Lorentz force formula. Lorentz force, definition, formula, physical meaning Lorentz force in si

The emergence of a force acting on an electric charge moving in an external electromagnetic field

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Description

The Lorentz force is the force acting on a charged particle moving in an external electromagnetic field.

The formula for the Lorentz force (F) was first obtained by generalizing the experimental facts of H.A. Lorentz in 1892 and presented in the work "Maxwell's electromagnetic theory and its application to moving bodies". It looks like:

F = qE + q, (1)

where q is a charged particle;

E - electric field strength;

B is the vector of magnetic induction, independent of the magnitude of the charge and the speed of its movement;

V is the velocity vector of the charged particle relative to the coordinate system in which the values ​​F and B are calculated.

The first term on the right side of equation (1) is the force acting on a charged particle in an electric field F E \u003d qE, the second term is the force acting in a magnetic field:

F m = q. (2)

Formula (1) is universal. It is valid for both constant and variable force fields, as well as for any value of the speed of a charged particle. It is an important relation of electrodynamics, since it allows one to relate the equations of an electromagnetic field with the equations of motion of charged particles.

In the nonrelativistic approximation, the force F, like any other force, does not depend on the choice of the inertial frame of reference. At the same time, the magnetic component of the Lorentz force F m changes when moving from one reference frame to another due to a change in speed, so the electric component F E will also change. In this regard, the division of the force F into magnetic and electric makes sense only with an indication of the reference system.

In scalar form, expression (2) has the form:

Fм = qVBsina , (3)

where a is the angle between the velocity and magnetic induction vectors.

Thus, the magnetic part of the Lorentz force is maximum if the direction of particle motion is perpendicular to the magnetic field (a = p /2), and is zero if the particle moves along the direction of the field B (a = 0).

The magnetic force F m is proportional to the vector product, i.e. it is perpendicular to the velocity vector of the charged particle and therefore does no work on the charge. This means that in a constant magnetic field, only the trajectory of a moving charged particle is bent under the action of a magnetic force, but its energy always remains unchanged, no matter how the particle moves.

The direction of the magnetic force for a positive charge is determined according to the vector product (Fig. 1).

The direction of the force acting on a positive charge in a magnetic field

Rice. one

For a negative charge (electron), the magnetic force is directed in the opposite direction (Fig. 2).

Direction of the Lorentz force acting on an electron in a magnetic field

Rice. 2

The magnetic field B is directed towards the reader perpendicular to the drawing. There is no electric field.

If the magnetic field is uniform and directed perpendicular to the velocity, a charge of mass m moves in a circle. The radius of the circle R is determined by the formula:

where is the specific charge of the particle.

The period of revolution of a particle (the time of one revolution) does not depend on the speed, if the speed of the particle is much less than the speed of light in vacuum. Otherwise, the period of revolution of the particle increases due to the increase in the relativistic mass.

In the case of a non-relativistic particle:

where is the specific charge of the particle.

In a vacuum in a uniform magnetic field, if the velocity vector is not perpendicular to the magnetic induction vector (a№p /2), a charged particle under the action of the Lorentz force (its magnetic part) moves along a helix with a constant velocity V. In this case, its movement consists of a uniform rectilinear movement along the direction of the magnetic field B with a speed and a uniform rotational movement in a plane perpendicular to the field B with a speed (Fig. 2).

The projection of the trajectory of the particle on the plane perpendicular to B is a circle of radius:

particle revolution period:

The distance h that the particle travels in time T along the magnetic field B (the step of the helical trajectory) is determined by the formula:

h = Vcos a T . (6)

The axis of the helix coincides with the direction of the field В, the center of the circle moves along the field line of force (Fig. 3).

The motion of a charged particle flying in at an angle a№p /2 into magnetic field B

Rice. 3

There is no electric field.

If the electric field E is 0, the motion is more complex.

In a particular case, if the vectors E and B are parallel, the velocity component V 11 , which is parallel to the magnetic field, changes during the movement, as a result of which the pitch of the helical trajectory (6) changes.

In the event that E and B are not parallel, the center of rotation of the particle moves, called drift, perpendicular to the field B. The direction of the drift is determined by the vector product and does not depend on the sign of the charge.

The effect of a magnetic field on moving charged particles leads to a redistribution of the current over the cross section of the conductor, which is manifested in thermomagnetic and galvanomagnetic phenomena.

The effect was discovered by the Dutch physicist H.A. Lorenz (1853-1928).

Timing

Initiation time (log to -15 to -15);

Lifetime (log tc 15 to 15);

Degradation time (log td -15 to -15);

Optimal development time (log tk -12 to 3).

Diagram:

Technical realizations of the effect

Technical implementation of the action of the Lorentz force

The technical implementation of an experiment on direct observation of the action of the Lorentz force on a moving charge is usually rather complicated, since the corresponding charged particles have a characteristic molecular size. Therefore, the observation of their trajectory in a magnetic field requires the working volume to be evacuated in order to avoid collisions that distort the trajectory. So, as a rule, such demonstration installations are not specially created. The easiest way to demonstrate is to use a standard Nier sector magnetic mass analyzer, see Effect 409005, which is entirely based on the Lorentz force.

Applying an effect

A typical application in engineering is the Hall sensor, which is widely used in measurement technology.

A plate of metal or semiconductor is placed in a magnetic field B. When an electric current of density j is passed through it in a direction perpendicular to the magnetic field, a transverse electric field arises in the plate, the strength of which E is perpendicular to both vectors j and B. According to the measurement data, V is found.

This effect is explained by the action of the Lorentz force on a moving charge.

Galvanomagnetic magnetometers. Mass spectrometers. Accelerators of charged particles. Magnetohydrodynamic generators.

Literature

1. Sivukhin D.V. General course of physics.- M.: Nauka, 1977.- V.3. Electricity.

2. Physical encyclopedic dictionary. - M., 1983.

3. Detlaf A.A., Yavorsky B.M. Course of physics.- M.: Higher school, 1989.

Keywords

• electric charge
• magnetic induction
• a magnetic field
• electric field strength
• Lorentz force
• particle speed
• circle radius
• circulation period
• step of the helical trajectory
• electron
• proton
• positron

Sections of natural sciences:

Along with the Ampère force, Coulomb interaction, electromagnetic fields, the concept of the Lorentz force is often encountered in physics. This phenomenon is one of the fundamental in electrical engineering and electronics, along with, and others. It acts on charges that move in a magnetic field. In this article, we will briefly and clearly consider what the Lorentz force is and where it is applied.

Definition

When electrons move through a conductor, a magnetic field develops around it. At the same time, if you place the conductor in a transverse magnetic field and move it, an EMF of electromagnetic induction will occur. If a current flows through a conductor that is in a magnetic field, the Ampere force acts on it.

Its value depends on the flowing current, the length of the conductor, the magnitude of the magnetic induction vector and the sine of the angle between the magnetic field lines and the conductor. It is calculated by the formula:

The force under consideration is somewhat similar to the one discussed above, but it does not act on a conductor, but on a moving charged particle in a magnetic field. The formula looks like:

Important! The Lorentz force (Fl) acts on an electron moving in a magnetic field, and Ampere acts on a conductor.

It can be seen from the two formulas that in both the first and second cases, the closer the sine of the angle alpha to 90 degrees, the greater the effect Fa or Fl has on the conductor or charge, respectively.

So, the Lorentz force characterizes not a change in the magnitude of the velocity, but what kind of influence occurs from the side of the magnetic field on a charged electron or a positive ion. When exposed to them, Fl does not do work. Accordingly, it is the direction of the velocity of the charged particle that changes, and not its magnitude.

As for the unit of measurement of the Lorentz force, as in the case of other forces in physics, such a quantity as Newton is used. Its components:

How is the Lorentz force directed?

To determine the direction of the Lorentz force, as with the Ampère force, the left hand rule works. This means, in order to understand where the value of Fl is directed, you need to open the palm of your left hand so that the lines of magnetic induction enter the hand, and the outstretched four fingers indicate the direction of the velocity vector. Then the thumb, bent at right angles to the palm, indicates the direction of the Lorentz force. In the picture below you see how to determine the direction.

Attention! The direction of the Lorentzian action is perpendicular to the motion of the particle and the lines of magnetic induction.

At the same time, to be more precise, for positively and negatively charged particles, the direction of the four extended fingers matters. The left hand rule described above is formulated for a positive particle. If it is negatively charged, then the lines of magnetic induction should be directed not to the open palm, but to its back side, and the direction of the Fl vector will be opposite.

Now we will tell in simple terms what this phenomenon gives us and what real effect it has on charges. Let us assume that an electron moves in a plane perpendicular to the direction of the lines of magnetic induction. We have already mentioned that Fl does not affect the speed, but only changes the direction of particle motion. Then the Lorentz force will have a centripetal effect. This is reflected in the figure below.

Application

Of all the areas where the Lorentz force is used, one of the largest is the movement of particles in the earth's magnetic field. If we consider our planet as a large magnet, then the particles that are near the north magnetic poles make an accelerated movement in a spiral. As a result of this, they collide with atoms from the upper atmosphere, and we see the northern lights.

However, there are other cases where this phenomenon applies. For example:

• cathode ray tubes. In their electromagnetic deflecting systems. CRTs have been used for more than 50 years in a variety of devices, ranging from the simplest oscilloscope to televisions of various shapes and sizes. It is curious that in matters of color reproduction and work with graphics, some still use CRT monitors.
• Electrical machines - generators and motors. Although the force of Ampere is more likely to act here. But these quantities can be considered as adjacent. However, these are complex devices during the operation of which the influence of many physical phenomena is observed.
• In charged particle accelerators in order to set their orbits and directions.

Conclusion

To sum up and outline the four main theses of this article in simple terms:

1. The Lorentz force acts on charged particles that move in a magnetic field. This follows from the main formula.
2. It is directly proportional to the speed of the charged particle and the magnetic induction.
3. Does not affect particle speed.
4. Affects the direction of the particle.

Its role is quite large in the "electric" areas. A specialist should not lose sight of the basic theoretical information about fundamental physical laws. This knowledge will be useful, as well as for those who are engaged in scientific work, design and just for general development.

Now you know what the Lorentz force is, what it is equal to, and how it acts on charged particles. If you have any questions, ask them in the comments below the article!

materials

The action exerted by a magnetic field on moving charged particles is very widely used in technology.

For example, the deflection of the 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.

Definition

Force acting on a moving charged particle in a magnetic field, equal to:

called Lorentz force (magnetic force).

Based on definition (1), the modulus of the force under consideration is:

where is the particle velocity vector, q is the particle charge, is the magnetic field induction vector at the point where the charge is located, is the angle between the vectors and . From expression (2) it follows that if the charge moves parallel to the magnetic field lines, then the Lorentz force is zero. Sometimes, trying to isolate the Lorentz force, they denote it using the index:

Direction of the Lorentz force

The Lorentz force (like any force) is a vector. Its direction is perpendicular to the velocity vector and the vector (that is, perpendicular to the plane in which the velocity and magnetic induction vectors are located) and is determined by the rule of the right gimlet (right screw) Fig. 1 (a). If we are dealing with a negative charge, the direction of the Lorentz force is opposite to the result of the cross product (Fig. 1(b)).

the vector is directed perpendicular to the plane of the drawings on us.

Consequences of the properties of the Lorentz force

Since the Lorentz force is always directed perpendicular to the direction of the charge velocity, its work on the particle is zero. It turns out that by acting on a charged particle with a constant magnetic field, it is impossible to change its energy.

If the magnetic field is uniform and directed perpendicular to the velocity of the charged particle, then the charge under the influence of the Lorentz force will move along a circle of radius R=const in a plane that is perpendicular to the magnetic induction vector. In this case, the radius of the circle is:

where m is the particle mass, |q| is the particle charge modulus, is the relativistic Lorentz factor, c is the speed of light in vacuum.

The Lorentz force is a centripetal force. According to the direction of deviation of an elementary charged particle in a magnetic field, a conclusion is made about its sign (Fig. 2).

Lorentz force formula in the presence of magnetic and electric fields

If a charged particle moves in space in which two fields (magnetic and electric) are located simultaneously, then the force that acts on it is equal to:

where is the electric field strength vector at the point where the charge is located. Expression (4) was empirically obtained by Lorentz. The force that enters formula (4) is also called the Lorentz force (Lorentz force). The division of the Lorentz force into components: electric and magnetic relatively, since it is connected with the choice of the inertial frame of reference. So, if the reference frame moves with the same speed as the charge, then in such a frame the Lorentz force acting on the particle will be equal to zero.

Lorentz force units

The basic unit of measure for the Lorentz force (as well as any other force) in the SI system is: [F]=H

In GHS: [F]=din

Examples of problem solving

Example

Exercise. What is the angular velocity of an electron moving in a circle in a magnetic field with induction B?

Solution. Since an electron (a particle with a charge) moves in a magnetic field, the Lorentz force of the form acts on it:

where q=q e is the electron charge. Since the condition says that the electron moves in a circle, this means that, therefore, the expression for the Lorentz force modulus will take the form:

The Lorentz force is centripetal and, in addition, according to Newton's second law, in our case it will be equal to:

Equate the right parts of expressions (1.2) and (1.3), we have:

From expression (1.3) we obtain the speed:

The period of revolution of an electron in a circle can be found as:

Knowing the period, you can find the angular velocity as:

Answer.

Example

Exercise. A charged particle (charge q, mass m) flies with a speed v into a region where there is an electric field with strength E and a magnetic field with induction B. The vectors and coincide in direction. What is the acceleration of the particle at the moment of the beginning of the movement in the fields, if ?

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.).