How to determine the modulus of the Lorentz force. Lorentz force

Why does history add some scholars to its pages in golden letters, and erase some without a trace? Everyone who comes to science is obliged to leave his mark on it. It is by the size and depth of this trace that history judges. So, Ampere and Lorentz made an invaluable contribution to the development of physics, which made it possible not only to develop scientific theories, but received a significant practical value. How did the telegraph appear? What are electromagnets? All these questions will be answered in today's lesson.

For science, the knowledge gained is of great value, which can subsequently find its practical application. 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.

First, it is the interaction of conductors with current. Two parallel conductors with currents are attracted to each other if the currents in them are co-directed, and repel if the currents in them are oppositely directed (Fig. 1).

Rice. 1. Conductors with current

Ampere's law reads:

The force of interaction of 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 of two parallel conductors,

The values ​​of the currents in the conductors,

- the length of the conductors,

Distance between conductors,

Magnetic constant.

The discovery of this law made it possible to introduce into the units of measurement the magnitude of the current strength, which did not exist until that time. So, if we proceed from the definition of the current strength as the ratio of the amount of charge transferred through the cross section of the conductor per unit of time, then we will get a fundamentally not measurable value, namely the amount of charge transferred through the cross section of the conductor. Based on this definition, we will not be able to enter a unit for measuring current strength. Ampere's law makes it possible to establish a relationship between the values ​​of the currents in conductors and the quantities that can be measured empirically: mechanical force and distance. Thus, it is possible to introduce into consideration the unit of current strength - 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 the other interact with Newton's force.

The law of interaction of currents - two parallel conductors in a vacuum, the diameters of which are much less 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 conductor with current. It is expressed primarily in the action of a magnetic field on a coil or frame with current. So, a moment of force acts on a loop with a current in a magnetic field, which tends to unfold this loop in such a way that its plane becomes perpendicular to the lines of the magnetic field. The turn angle of the turn is directly proportional to the amount of current in the turn. If the external magnetic field in the loop is constant, then the value of the magnetic induction modulus is also constant. The area of ​​the loop at not very large currents can also be considered constant, therefore, it is true that the current strength is equal to the product of the moment of forces unfolding the loop with the current by some constant value under constant conditions.

- current strength,

- the moment of forces unfolding the coil with current.

Consequently, it becomes possible to measure the current strength by the value of the angle of rotation of the frame, which is implemented in a measuring device - an ammeter (Fig. 2).

Rice. 2. Ammeter

After the discovery of the action of a magnetic field on a conductor with a current, Ampere realized that this discovery could be used to make the 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

Consider a simplified model of the motor, which consists of a fixed part with magnets attached to it - a stator. Inside the stator, a frame made of conductive material, called a rotor, can rotate freely. In order for an 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 DC source in a circuit with a voltmeter, then when the circuit is closed, the frame with current will start rotating.

Rice. 4. The principle of operation of the electric motor

In 1269, the French naturalist Pierre de Maricourt wrote a work called "The 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 the amazing properties of magnets. How successful his attempts were is unknown, but it is certain that Jacobi used his electric motor to propel the boat, while he was able 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. The ampere showed that the current coil behaves like a permanent magnet. This means that you can construct 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 arrows, you can 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 apparatus and, most importantly, developed a binary alphabet consisting of dots and dashes, which is called Morse code.

From the transmitting telegraph apparatus with the help of the Morse key, which closes the electrical circuit, short or long electrical signals are formed in the communication line, corresponding to dots or dashes of the Morse code. On the receiving telegraph apparatus (writing device), for the duration of the passage of the signal (electric current), the electromagnet attracts the anchor, with which the writing metal wheel or scribe is rigidly connected, which leave an ink mark on the paper tape (Fig. 7).

Rice. 7. Scheme of the telegraph

The mathematician Gauss, when he became acquainted with Ampere's research, proposed to create an original cannon (Fig. 8), operating on the principle of the action of a magnetic field on an iron ball - a projectile.

Rice. 8. Gauss Cannon

It is necessary to pay attention to what historical era these discoveries were made. In the first half of the 19th century, Europe was taking 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 a telegraph, which are still widely used today.

Let's highlight the main discoveries of Lorentz.

Lorentz established that a magnetic field acts on a particle moving in it, forcing it to move along an arc of a circle:

The Lorentz force is a centripetal force perpendicular to the direction of velocity. First of all, the law discovered by Lorentz makes it possible to determine such an important characteristic as the ratio of charge to mass - specific charge.

The specific charge value is a value that is unique for each charged particle, which allows them to be identified, be it an electron, proton or any other particle. Thus, scientists have 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 nucleus of a helium atom - and beta particles - electrons.

In the twentieth century, accelerators appeared, the work of which is based on the fact that charged particles are accelerated in a magnetic field. The magnetic field bends the trajectories of the particles (Fig. 9). The direction of the wake bending makes it possible to judge the sign of the particle charge; by measuring the radius of the trajectory, it is possible to determine the speed of the particle if its mass and charge are known.

Rice. 9. Curvature of the trajectory of particles in a magnetic field

The Large Hadron Collider was developed on this principle (Fig. 10). Thanks to the discoveries of Lorentz, science 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 recall that the expression for the Lorentz force implies the possibility of calculating the radius of curvature of the trajectory of a particle that moves in a constant magnetic field. Under constant external conditions, this radius depends on the mass of the particle, its velocity and charge. Thus, we get the opportunity to classify charged particles by 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 arcs of circles with different radii - the particles will leave the field at different points, and it remains only to fix these points of departure, which is implemented using a screen covered with a phosphor , which glows when charged particles hit it. It is according to this scheme that mass analyzer(fig. 11) . Mass analyzers are widely used in physics and chemistry to analyze the composition of mixtures.

Rice. 11. Mass analyzer

These are not all technical devices that work on the basis of the developments and discoveries of Ampere and Lorenz, because scientific knowledge sooner or later 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 life more comfortable.

Bibliography

1. Kasyanov V.A., Physics 11th grade: Textbook. for general education. institutions. - 4th ed., Stereotype. - M .: Bustard, 2004 .-- 416s.: Ill., 8 p. color incl.
2. Gendenshtein L.E., Dick Yu.I., Physics 11. - M .: Mnemosyne.
3. Tikhomirova S.A., Yavorskiy B.M., Physics 11. - M .: Mnemosina.

1. Internet portal "Chip and Dip" ().
2. Internet portal "Kiev City Library" ().
3. Internet portal "Institute of Distance Education" ().

Homework

1. Kasyanov VA, Physics 11th grade: Textbook. for general education. institutions. - 4th ed., Stereotype. - M .: Bustard, 2004 .-- 416s.: Ill., 8 p. color incl., art. 88, c. 1-5.

2. In the Wilson chamber, which is placed in a uniform magnetic field with an induction of 1.5 T, an alpha particle, flying in perpendicular to the lines of induction, leaves a trace in the form of an arc of a circle 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 lines of magnetic induction. The beam consists of two types of ions with molecular weights of 0.02 kg / mol and 0.022 kg / mol. All ions have a charge of 1.6 ∙ 10 -19 C. Ions fly out of the field in two beams (Fig. 5). Find the distance between the beams of ions that are emitted.

4. * With the help of a DC motor, they lift a load on a rope. If the motor is disconnected from the voltage source and the rotor is short-circuited, the load will descend at a constant speed. Explain this phenomenon. What form does the potential energy of the load take?

In the article, we will talk about the Lorentz magnetic force, how it acts on a conductor, consider the left-hand rule for the Lorentz force and the moment of force acting on a circuit with a 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 in relation to the 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 moves in the field B, a force appears, which is expressed by the formula, which can be considered as the definition of the field B:

Thus, it can be seen 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 the diagram:

On the q diagram, there is a positive charge!

The units of the field B can be obtained from the Lorentz equation. Thus, the SI unit of B is equal to 1 Tesla (1T). In the CGS system, the field unit is Gauss (1G). 1T = 10 4 G

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

When the field B covers a large area, charge q moving perpendicular to the direction of the vector B, stabilizes its movement along a circular path. However, when the vector v has a component parallel to a 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 a section with a guide of 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 current is expressed by the formula

To account for the dependence of strength F from the angle between the vector B and the axis of the segment, the length of the segment l was given by the characteristics of the vector.

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

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

If the left hand is positioned so that the magnetic field lines are perpendicular to the inner surface of the hand (so that they penetrate into the hand), and all fingers - except the thumb - indicate the direction of positive current flow (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 electromagnetic force is to place your thumb, index, and middle fingers at right angles. With this arrangement, the index finger shows the direction of the magnetic field lines, the direction of the middle finger shows the direction of current movement, as well as the direction of the thumb force.

The moment of force acting on a circuit with a current in a magnetic field

The moment of force acting on a circuit with a current in a magnetic field (for example, on a wire coil in an electric 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 a current I, then two unbalanced forces F appear, acting sideways from the frame parallel to the axis of rotation.

Determination of the strength of the magnetic force

Definition

If a charge moves in a magnetic field, then a force ($ \ overrightarrow (F) $) acts on it, which depends on the magnitude of the charge (q), the speed of the particle ($ \ overrightarrow (v) $) relative to the magnetic field, and the magnetic induction fields ($ \ overrightarrow (B) $). This force has been established experimentally, it is called magnetic force.

And it has the form in the SI system:

\ [\ overrightarrow (F) = q \ left [\ overrightarrow (v) \ overrightarrow (B) \ right] \ \ left (1 \ right). \]

The modulus of force in accordance with (1) is equal to:

where $ \ alpha $ is the angle between the vectors $ \ overrightarrow (v \) and \ \ overrightarrow (B) $. From equation (2) it follows that if a charged particle moves along the line of the magnetic field, then it does not experience the action of a magnetic force.

Direction of magnetic force

Based on (1), the magnetic force is directed perpendicular to the plane in which the vectors $ \ overrightarrow (v \) and \ \ overrightarrow (B) $ lie. Its direction coincides with the direction of the vector product $ \ overrightarrow (v \) and \ \ overrightarrow (B) $ if the value of the moving charge is greater than zero, and is directed in the opposite direction if $ q

Strength properties of magnetic force

The magnetic force does not work on the particle, since it is always directed perpendicular to the speed of its movement. It follows from this statement that by acting on a charged particle with the help of a constant magnetic field, its energy cannot be changed.

If a particle with a charge is acted upon simultaneously by an electric and a magnetic field, then the resultant force can be written as:

\ [\ overrightarrow (F) = q \ overrightarrow (E) + q \ left [\ overrightarrow (v) \ overrightarrow (B) \ right] \ \ left (3 \ right). \]

The force indicated in expression (3) is called the Lorentz force. The part $ q \ overrightarrow (E) $ is the force acting on the charge from the electric field, $ q \ left [\ overrightarrow (v) \ overrightarrow (B) \ right] $ characterizes the force of the magnetic field on the charge. The Lorentz force is manifested when electrons and ions move in magnetic fields.

Example 1

Task: A proton ($ p $) and an electron ($ e $), accelerated by the same potential difference, fly into a uniform magnetic field. How many times the radius of curvature of the proton trajectory $ R_p $ differs from the radius of curvature of the electron trajectory $ R_e $. The angles at which particles fly into the field are the same.

\ [\ frac (mv ^ 2) (2) = qU \ left (1.3 \ right). \]

From formula (1.3) we express the speed of the particle:

Substituting (1.2), (1.4) into (1.1), we express the radius of curvature of the trajectory:

Substituting the data for different particles, we find the ratio $ \ frac (R_p) (R_e) $:

\ [\ frac (R_p) (R_e) = \ frac (\ sqrt (2Um_p)) (B \ sqrt (q_p) sin \ alpha) \ cdot \ frac (B \ sqrt (q_e) sin \ alpha) (\ sqrt ( 2Um_e)) = \ frac (\ sqrt (m_p)) (\ sqrt (m_e)). \]

The charges of a proton and an electron are equal in absolute value. The mass of an electron is $ m_e = 9.1 \ cdot (10) ^ (- 31) kg, m_p = 1.67 \ cdot (10) ^ (- 27) kg $.

Let's carry out the calculations:

\ [\ frac (R_p) (R_e) = \ sqrt (\ frac (1.67 \ cdot (10) ^ (- 27)) (9.1 \ cdot (10) ^ (- 31))) \ approx 42 . \]

Answer: The radius of curvature of a proton is 42 times greater than the radius of curvature of an electron.

Example 2

Task: Find the strength of the electric field (E) if the proton moves in a straight line in the crossed magnetic and electric fields. He flew into these fields, passing an accelerating potential difference equal to U. The fields are crossed at right angles. The magnetic induction is B.

According to the conditions of the problem, the particle is acted upon by the Lorentz force, which has two components: magnetic and electric. The first component is magnetic, it is equal to:

\ [\ overrightarrow (F_m) = q \ left [\ overrightarrow (v) \ overrightarrow (B) \ right] \ \ left (2.1 \ right). \]

$ \ overrightarrow (F_m) $ - directed perpendicular to $ \ overrightarrow (v \) and \ \ overrightarrow (B) $. The electrical component of the Lorentz force is:

\ [\ overrightarrow (F_q) = q \ overrightarrow (E) \ left (2.2 \ right). \]

The power $ \ overrightarrow (F_q) $ - is directed according to the tension $ \ overrightarrow (E) $. We remember that the proton has a positive charge. In order for the proton to move in a straight line, it is necessary that the magnetic and electrical components of the Lorentz force balance each other, that is, their geometric sum is equal to zero. Let us depict the forces, fields and velocity of the proton, fulfilling the conditions for their orientation in Fig. 2.

From Fig. 2 and the conditions for the balance of forces, we write:

We find the speed from the law of conservation of energy:

\ [\ frac (mv ^ 2) (2) = qU \ to v = \ sqrt (\ frac (2qU) (m)) \ left (2.5 \ right). \]

Substituting (2.5) into (2.4), we get:

Answer: $ E = B \ sqrt (\ frac (2qU) (m)). $

Nowhere else does the school course of physics have such a strong resemblance to big science as in electrodynamics. In particular, its cornerstone - the impact on charged particles from the electromagnetic field, has found wide application in electrical engineering.

Lorentz force formula

The formula describes the relationship between the magnetic field and the main characteristics of a moving charge. But first you need to figure out what it is.

Definition and formula of the Lorentz force

At school, they often show the experience with a magnet and iron filings on a sheet of paper. If you place it under the paper and shake it slightly, then the sawdust will line up along the lines, which are usually called the lines of magnetic tension. In simple terms, it is the force field of a magnet that surrounds it like a cocoon. It is closed on itself, that is, it has neither beginning nor end. This is a vector quantity that is directed from the south pole of the magnet to the north.

If a charged particle flew into it, the field would affect it in a very curious way. She would not slow down and accelerate, but only deviated to the side. The faster it is and the stronger the field, the more this force acts on it. It was named the Lorentz force in honor of the physicist who first discovered this property of the magnetic field.

Calculate it using a special formula:

here q is the value of the charge in Coulomb, v is the speed with which the charge moves, in m / s, and B is the magnetic field induction in the unit of measurement T (Tesla).

Lorentz force direction

Scientists have noticed that there is a certain pattern between how a particle flies into a magnetic field and where it deflects it. To make it easier to remember, they developed a special mnemonic rule. To memorize it, you need very little effort, because it uses what is always at hand - the hand. More precisely, the left hand, in honor of which it is called the left hand rule.

So, the palm should be open, four fingers are looking forward, the thumb is protruding to the side. The angle between them is 900. Now it is necessary to imagine that the magnetic flux is an arrow that digs into the palm from the inside and comes out from the back. At the same time, the fingers look in the same direction as the imaginary particle is flying. In this case, the thumb will show where it will deviate.

Interesting!

It is important to note that the left-hand rule only applies to particles with a plus sign. To find out where the negative charge will deviate, you need to point four fingers in the direction from which the particle is flying. All other manipulations remain the same.

Consequences of the properties of the Lorentz force

The body flies in a magnetic field at a certain angle. It is intuitively clear that its value has some significance on the nature of the impact of the field on it, here you need a mathematical expression to make it clearer. You should know that both force and speed are vector quantities, that is, they have a direction. The same is true for magnetic tension lines. Then the formula can be written as follows:

sin α here is the angle between two vector quantities: velocity and flux of the magnetic field.

As you know, the sine of the zero angle is also zero. It turns out that if the trajectory of the particle moves along the lines of force of the magnetic field, then it does not deviate anywhere.

In a uniform magnetic field, the lines of force have the same and constant distance from each other. Now imagine that a particle moves perpendicular to these lines in such a field. In this case, the Lawrence force will make it move along a circle in a plane perpendicular to the lines of force. To find the radius of this circle, you need to know the mass of the particle:

The value of the charge is not accidentally taken as a module. This means that it doesn't matter if a negative or a positive particle enters the magnetic field: the radius of curvature will be the same. Only the direction in which it flies will change.

In all other cases, when the charge has a certain angle α with the magnetic field, it will move along a trajectory that resembles a spiral with a constant radius R and a step h. It can be found by the formula:

Another consequence of the properties of this phenomenon is the fact that she does not do any work. That is, it does not give or take away energy from the particle, but only changes the direction of its motion.

The most striking illustration of this effect of the interaction of a magnetic field and charged particles is the aurora borealis. The magnetic field surrounding our planet deflects charged particles arriving from the Sun. But since it is weakest at the magnetic poles of the Earth, electrically charged particles penetrate there, causing the atmosphere to glow.

The centripetal acceleration given to particles is used in electric machines - electric motors. Although it is more appropriate here to talk about the Ampere force - a particular manifestation of Lawrence's force, which acts on the conductor.

The principle of operation of particle accelerators is also based on this property of the electromagnetic field. Superconducting electromagnets deflect particles from linear motion, forcing them to move in a circle.

The most curious thing is that the Lorentz force does not obey Newton's third law, which states that every action has its own opposition. This is due to the fact that Isaac Newton believed that any interaction at any distance occurs instantly, but this is not the case. It actually happens through fields. Fortunately, the embarrassment was avoided, since physicists managed to rework the third law into the law of conservation of momentum, which is also true for the Lawrence effect.

Lorentz force formula in the presence of magnetic and electric fields

A magnetic field is present not only in permanent magnets, but also in any conductor of electricity. Only in this case, in addition to the magnetic component, there is also an electrical component in it. However, even in this electromagnetic field, the Lawrence effect continues to be influenced and is determined by the formula:

where v is the speed of an electrically charged particle, q is its charge, B and E are the strengths of the magnetic and electric fields of the field.

Lorentz Force Units

Like most other physical quantities that act on a body and change its state, it is measured in newtons and is denoted by the letter N.

Electric field strength concept

The electromagnetic field actually consists of two halves - electric and magnetic. They are exactly twins, who have everything the same, but their character is different. And if you look closely, you will notice small differences in appearance.

The same goes for force fields. The electric field also has an intensity - a vector quantity, which is a power characteristic. It affects the particles that are motionless in it. By itself, it is not a Lorentz force, it just needs to be taken into account when calculating the effect on a particle in the presence of electric and magnetic fields.

Electric field strength

The electric field strength affects only a stationary charge and is determined by the formula:

The unit of measurement is N / C or V / m.

Task examples

Problem 1

On a charge of 0.005 C, which moves in a magnetic field with an induction of 0.3 T, the Lorentz force acts. Calculate it if the speed of the charge is 200 m / s, and it moves at an angle of 450 to the lines of magnetic induction.

Problem 2

Determine the speed of a body that has a charge and which moves in a magnetic field with an induction of 2 T at an angle of 900. The value with which the field acts on the body is 32 N, the charge of the body is 5 × 10-3 C.

Problem 3

An electron moves in a uniform magnetic field at an angle of 900 to its lines of force. The magnitude with which the field acts on the electron is 5 × 10-13 N. The magnitude of the magnetic induction is 0.05 T. Determine the acceleration of an electron.

ac = v2R = 6 × 10726.8 × 10-3 = 5 × 1017ms2

Electrodynamics operates with concepts that are difficult to find an analogy in the ordinary world. But this does not mean at all that it is impossible to comprehend them. With the help of various visual experiments and natural phenomena, the process of understanding the world of electricity can become truly exciting.

Definition

The force acting on a moving charged particle in a magnetic field is equal to:

called Lorentz force (magnetic force).

Based on definition (1), the modulus of the considered force:

where is the particle velocity vector, q is the particle charge, is the magnetic induction vector of the field 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 lines of force of the magnetic field, then the Lorentz force is zero. Sometimes, trying to isolate the Lorentz force, they denote using the index:

Lorentz force direction

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 thumb (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 vector product (Fig. 1 (b)).

the vector is directed perpendicular to the plane of the drawings at 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 acting on a charged particle with the help of a constant magnetic field cannot change its energy.

If the magnetic field is uniform and directed perpendicular to the speed of motion of a 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 vector of magnetic induction. In this case, the radius of the circle is:

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

The Lorentz force is a centripetal force. In the direction of deflection 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 there are simultaneously two fields (magnetic and electric), then the force that acts on it is equal to:

where is the vector of the electric field strength at the point at which the charge is located. Expression (4) was empirically obtained by Lorentz. The force that is included in formula (4) is also called the Lorentz force (Lorentz force). Division of the Lorentzian force into components: electric and magnetic relatively, since it is associated with the choice of the inertial frame of reference. So, if the frame of reference 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 measurement of the Lorentz force (like any other force) in the SI system is: [F] = H

In the SGS: [F] = ding

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, it is acted upon by a Lorentz force of the form:

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 modulus of the Lorentz force 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:

Equating the right-hand sides of expressions (1.2) and (1.3), we have:

From expression (1.3) we get 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) with speed v flies into the region where there is an electric field of strength E and a magnetic field with induction B. Vectors and coincide in direction. What is the acceleration of a particle at the moment of the beginning of movement in the fields, if?