic radiation. Under its influence, charged particles falling on the planet deviate from a straight path due to the presence of the Earth's magnetic field, causing auroras.

In engineering, the Lorentz force is used very often: in all engines and generators, it is she who drives the rotor under the influence of the electromagnetic field of the stator.

Thus, in any electric motors and electric drives, the Lorentz force is the main type of force. In addition, it is used in particle accelerators, as well as in electron guns, which were previously installed in tube televisions. In a kinescope, the electrons emitted by the gun are deflected under the influence of an electromagnetic field, which occurs with the participation of the Lorentz force.

In addition, this force is used in mass spectrometry and mass electrography for instruments capable of sorting charged particles based on their specific charge (the ratio of charge to particle mass). This makes it possible to determine the mass of particles with high accuracy. It also finds application in other instrumentation, for example, in a non-contact method for measuring the flow of electrically conductive liquid media (flowmeters). This is very important if the liquid medium has a very high temperature (melt of metals, glass, etc.).

In the article we will talk about the Lorentz magnetic force, how it acts on the conductor, consider the left hand rule for the Lorentz force and the moment of force acting on the circuit with current.

The Lorentz force is the force that acts on a charged particle falling at a certain speed into a magnetic field. The magnitude of this force depends on the magnitude of the magnetic induction of the magnetic field B, the electric charge of the particle q and speed v, from which the particle falls into the field.

The way the magnetic field B behaves with respect to a load completely different from how it is observed for an electric field E. First of all, the field B does not respond to load. However, when the load is moved into the field B, a force appears, which is expressed by a formula that can be considered as a definition of the field B:

Thus, it is clear that the field B acts as a force perpendicular to the direction of the velocity vector V loads and vector direction B. This can be illustrated in a diagram:

In the q diagram, there is a positive charge!

The units of the field B can be obtained from the Lorentz equation. Thus, in the SI system, the unit of B is equal to 1 tesla (1T). In the CGS system, the field unit is Gauss (1G). 1T=104G

For comparison, an animation of the movement of both positive and negative charges is shown.

When the field B covers a large area, a charge q moving perpendicular to the direction of the vector b, stabilizes its movement along a circular trajectory. However, when the vector v has a component parallel to the vector b, then the charge path will be a spiral as shown in the animation

Lorentz force on a conductor with current

The force acting on a conductor with current is the result of the Lorentz force acting on moving charge carriers, electrons or ions. If in the section of the guide length l, as in the drawing

the total charge Q moves, then the force F acting on this segment is equal to

The quotient Q / t is the value of the flowing current I and, therefore, the force acting on the section with the current is expressed by the formula

To take into account the dependence of the force F from the angle between the vector B and the axis of the segment, the length of the segment l was is given by the characteristics of the vector.

Only electrons move in a metal under the action of a potential difference; metal ions remain motionless in the crystal lattice. In electrolyte solutions, anions and cations are mobile.

Left hand rule Lorentz force is the determining direction and return of the magnetic (electrodynamic) energy vector.

If the left hand is positioned so that the magnetic field lines are directed perpendicular to the inner surface of the hand (so that they penetrate the inside of the hand), and all fingers - except the thumb - point in the direction of the flow of positive current (a moving molecule), the deflected thumb indicates the direction of the electrodynamic force acting on a positive electric charge placed in this field (for a negative charge, the force will be opposite).

The second way to determine the direction of the electromagnetic force is to place the thumb, index and middle fingers at a right angle. In this arrangement, the index finger shows the direction of the magnetic field lines, the direction of the middle finger the direction of current flow, and the direction of the force thumb.

Moment of force acting on a circuit with current in a magnetic field

The moment of force acting on a circuit with current in a magnetic field (for example, on a wire coil in a motor winding) is also determined by the Lorentz force. If the loop (marked in red in the diagram) can rotate around an axis perpendicular to the field B and conducts current I, then two unbalanced forces F appear, acting away from the frame, parallel to the axis of rotation.

Dutch physicist X. A. Lorentz at the end of the 19th century. found that the force acting from the magnetic field on a moving charged particle is always perpendicular to the direction of particle motion and the lines of force of the magnetic field in which this particle moves. The direction of the Lorentz force can be determined using the left hand rule. If you place the palm of your left hand so that four outstretched fingers indicate the direction of movement of the charge, and the vector of the magnetic induction of the field enters the retracted thumb, it will indicate the direction of the Lorentz force acting on the positive charge.

If the charge of the particle is negative, then the Lorentz force will be directed in the opposite direction.

The Lorentz force modulus is easily determined from Ampère's law and is:

F = | q| vB sin?,

Where q is the charge of the particle, v- the speed of its movement, ? - the angle between the velocity and induction vectors of the magnetic field.

If, in addition to the magnetic field, there is also an electric field, which acts on a charge with a force , then the total force acting on the charge is:

.

Often this force is called the Lorentz force, and the force expressed by the formula ( F = | q| vB sin?) are called the magnetic part of the Lorentz force.

Since the Lorentz force is perpendicular to the direction of motion of the particle, it cannot change its speed (it does not do work), but can only change the direction of its motion, i.e., bend the trajectory.

Such a curvature of the trajectory of electrons in the TV kinescope is easy to observe if you bring a permanent magnet to its screen - the image will be distorted.

Movement of a charged particle in a uniform magnetic field. Let a charged particle fly in with a speed v into a uniform magnetic field perpendicular to the lines of tension.

The force exerted by the magnetic field on the particle will cause it to rotate uniformly in a circle of radius r, which is easy to find using Newton's second law, the purposeful acceleration expression, and the formula ( F = | q| vB sin?):

.

From here we get

.

Where m is the mass of the particle.

Application of the Lorentz force.

The action of a magnetic field on moving charges is used, for example, in mass spectrographs, which make it possible to separate charged particles according to their specific charges, i.e., according to the ratio of the charge of a particle to its mass, and, based on the results obtained, accurately determine the masses of particles.

The vacuum chamber of the device is placed in a field (the induction vector is perpendicular to the figure). Charged particles (electrons or ions) accelerated by an electric field, having described an arc, fall on a photographic plate, where they leave a trace, which makes it possible to measure the radius of the trajectory with great accuracy r. The specific charge of the ion is determined from this radius. Knowing the charge of an ion, you can easily calculate its mass.