« 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 the side magnetic field, 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 vector magnetic induction.
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 bent by 90° thumb will indicate the direction of the Lorentz force F l acting on the charge
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.
Dutch physicist H. A. Lorenz in late XIX V. established that the force exerted by a magnetic field on a moving charged particle is always perpendicular to the direction of motion of the particle 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 position the palm of your left hand so that the four extended fingers indicate the direction of movement of the charge, and the vector of the magnetic induction field enters the outstretched 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 modulus of the Lorentz force is easily determined from Ampere's law and is:
F = | q| vB sin?,
Where q- particle charge, v- the speed of its movement, ? - the angle between the vectors of speed and magnetic field induction.
If, in addition to the magnetic field, there is also an electric field, which acts on the charge with force 
, then the total force acting on the charge is equal to:
.
Often this force is called the Lorentz force, and the force expressed by the formula ( F = | q| vB sin?) are called 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 a TV picture tube is easy to observe if you bring a permanent magnet to its screen - the image will be distorted.
Motion of a charged particle in a uniform magnetic field. Let a charged particle fly in at a speed v into a uniform magnetic field perpendicular to the tension lines.
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 expression for purposeful acceleration and the formula ( F = | q| vB sin?):
.
From here we get
.
Where m- particle mass.
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 by their specific charges, i.e., by the ratio of the charge of a particle to its mass, and from the results obtained to accurately determine the masses of the particles.
The vacuum chamber of the device is placed in the 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 the photographic plate, where they leave a trace that allows one to measure the radius of the trajectory with great accuracy r. This radius determines the specific charge of the ion. Knowing the charge of an ion, you can easily calculate its mass.
Electric charges moving in a certain direction create a magnetic field around themselves, the speed of propagation of which in a vacuum is equal to the speed of light, and in other media is slightly less. If the movement of a charge occurs in an external magnetic field, then an interaction occurs between the external magnetic field and the magnetic field of the charge. Since electric current is the directed movement of charged particles, the force that will act in a magnetic field on a current-carrying conductor will be the result of individual (elementary) forces, each of which is applied to an elementary charge carrier.
The processes of interaction between an external magnetic field and moving charges were studied by G. Lorentz, who, as a result of many of his experiments, derived a formula for calculating the force acting on a moving charged particle from the magnetic field. That is why the force that acts on a charge moving in a magnetic field is called the Lorentz force.
The force acting on the conductor by the drain (from Ampere's law) will be equal to:
By definition, the current strength is equal to I = qn (q is the charge, n is the number of charges passing through the cross section of the conductor in 1 s). It follows from this:
Where: n 0 is the number of charges contained in a unit volume, V is their speed of movement, S is the cross-sectional area of the conductor. Then:
Substituting this expression into Ampere's formula, we get:
This force will act on all charges located in the volume of the conductor: V = Sl. The number of charges present in a given volume will be equal to:
Then the expression for the Lorentz force will look like:
From this we can conclude that the Lorentz force acting on a charge q, which moves in a magnetic field, is proportional to the charge, the magnetic induction of the external field, the speed of its movement and the sine of the angle between V and B, that is:
The direction of movement of charged particles is taken to be the direction of movement of positive charges. Therefore, the direction of a given force can be determined using the left-hand rule.
The force acting on negative charges will be directed in the opposite direction.
The Lorentz force is always directed perpendicular to the speed V of the charge and therefore does not do any work. It only changes the direction of V, and the kinetic energy and velocity of the charge as it moves in a magnetic field remain unchanged.
When a charged particle moves simultaneously in magnetic and electric fields, it will be acted upon by a force:
Where E is the electric field strength.
Let's look at a small example:
An electron that has passed through an accelerating potential difference of 3.52∙10 3 V enters a uniform magnetic field perpendicular to the induction lines. Trajectory radius r = 2 cm, field induction 0.01 T. Determine the specific charge of the electron.
Specific charge is a value equal to the ratio of charge to mass, that is, e/m.
In a magnetic field with induction B, a charge moving with a speed V perpendicular to the induction lines is subject to the Lorentz force F L = BeV. Under its influence, the charged particle will move along a circular arc. Since in this case the Lorentz force will cause centripetal acceleration, then according to Newton’s 2nd law we can write:
The electron acquires kinetic energy, which will be equal to mV 2 /2, due to the work A of the electric field forces (A = eU), substituting it into the equation we get.
The force exerted by a magnetic field on a moving electrically charged particle.
where q is the charge of the particle;
V - charge speed;
a is the angle between the charge velocity vector and the magnetic induction vector.
The direction of the Lorentz force is determined according to the left hand rule:
If you place your left hand so that the component of the induction vector perpendicular to the speed enters the palm, and the four fingers are located in the direction of the speed of movement of the positive charge (or against the direction of the speed of the negative charge), then the bent thumb will indicate the direction of the Lorentz force:
Since the Lorentz force is always perpendicular to the speed of the charge, it does not do work (that is, it does not change the value of the charge speed and its kinetic energy).
If a charged particle moves parallel to the magnetic field lines, then Fl = 0, and the charge in the magnetic field moves uniformly and rectilinearly.
If a charged particle moves perpendicular to the magnetic field lines, then the Lorentz force is centripetal:
and creates a centripetal acceleration equal to:
In this case, the particle moves in a circle.
According to Newton's second law: the Lorentz force is equal to the product of the mass of the particle and the centripetal acceleration:
then the radius of the circle:
and the period of charge revolution in a magnetic field:
Since electric current represents the ordered movement of charges, the action of a magnetic field on a conductor with current is the result of its action on individual moving charges. If we introduce a current-carrying conductor into a magnetic field (Fig. 96a), we will see that as a result of the addition of the magnetic fields of the magnet and the conductor, the resulting magnetic field will increase on one side of the conductor (in the drawing above) and the magnetic field will weaken on the other side conductor (in the drawing below). As a result of the action of two magnetic fields, the magnetic lines will bend and, trying to contract, they will push the conductor down (Fig. 96, b).
The direction of the force acting on a current-carrying conductor in a magnetic field can be determined by the “left-hand rule.” If the left hand is placed in a magnetic field so that the magnetic lines coming out of the north pole seem to enter the palm, and the four extended fingers coincide with the direction of the current in the conductor, then the large bent finger of the hand will show the direction of the force. Ampere force acting on an element of the length of the conductor depends on: the magnitude of the magnetic induction B, the magnitude of the current in the conductor I, the element of the length of the conductor and the sine of the angle a between the direction of the element of the length of the conductor and the direction of the magnetic field.
This dependence can be expressed by the formula:
For a straight conductor of finite length, placed perpendicular to the direction of a uniform magnetic field, the force acting on the conductor will be equal to:
From the last formula we determine the dimension of magnetic induction.
Since the dimension of force is:
i.e., the dimension of induction is the same as what we obtained from Biot and Savart’s law.
Tesla (unit of magnetic induction)
Tesla, unit of magnetic induction International System of Units, equal magnetic induction, at which the magnetic flux through a cross section of area 1 m 2 equals 1 Weber. Named after N. Tesla. Designations: Russian tl, international T. 1 tl = 104 gs(gauss).
Magnetic torque, magnetic dipole moment- the main quantity characterizing the magnetic properties of a substance. The magnetic moment is measured in A⋅m 2 or J/T (SI), or erg/Gs (SGS), 1 erg/Gs = 10 -3 J/T. The specific unit of elementary magnetic moment is the Bohr magneton. In the case of a flat circuit with electric current, the magnetic moment is calculated as
where is the current strength in the circuit, is the area of the circuit, is the unit vector normal to the plane of the circuit. The direction of the magnetic moment is usually found according to the gimlet rule: if you rotate the handle of the gimlet in the direction of the current, then the direction of the magnetic moment will coincide with the direction of the translational movement of the gimlet.
For an arbitrary closed loop, the magnetic moment is found from:
where is the radius vector drawn from the origin to the contour length element
In the general case of arbitrary current distribution in a medium:
where is the current density in the volume element.
So, a torque acts on a current-carrying circuit in a magnetic field. The contour is oriented at a given point in the field in only one way. Let's take the positive direction of the normal to be the direction of the magnetic field at a given point. Torque is directly proportional to current I, contour area S and the sine of the angle between the direction of the magnetic field and the normal.
Here M - torque , or moment of force , - magnetic moment circuit (similarly - the electric moment of the dipole).
In an inhomogeneous field (), the formula is valid if the contour size is quite small(then the field can be considered approximately uniform within the contour). Consequently, the circuit with current still tends to turn around so that its magnetic moment is directed along the lines of the vector.
But, in addition, a resultant force acts on the circuit (in the case of a uniform field and . This force acts on a circuit with current or on a permanent magnet with a moment and draws them into a region of a stronger magnetic field.
Work on moving a circuit with current in a magnetic field.
It is easy to prove that the work of moving a circuit with current in a magnetic field is equal to , where and are the magnetic fluxes through the area of the circuit in the final and initial positions. This formula is valid if the current in the circuit is constant, i.e. When moving the circuit, the phenomenon of electromagnetic induction is not taken into account.
The formula is also valid for large circuits in a highly inhomogeneous magnetic field (provided I= const).
Finally, if the circuit with current is not displaced, but the magnetic field is changed, i.e. change the magnetic flux through the surface covered by the circuit from value to then for this you need to do the same work. This work is called the work of changing the magnetic flux associated with the circuit. Magnetic induction vector flux (magnetic flux) through the pad dS is called scalar physical quantity, which is equal
where B n =Вcosα is the projection of the vector IN to the direction of the normal to the site dS (α is the angle between the vectors n And IN), d S= dS n- a vector whose module is equal to dS, and its direction coincides with the direction of the normal n to the site. Flow vector IN can be either positive or negative depending on the sign of cosα (set by choosing the positive direction of the normal n). Flow vector IN usually associated with a circuit through which current flows. In this case, we specified the positive direction of the normal to the contour: it is associated with the current by the rule of the right screw. This means that the magnetic flux that is created by the circuit through the surface limited by itself is always positive.
The flux of the magnetic induction vector Ф B through an arbitrary given surface S is equal to
For a uniform field and a flat surface, which is located perpendicular to the vector IN, B n =B=const and
This formula gives the unit of magnetic flux weber(Wb): 1 Wb is a magnetic flux that passes through a flat surface with an area of 1 m 2, which is located perpendicular to a uniform magnetic field and whose induction is 1 T (1 Wb = 1 T.m 2).
Gauss's theorem for field B: the flux of the magnetic induction vector through any closed surface is zero:
This theorem is a reflection of the fact that no magnetic charges, as a result of which the lines of magnetic induction have neither beginning nor end and are closed.
Therefore, for streams of vectors IN And E through a closed surface in the vortex and potential fields, different formulas are obtained.
As an example, let's find the vector flow IN through the solenoid. The magnetic induction of a uniform field inside a solenoid with a core with magnetic permeability μ is equal to
The magnetic flux through one turn of the solenoid with area S is equal to
and the total magnetic flux, which is linked to all turns of the solenoid and is called flux linkage,
DEFINITION
Lorentz force– force acting on a point charged particle moving in a magnetic field.
It is equal to the product of the charge, the modulus of the particle velocity, the modulus of the magnetic field induction vector and the sine of the angle between the magnetic field vector and the particle velocity.
Here is the Lorentz force, is the particle charge, is the magnitude of the magnetic field induction vector, is the particle velocity, is the angle between the magnetic field induction vector and the direction of motion.
Unit of force – N (newton).
The Lorentz force is a vector quantity. The Lorentz force takes its greatest value when the induction vectors and direction of the particle velocity are perpendicular ().
The direction of the Lorentz force is determined by the left-hand rule:
If the magnetic induction vector enters the palm of the left hand and four fingers are extended towards the direction of the current movement vector, then the thumb bent to the side shows the direction of the Lorentz force.
In a uniform magnetic field, the particle will move in a circle, and the Lorentz force will be a centripetal force. In this case, no work will be done.
Examples of solving problems on the topic “Lorentz force”
EXAMPLE 1
EXAMPLE 2
| Exercise | Under the influence of the Lorentz force, a particle of mass m with charge q moves in a circle. The magnetic field is uniform, its strength is equal to B. Find the centripetal acceleration of the particle.
|
| Solution | Let us recall the Lorentz force formula: In addition, according to Newton's 2nd law: IN in this case The Lorentz force is directed towards the center of the circle and the acceleration created by it is directed there, that is, this is centripetal acceleration. Means: |
