« Physics - 11th grade"
A magnetic field acts with force on moving charged particles, including current-carrying conductors.
What is the force acting on one particle?
1.
The force acting on a moving charged particle from a magnetic field is called Lorentz force in honor of the great Dutch physicist H. Lorentz, who created the electronic theory of the structure of matter.
The Lorentz force can be found using Ampere's law.
Lorentz force modulus is equal to the ratio of the modulus of force F acting on a section of a 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 (Ampere force) acting on a section of a conductor from the magnetic field
equal to F = | I | BΔl sin α,
and the current strength in the conductor is equal to I = qnvS
Where
q - particle charge
n - particle concentration (i.e. the number of charges per unit volume)
v - particle speed
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.
Lorentz force direction
The direction of the Lorentz force is determined using the same left hand rules, which is the same as the direction of the Ampere force:
If the left hand is positioned so that the component of magnetic induction, perpendicular to the speed of the charge, enters the palm, and the four extended fingers are directed along the movement of the positive charge (against the movement of the negative), then the thumb bent 90° will indicate the direction of the Lorentz force F acting on the charge l
3.
If in the space where a charged particle is moving, there is both an electric field and a magnetic field at the same time, then the total force acting on the charge is equal to: = el + l where the force with which the electric field acts on charge q is equal to F el = q .
4.
The Lorentz force does no work, because it is perpendicular to the particle velocity vector.
This means that the Lorentz force does not change the kinetic energy of the particle and, therefore, the modulus of its velocity.
Under the influence of the Lorentz force, only the direction of the particle's velocity changes.
5.
Motion of a charged particle in a uniform magnetic field
Eat homogeneous magnetic field directed perpendicular to the initial velocity of the particle. 
The Lorentz force depends on the absolute values 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 also remains unchanged.
The Lorentz force is perpendicular to the speed and, therefore, determines the centripetal acceleration of the particle.
The invariance in absolute value of the centripetal acceleration of a particle moving with a constant velocity in absolute value means that
In a uniform magnetic field, a charged particle moves uniformly in 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 a particle to make a complete revolution (orbital period) is equal to: 
6.
Using the action of a magnetic field on a moving charge.
The effect of a magnetic field on a moving charge is used in television picture tubes, in which electrons flying towards the screen are deflected using a magnetic field created by special coils.
The Lorentz force is used in a cyclotron - a charged particle accelerator to produce particles with high energies.
The device of mass spectrographs, which make it possible to accurately determine the masses of particles, is also based on the action of a magnetic field.
Determination of Magnetic Force Strength
Definition
If a charge moves in a magnetic field, then it is acted upon by a force ($\overrightarrow(F)$), 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 and is called magnetic force.
And in the SI system it has the form:
\[\overrightarrow(F)=q\left[\overrightarrow(v)\overrightarrow(B)\right]\ \left(1\right).\]
The force modulus 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 a magnetic field line, it does not experience the action of a magnetic force.
Direction of magnetic force
The magnetic force, based on (1), 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 magnitude of the moving charge is greater than zero, and is directed in the opposite direction if $q
Properties of magnetic force
The magnetic force does not do any work on the particle, since it is always directed perpendicular to the speed of its movement. From this statement it follows that by influencing a charged particle with a constant magnetic field, its energy cannot be changed.
If a particle with a charge is simultaneously acted upon by electric and magnetic fields, 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. Part $q\overrightarrow(E)$ is the force exerted by the electric field on the charge, $q\left[\overrightarrow(v)\overrightarrow(B)\right]$ characterizes the force of the magnetic field on the charge. The Lorentz force manifests itself 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 does the radius of curvature of the proton trajectory $R_p$ differ 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:
Let us substitute (1.2), (1.4) into (1.1), and express the radius of curvature of the trajectory:
Let's substitute the data for different particles and 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 electron are equal in absolute value. Electron mass $m_e=9.1\cdot (10)^(-31)kg,m_p=1.67\cdot (10)^(-27)kg$.
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 electric field strength (E) if a proton moves in a straight line in crossed magnetic and electric fields. He flew into these fields, passing through an accelerating potential difference equal to U. The fields are crossed at right angles. The magnetic field 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 magnetic component 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 equal to:
\[\overrightarrow(F_q)=q\overrightarrow(E)\left(2.2\right).\]
The force $\overrightarrow(F_q)$- is directed along the tension $\overrightarrow(E)$. We remember that a proton has a positive charge. In order for a proton to move in a straight line, it is necessary that the magnetic and electric components of the Lorentz force balance each other, that is, their geometric sum is equal to zero. Let us depict the forces, fields and speed of proton motion, fulfilling the conditions for their orientation in Fig. 2.
From Fig. 2 and the conditions of equilibrium 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: $E=B\sqrt(\frac(2qU)(m)).$
Definition of Lorentz force
The Lorentz force is a combination of magnetic and electric force on a point charge that is caused by electromagnetic fields. Or in other words, the Lorentz force is a force acting on any charged particle that falls in a magnetic field at a certain speed. Its value depends on the magnitude of magnetic induction IN, electric charge of the particle q and the speed with which the particle falls into the field – V. Read on to learn about the formula for calculating the Lorentz force, as well as its practical significance in physics.
A little history
The first attempts to describe electromagnetic force were made back in the 18th century. Scientists Henry Cavendish and Tobias Mayer proposed that the force on magnetic poles and electrically charged objects obeys the inverse square law. However, the experimental proof of this fact was not complete and convincing. It was only in 1784 that Charles Augustine de Coulomb, using his torsion balance, was able to finally prove this assumption.
In 1820, the physicist Oersted discovered the fact that a volt current acts on the magnetic needle of a compass, and Andre-Marie Ampere in the same year was able to develop a formula for the angular dependence between two current elements. In fact, these discoveries became the foundation of the modern concept of electric and magnetic fields. The concept itself was further developed in the theories of Michael Faraday, especially in his idea of lines of force. Lord Kelvin and James Maxwell added detailed mathematical descriptions to Faraday's theories. In particular, Maxwell created the so-called “Maxwell field equation” - which is a system of differential and integral equations that describe the electromagnetic field and its relationship with electric charges and currents in vacuum and continuous media.
JJ Thompson was the first physicist to try to derive from Maxwell's field equation the electromagnetic force that acts on a moving charged object. In 1881, he published his formula F = q/2 v x B. But due to some miscalculations and an incomplete description of the bias current, it turned out to be not entirely correct.
And finally, in 1895, the Dutch scientist Hendrik Lorentz derived the correct formula, which is still used today, and also bears his name, just as the force that acts on a flying particle in a magnetic field is now called the “Lorentz force.”
Lorentz force formula
The formula for calculating the Lorentz force is as follows:
Where q is the electric charge of the particle, V is its speed, and B is the magnitude of the magnetic induction of the magnetic field.
In this case, field B acts as a force perpendicular to the direction of the velocity vector V of the loads and the direction of vector B. This can be illustrated in the diagram:
The left-hand rule allows physicists to determine the direction and return of the vector of magnetic (electrodynamic) energy. Imagine that our left hand is positioned in such a way that the magnetic field lines are directed perpendicular to the inner surface of the hand (so that they penetrate inside the hand), and all fingers except the thumb point in the direction of positive current flow, the deflected thumb indicates the direction of the electrodynamic force acting on a positive charge placed in this field.
This is how it will look schematically.
There is also a second way to determine the direction of the electromagnetic force. It consists of placing the thumb, index and middle fingers at right angles. In this case, the index finger will show the direction of the magnetic field lines, the middle finger will show the direction of current movement, and the thumb will show the direction of the electrodynamic force.
Application of Lorentz force
The Lorentz force and its calculations have their practical application in the creation of both special scientific instruments - mass spectrometers, used to identify atoms and molecules, and in the creation of many other devices of a wide variety of applications. The devices include electric motors, loudspeakers, and rail guns.
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 current-carrying circuit.
The Lorentz force is a 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, electric charge of the particle q and speed v, from which the particle falls into the field.
The way a magnetic field B behaves in relation to the load completely different from how it is observed for the electric field E. First of all, the field B does not respond to load. However, when the load moves 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 diagram q has a positive charge!
The units of the B field can be obtained from the Lorentz equation. Thus, in the SI system, the unit B is equal to 1 tesla (1T). In the CGS system, the field unit is Gauss (1G). 1T = 10 4 G
For comparison, an animation of the movement 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 the vector B, then the charge path will be a spiral as shown in the animation
Lorentz force on a current-carrying conductor
The force acting on a current-carrying conductor is the result of the Lorentz force acting on moving charge carriers, electrons or ions. If the guide section has a length l, as in the drawing
the total charge Q is moving, then the force F acting on this segment is
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, length of the segment l was given by the characteristics of the vector.
Only electrons move in the metal under the influence of potential differences; metal ions remain immobile in the crystal lattice. In electrolyte solutions, anions and cations are mobile.
Left hand rule Lorentz force— determining the direction and return of the vector of magnetic (electrodynamic) energy.
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 into the hand), and all fingers - except the thumb - point in the direction of positive current flow (moving molecule), the deflected thumb indicates the direction of the electrodynamic force acting to a positive electric charge placed in this field (for a negative charge, the force will be the opposite).
The second way to determine the direction of the electromagnetic force is to position the 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 flow, and also the direction of the force with the thumb.
Moment of force acting on a current-carrying circuit 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 the winding of an electric motor) 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 to the sides of the frame parallel to the axis of rotation.
but what does the current have to do with it, 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 positive charge moving at speed(here is the speed of ordered movement of positive charge carriers). Lorentz force modulus:
| , | (2.5.3) |
where α is the angle between And .
From (2.5.4) it is clear that a charge moving along the line is not affected by force ().
| Lorenz Hendrik Anton(1853–1928) – Dutch theoretical physicist, creator of classical electronic theory, member of the Netherlands Academy of Sciences. He derived a formula relating the dielectric constant to the density of the 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, and developed the theory of light dispersion. Developed the electrodynamics of moving bodies. In 1904, he derived formulas connecting the coordinates and time of the same event in two different inertial reference systems (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 force for a negative charge is opposite, therefore, to The 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 . Consequently, 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 and changes its energy.
Everyday we observe the effect of magnetic force on a moving charge on a television screen (Fig. 2.7).
The movement of the electron beam along the screen plane is stimulated by the magnetic field of the deflection coil. If you bring a permanent magnet close to the plane of the screen, you can easily 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.