Movement of charged particles in electric and magnetic fields. The movement of charged particles in an electric and magnetic field - laboratory work. The speed of particles in an electric field.

Let a particle of mass m and charge e fly with speed v into the electric field of a flat capacitor. The length of the capacitor is x, the field strength is equal to E. Shifting upward in the electric field, the electron will fly through the capacitor along a curved path and fly out of it, deviating from the original direction by y. Under the influence of the field force, F = eE = ma, the particle moves accelerated vertically, therefore . The time of movement of a particle along the x axis at a constant speed. Then . And this is the equation of a parabola. That. a charged particle moves in an electric field along a parabola.

3. Movement of charged particles in a magnetic field.

Let's consider the movement of a charged particle in a magnetic field of strength N. The field lines are depicted by dots and are directed perpendicular to the plane of the drawing (towards us).

A moving charged particle represents an electric current. Therefore, the magnetic field deflects the particle upward from its original direction of motion (the direction of motion of the electron is opposite to the direction of the current)

According to Ampere's formula, the force deflecting a particle at any section of the trajectory is equal to , current, where t is the time during which charge e passes along section l. That's why . Considering that , we get

The force F is called the Lorentz force. The directions F, v and H are mutually perpendicular. The direction of F can be determined by the left hand rule.

Being perpendicular to the velocity, the Lorentz force changes only the direction of the particle's velocity, without changing the magnitude of this velocity. It follows that:

1. The work done by the Lorentz force is zero, i.e. a constant magnetic field does not do work on a charged particle moving in it (does not change the kinetic energy of the particle).

Let us recall that, unlike a magnetic field, an electric field changes the energy and speed of a moving particle.

2. The trajectory of a particle is a circle on which the particle is held by the Lorentz force, which plays the role of a centripetal force.

We determine the radius r of this circle by equating the Lorentz and centripetal forces:

Where .

That. The radius of the circle along which the particle moves is proportional to the speed of the particle and inversely proportional to the magnetic field strength.

The period of revolution of a particle T is equal to the ratio of the circumference S to the particle velocity v: . Taking into account the expression for r, we obtain . Consequently, the period of revolution of a particle in a magnetic field does not depend on its speed.

If a magnetic field is created in the space where a charged particle is moving, directed at an angle to its speed, then the further movement of the particle will be the geometric sum of two simultaneous movements: rotation in a circle with a speed in a plane perpendicular to the lines of force, and movement along the field with a speed . Obviously, the resulting trajectory of the particle will be a helical line.

4. Electromagnetic blood speed meters.

The operating principle of an electromagnetic meter is based on the movement of electric charges in a magnetic field. There is a significant amount of electrical charges in the blood in the form of ions.

Let us assume that a certain number of singly charged ions move inside the artery at a speed of . If an artery is placed between the poles of a magnet, the ions will move in the magnetic field.

For directions and B shown in Fig. 1, the magnetic force acting on positively charged ions is directed upward, and the force acting on negatively charged ions is directed downward. Under the influence of these forces, the ions move to the opposite walls of the artery. This polarization of arterial ions creates a field E (Fig. 2) equivalent to the uniform field of a parallel-plate capacitor. Then the potential difference in an artery U with diameter d is related to E by the formula. This electric field, acting on the ions, creates electric forces and, the direction of which is opposite to the direction and, as shown in Fig. 2.

The concentration of charges on the opposite walls of the artery will continue until the electric field increases so much that = .

For the equilibrium state, we can write ; , where .

Thus, blood velocity is proportional to the tension increasing across the artery. Knowing the voltage, as well as the values ​​of B and d, the blood speed can be determined.

Examples of problem solving

1. Calculate the radius of the circular arc that a proton describes in a magnetic field with an induction of 15 mT, if the speed of the proton is 2 Mm/s.

The radius of the circular arc is determined by the formula

2. A proton, having passed through an accelerating potential difference U = 600 V, flew into a uniform magnetic field with induction B = 0.3 T and began to move in a circle. Calculate the radius R of the circle.

The work done by the electric field when a proton passes through an accelerating potential difference turns into kinetic energy proton:

The radius of a circle can be found using the formula

Let's find v from (1): Substitute this into (2):

3. What energy will an electron acquire after making 40 revolutions in the magnetic field of a cyclotron used for radiation therapy, if the maximum value of the variable potential difference between the dees is U = 60 kV? What speed will the proton acquire?

During 1 revolution, a proton will pass between the dees of the cyclotron twice and acquire an energy of 2eU. For N revolutions the energy is T = 2eUN = 4.8 MeV.

The proton speed can be determined from the relation, from where

Lecture No. 7

1. Electromagnetic induction. Faraday's law. Lenz's rule.

2. Mutual induction and self-induction. Magnetic field energy.

3. Alternating current. AC operation and power.

4. Capacitive and inductive reactance.

5. The use of alternating current in medical practice, its effect on the body.

1. Electromagnetic induction. Faraday's law. Lenz's rule.

The current excited by a magnetic field in a closed circuit is called induction current, and the very phenomenon of excitation of current through a magnetic field is called electromagnetic induction.

The electromotive force that causes the induction current is called the electromotive force of induction.

In a closed circuit, a current is induced in all cases when there is a change in the flux of magnetic induction through the area limited by the circuit - this is Faraday's law.

Magnitude induced emf is proportional to the rate of change of magnetic induction flux:

The direction of the induction current is determined by Lenz's rule:

The induced current has such a direction that its own magnetic field compensates for the change in the flux of magnetic induction that causes this current:

2. Mutual induction and self-induction are special cases electromagnetic induction.

By mutual induction is called the excitation of current in a circuit when the current in another circuit changes.

Let us assume that current I 1 flows in circuit 1. Magnetic flux Ф 2 associated with circuit 2 is proportional to the magnetic flux associated with circuit 1.

In turn, the magnetic flux associated with circuit 1 is ~ I 1, therefore

where M is the mutual induction coefficient. Let us assume that during the time dt the current in circuit 1 changes by the amount dI 1. Then, according to formula (3), the magnetic flux associated with circuit (2) will change by the amount , as a result of which a mutual induction emf will appear in this circuit (according to Faraday’s law)

Formula (4) shows that the electromotive force of mutual induction arising in a circuit is proportional to the rate of change of current in the adjacent circuit and depends on the mutual inductance of these circuits.

From formula (3) it follows that

Those. The mutual inductance of two circuits is equal to the magnetic flux associated with one of the circuits when a current of unity flows in the other circuit. M is measured in Henry [G = Wb/A].

Mutual inductance depends on the shape, size and relative position circuits and on the magnetic permeability of the medium, but does not depend on the current strength in the circuit.

A circuit in which the current changes induces a current not only in other, neighboring circuits, but also in itself: this phenomenon is called self-induction.

The magnetic flux Ф associated with the circuit is proportional to the current I in the circuit, therefore

Where L- coefficient of self-induction, or loop inductance.

Let us assume that during the time dt the current in the circuit changes by the amount dI. Then from (6), as a result of which a self-induction EMF will appear in this circuit:

From (6) it follows that . Those. the inductance of a circuit is equal to the magnetic flux associated with it if a current equal to unity flows in the circuit.

The phenomenon of electromagnetic induction is based on the mutual transformation of energies electric current and magnetic field.

Let a current be switched on in a certain circuit with inductance L. Increasing from 0 to I, it creates a magnetic flux.

A change in dI by a small value is accompanied by a change in magnetic flux by a small amount

In this case, the current does work dA = IdФ, i.e. . Then

. (9)

1. Alternating current. AC operation and power.

A sinusoidal emf occurs in a frame that rotates at an angular velocity in a uniform magnetic field of induction B.

Since magnetic flux

where is the angle between the normal to the frame n and the magnetic induction vector B, directly proportional to time t.

According to Faraday's law of electromagnetic induction

where is the rate of change of the electromagnetic induction flux. Then

where is the amplitude value of the induced emf.

This EMF creates a sinusoidal alternating current in the circuit with a force of:

, (13)

where the maximum current value, R 0 is the ohmic resistance of the circuit.

The change in emf and current occurs in the same phases.

The effective strength of an alternating current is equal to the strength of a direct current that has the same power as a given alternating current:

The effective (effective) voltage value is calculated similarly:

AC work and power are calculated using the following expressions:

(16)

(17)

4. Capacitive and inductive reactance.

Capacitance. In a DC circuit, a capacitor represents an infinitely large resistance: D.C. does not pass through the dielectric separating the capacitor plates. The capacitor does not break the alternating current circuit: by alternately charging and discharging, it ensures the movement of electrical charges, i.e. supports alternating current in the external circuit. Thus, for alternating current, the capacitor represents a finite resistance called capacitance. Its value is determined by the expression:

where is the circular frequency of alternating current, C is the capacitance of the capacitor

Inductive reactance. It is known from experience that the alternating current strength in a conductor coiled in the form of a coil is significantly less than in a straight conductor of the same length. This means that in addition to ohmic resistance, the conductor also has additional resistance, which depends on the inductance of the conductor and is therefore called inductive reactance. Its physical meaning is the occurrence of self-induction EMF in the coil, which prevents changes in the current in the conductor, and, consequently, reduces the effective current. This is equivalent to the appearance of additional (inductive) resistance. Its value is determined by the expression:

where L is the inductance of the coil. Capacitive and inductive reactance are called reactance. Reactive resistance does not consume electricity, which makes it significantly different from active resistance. The human body has only capacitive properties.

The total resistance of a circuit containing active, inductive and capacitive resistance is equal to: .

5. The use of alternating current in medical practice, its effect on the body.

The effect of alternating current on the body depends significantly on its frequency. At low, sound and ultrasonic frequencies, alternating current, like direct current, causes an irritating effect on biological tissues. This is due to the displacement of ions in electrolyte solutions, their separation, and changes in their concentration in different parts of the cell and intercellular space. Tissue irritation also depends on the shape of the pulse current, the pulse duration and its amplitude.

Since the specific physiological effect of electric current depends on the shape of the impulses, in medicine for stimulation nervous system(electrosleep, electronarcosis), neuromuscular system (pacemakers, defibrillators), etc. use currents with different time dependences.

By affecting the heart, the current can cause ventricular fibrillation, which leads to the death of a person. Passing high frequency current through tissue is used in physiotherapeutic procedures called diathermy and local darsonvalization.

High frequency currents are also used for surgical purposes (electrosurgery). They allow you to cauterize, “weld” tissues (diathermocoagulation) or cut them (diathermotomy).

Examples of problem solving

1. In a uniform magnetic field with induction B = 0.1 T, a frame containing N = 1000 turns rotates uniformly. Frame area S=150cm2. The frame rotates with a frequency. Determine the instantaneous value of the emf corresponding to the frame rotation angle of 30º. =-

Substituting the expression for L from (2) into (1), we obtain:

Substituting the volume of the core into (3) as V = Sl, we obtain:

(4)

Let's substitute numerical values ​​into (4).

Familiarize yourself with the theory in the notes and textbook (Savelyev, vol. 2, § 5, § 73). Launch the program. Select "Electricity and Magnetism" and "Charge Movement in an Electric Field." Click the button with the page image at the top of the inner window. Read the brief theoretical information. Write down what is necessary in your notes. (If you have forgotten how to operate the computer simulation system, read the INTRODUCTION page 5 again.)

GOAL OF THE WORK:

* Acquaintance with the model of the process of charge movement in a uniform electric field.

* Experimental study patterns of motion of a point charge in a uniform electric field.

* Experimental determination of the specific charge of a particle.

BRIEF THEORY:

The movement of charged particles in an electric field is widely used in modern electronic devices, in particular, in cathode ray tubes with an electrostatic system for deflecting the electron beam.

ELECTRIC CHARGE is a quantity that characterizes the ability of an object to create an electric field and interact with an electric field.

A POINT CHARGE is an abstract object (model) that has the form material point, carrying an electric charge (charged MT).

An ELECTRIC FIELD is something that exists in a region of space in which a force called electric force acts on a charged object.

THE MAIN PROPERTIES of the charge are:

Additivity (summability);

· invariance (sameness in all inertial reference systems);

discreteness (presence elementary charge, denoted e, and the multiplicity of any charge to this elementary: q = Ne, Where N- any positive or negative integer);

· obedience to the law of conservation of charge (the total charge of an electrically isolated system, through the boundaries of which charged particles cannot penetrate, is conserved);

· the presence of positive and negative charges (charge is an algebraic quantity).

COULLOMB'S LAW determines the force of interaction between two point charges: , where is the unit vector directed from the first charge q 1 to 2 q 2 .

TENSION is called a vector characteristic fields, numerically equal to the ratio of the force acting on a point charge to the value q of this charge: . If the tension is given electric field, then the force acting on the charge will be determined by the formula .

A field is called HOMOGENEOUS, the intensity of which at all points is the same both in magnitude and direction. The force acting on a charged particle in a uniform field is the same everywhere, therefore the acceleration of the particle, determined by Newton’s second law (at low speeds of movement) will also remain unchanged V« c, Where With– speed of light in vacuum): = const. Then Y = , And

V Y= , Where Y– vertical displacement of the particle and V Y is the vertical component of the velocity at the moment of time when the particle leaves the capacitor.

METHOD AND PROCEDURE OF MEASUREMENTS

Close the theory window. Carefully examine the drawing, find all the controls and other main elements.

Draw the experimental field and the trajectory of the particle. By pressing the “Start” button, observe the movement of the particle on the screen.

If a particle with charge e moves in space where there is an electric field with intensity E, then it is acted upon by a force eE. If, in addition to the electric field, there is a magnetic field, then the Lorentz force equal to e also acts on the particle, where u is the speed of the particle relative to the field, B is the magnetic induction. Therefore, according to Newton’s second law, the equation of particle motion has the form:

The written vector equation breaks down into three scalar equations, each of which describes movement along the corresponding coordinate axis.

In what follows we will be interested only in some special cases of motion. Let us assume that charged particles, initially moving along the X axis with speed, enter the electric field of a flat capacitor.

If the gap between the plates is small compared to their length, then edge effects can be neglected and the electric field between the plates can be considered uniform. By directing the Y axis parallel to the field, we have: . Since there is no magnetic field, then . In the case under consideration, the charged particles are only affected by the force from the electric field, which, for the chosen direction of the coordinate axes, is entirely directed along the Y axis. Therefore, the trajectory of the particles lies in the XY plane and the equations of motion take the form:

The movement of particles in this case occurs under the influence of a constant force and is similar to the movement of a horizontally thrown body in a gravitational field. Therefore, it is clear without further calculations that the particles will move along parabolas.

Let us calculate the angle by which the particle beam will deviate after passing through the capacitor. Integrating the first of equations (3.2), we find:

Integrating the second equation gives:

Since at t=0 (the moment the particle enters the capacitor) u(y)=0, then c=0, and therefore

From here we get for the deflection angle:

We see that the beam deflection significantly depends on the specific particle charge e/m

§ 72. Motion of a charged particle in a uniform magnetic field

Let's imagine a charge moving in a uniform magnetic field with a speed v perpendicular to V. The magnetic force imparts to the charge an acceleration perpendicular to the speed

(see formula (43.3); the angle between v and B is a straight line). This acceleration only changes the direction of the speed, but the magnitude of the speed remains unchanged. Consequently, the acceleration (72.1) will be constant in magnitude. Under these conditions, a charged particle moves uniformly in a circle, the radius of which is determined by the relation. Substituting here the value (72.1) for and solving the resulting equation for R, we obtain

So, in the case when a charged particle moves in a uniform magnetic field perpendicular to the plane in which the movement occurs, the trajectory of the particle is a circle. The radius of this circle depends on the speed of the particle, the magnetic induction of the field and the ratio of the particle's charge to its mass. The ratio is called specific charge.

Let us find the time T spent by the particle on one revolution. To do this, divide the circumference by the velocity of the particle v. As a result we get

From (72.3) it follows that the period of revolution of a particle does not depend on its speed; it is determined only by the specific charge of the particle and the magnetic induction of the field.

Let us find out the nature of the motion of a charged particle in the case when its speed forms an angle a other than a straight line with the direction of a uniform magnetic field. Let us decompose the vector v into two components; - perpendicular to B and - parallel to B (Fig. 72.1). The modules of these components are equal

Magnetic force has a modulus

and lies in a plane perpendicular to B. The acceleration created by this force is normal for the component.

The component of the magnetic force in direction B is zero; therefore, this force cannot affect the value. Thus, the movement of a particle can be represented as the superposition of two movements: 1) movement along direction B at a constant speed and 2) uniform motion around a circle in a plane perpendicular to vector B. The radius of the circle is determined by formula (72.2) with v replaced by . The trajectory of motion is a helical line, the axis of which coincides with direction B (Fig. 72.2). The line step can be found by multiplying the rotation period T determined by formula (72.3):

The direction in which the trajectory twists depends on the sign of the particle's charge. If the charge is positive, the trajectory spins counterclockwise. The trajectory along which a negatively charged particle moves twists clockwise (it is assumed that we are looking at the trajectory along direction B; the particle flies away from us, if, and towards us, if).

16. Movement of charged particles in an electromagnetic field. Application of electron beams in science and technology: electron and ion optics, electron microscope. Charged particle accelerators.

Let's introduce the conceptelementary particle as an object, the mechanical state of which is completely described by specifying three coordinates and three components of the speed of its movement as a whole. Studyinteractions of elementary particles with em.m. Let us preface the field with some general considerations related to the concept of “particle” in relativistic mechanics.

Particle interaction with each other is described (and was described before the theory of relativity) using the concept of a force field. Each particle creates a field around itself. Every other particle in this field is subject to a force. This applies to both charged particles interacting with em. field, and massive particles that do not have a charge and are in a gravitational field.

In classical mechanics, the field was only a way of describing the interaction of particles as a physical phenomenon. The situation is changing significantly in the theory of relativity due to the finite speed of field propagation. The forces currently acting on a particle are determined by their location at the previous time. A change in the position of one of the particles is reflected in other particles only after a certain period of time. The field becomes physical reality through which the interaction of particles occurs. We cannot talk about the direct interaction of particles located at a distance from each other. Interaction can occur at any moment only between neighboring points in space (short-range interaction). That's why we can talk about the interaction of a particle with a field and the subsequent interaction of the field with another particle .

In classical mechanics, you can introduce the concept of an absolutely rigid body, which under no circumstances can be deformed. However, in the impossibility of existence absolutely rigid body can be easily verified using the following reasoning based on theory of relativity.

Let a rigid body be set in motion at any one point by an external influence. If there was a body absolutely solid, then all its points would have to move simultaneously with the one that was affected. (Otherwise the body would have to deform). The theory of relativity, however, makes this impossible, since the impact from a given point is transmitted to others at a finite speed, and therefore all points of the body cannot simultaneously begin to move. Therefore, under absolutely solid body we should mean a body, all dimensions of which remain unchanged in the frame of reference where it is at rest.

From the above, certain conclusions regarding the consideration of elementary particles . It is obvious that in relativistic mechanics particles, which we consider as elementary , cannot be assigned finite dimensions. In other words, within the strict special theory of relativityelementary particles should not have finite dimensions and, therefore, should be considered as point ones.

17. Own electromagnetic oscillations. Differential equation of natural electromagnetic oscillations and its solution.

Electromagnetic vibrations are called periodic changes in tension E and induction B.

Electromagnetic waves include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, x-rays, and gamma rays.

In unlimited space or in systems with energy losses (dissipative), eigenelectric circuits with a continuous frequency spectrum are possible.

18. Damped electromagnetic oscillations. Differential equation of damped electromagnetic oscillations and its solution. Attenuation coefficient. Logarithmic damping decrement. Good quality.

electromagnetic damped oscillations arise in e electromagnetic oscillatory system, called LCR - circuit (Figure 3.3).

Figure 3.3.

Differential equation we obtain using Kirchhoff’s second law for a closed LCR circuit: the sum of the voltage drops across the active resistance (R) and capacitor (C) is equal to the induced emf developed in the circuit circuit:

attenuation coefficient

This is a differential equation that describes the fluctuations in the charge of a capacitor. Let us introduce the following notation: The value β, as in the case of mechanical vibrations, is called attenuation coefficient, and ω 0 – natural cyclic frequency hesitation. With the introduced notation, equation (3.45) takes the form Equation (3.47) completely coincides with the differential equation of a harmonic oscillator with viscous friction (formula (4.19) from the section “Physical foundations of mechanics”). The solution to this equation describes damped oscillations of the form q(t) = q 0 e -bt cos(wt + j) (3.48) where q 0 is the initial charge of the capacitor, ω = is the cyclic oscillation frequency, φ is initial phase hesitation. In Fig. Figure 3.17 shows the form of the function q(t). The dependence of the voltage on the capacitor on time has the same form, since U C = q/C.

DECREMENT DECREMENT

(from Latin decrementum - decrease, decrease) (logarithmic attenuation decrement) - a quantitative characteristic of the rate of attenuation of oscillations in linear system; represents the natural logarithm of the ratio of two subsequent maximum deviations of a fluctuating quantity in the same direction. Because in a linear system, the oscillating value changes according to the law (where the constant value is the damping coefficient) and the two subsequent maximum. deviations in one direction X 1 and X 2 (conventionally called “amplitudes” of oscillations) are separated by a period of time (conventionally called “period” of oscillations), then , and D. z..

So, for example, for mechanical oscillate system consisting of mass T, held in the equilibrium position by a spring with a coefficient. elasticity k and frictional force F T , proportional speed v(F T =-bv, Where b- coefficient proportionality), D. z.

At low attenuation. Likewise for electric. circuit consisting of inductance L, active resistance R and containers WITH, D. z.

.

At low attenuation.

For nonlinear systems, the law of damping of oscillations is different from the law, i.e., the ratio of two subsequent “amplitudes” (and the logarithm of this ratio) does not remain constant; therefore D. z. does not have such a definition. meaning, as for linear systems.

Good quality- a parameter of the oscillatory system that determines the width of the resonance and characterizes how many times the energy reserves in the system are greater than the energy losses during one oscillation period. Indicated by the symbol from English. quality factor.

The quality factor is inversely proportional to the rate of decay of natural oscillations in the system. That is, the higher the quality factor of the oscillatory system, the less energy loss for each period and the slower the oscillations decay.

19. Forced electromagnetic oscillations. Differential equation of forced electromagnetic oscillations and its solution. Resonance.

Forced electromagnetic oscillations are called periodic changes in current and voltage in an electrical circuit that occur under the influence of an alternating emf from an external source. An external source of EMF in electrical circuits are alternating current generators operating at power plants.

In order to carry out undamped oscillations in a real oscillatory system, it is necessary to compensate for the energy loss in some way. Such compensation is possible if we use any periodically acting factor X(t), which changes according to a harmonic law: When considering mechanical vibrations, then the role of X(t) is played by the external driving force (1) Taking into account (1), the law of motion for the spring pendulum (formula (9) of the previous section) will be written as Using the formula for the cyclic frequency of free undamped oscillations of the spring pendulum and (10) of the previous section , we obtain equation (2) When considering an electric oscillatory circuit, the role of X(t) is played by the external emf supplied to the circuit, which periodically changes according to the harmonic law. or alternating voltage (3) Then the differential equation of oscillations of charge Q in the simplest circuit, using (3), can be written as Knowing the formula for the cyclic frequency of free oscillations of the oscillatory circuit and the formula of the previous section (11), we arrive at differential equation(4) Oscillations that arise under the influence of an external periodically varying force or an external periodically varying emf are called respectively forced mechanical And forced electromagnetic oscillations. Equations (2) and (4) will be reduced to a linear inhomogeneous differential equation (5) and further we will apply its solution for forced vibrations depending on the specific case (x 0 if mechanical vibrations is equal to F 0 /m, in the case of electromagnetic vibrations - U m/L). The solution to equation (5) will be equal (as is known from the course on differential equations) to the sum general solution(5) homogeneous equation (1) and a particular solution to the inhomogeneous equation. We are looking for a private solution in complex form. Let's replace the right-hand side of equation (5) with the complex variable x 0 e iωt: (6) We will look for a particular solution to this equation in the form Substituting the expression for s and its derivatives (and) into expression (6), we will find (7) Since this equality should be true for all times, then time t must be excluded from it. This means η=ω. Taking this into account, from formula (7) we find the value s 0 and multiply its numerator and denominator by (ω 0 2 - ω 2 - 2iδω) We represent this complex number in exponential form: where (8) (9) This means that the solution to equation (6) in complex form will have the form Its real part, which is the solution to equation (5), is equal to (10) where A and φ are determined by formulas (8) and (9), respectively. Consequently, a particular solution to the inhomogeneous equation (5) is equal to (11) The solution to equation (5) is the sum of the general solution to the homogeneous equation (12) and the particular solution to equation (11). Term (12) plays a significant role only in the initial stage of the process (when oscillations are established) until the amplitude of forced oscillations reaches the value determined by equality (8). Graphically forced oscillations are shown in Fig. 1. This means that in a steady state, forced oscillations occur with a frequency ω and are harmonic; the amplitude and phase of the oscillations, which are determined by equations (8) and (9), also depend on ω.

Fig.1

Let us write down expressions (10), (8) and (9) for electromagnetic oscillations, taking into account that ω 0 2 = 1/(LC) and δ = R/(2L) : (13) Differentiating Q=Q m cos(ωt–α) with respect to t, we obtain the current strength in the circuit during steady oscillations: (14) where (15) Equation (14) can be written as where φ = α – π/2 - phase shift between current and applied voltage (see (3)). In accordance with equation (13) (16) From (16) it follows that the current lags in phase with the voltage (φ>0) if ωL>1/(ωС), and leads the voltage (φ<0), если ωL<1/(ωС). Выражения (15) и (16) можно также вывести с помощью векторной диаграммы. Это будет осуществлено далее для переменных токов.

Resonance(fr. resonance, from lat. resono“I respond”) is the phenomenon of a sharp increase in the amplitude of forced oscillations, which occurs when the frequency of natural oscillations coincides with the oscillation frequency of the driving force. An increase in amplitude is only a consequence of resonance, and the reason is the coincidence of the external (exciting) frequency with some other frequency determined from the parameters of the oscillatory system, such as the internal (natural) frequency, viscosity coefficient, etc. Usually the resonant frequency is not much different from own normal, but not in all cases we can talk about their coincidence.

20. Electromagnetic waves. Electromagnetic wave energy. Energy flux density. Umov-Poynting vector. Wave intensity.

ELECTROMAGNETIC WAVES, electromagnetic oscillations propagating in space at a finite speed, depending on the properties of the medium. An electromagnetic wave is a propagating electromagnetic field ( cm. ELECTROMAGNETIC FIELD).

We strengthen our skills in solving and visualizing differential equations using the example of one of the most common evolutionary equations, remember the good old Scilab and try to understand whether we need it... Pictures under the cut (700 kilobytes)

Let's make sure the software is fresh

julia>] (v1.0) pkg>update #will you have time to make tea (v1.0) pkg> status Status `C:\Users\Igor\.julia\environments\v1.0\Project.toml` AbstractPlotting v0.9.0 Blink v0.8.1 Cairo v0.5.6 Colors v0.9.5 Conda v1.1.1 DifferentialEquations v5.3.1 Electron v0.3.0 FileIO v1.0.2 GMT v0.5.0 GR v0.35.0 Gadfly v1.0.0+ #master (https://github.com /GiovineItalia/Gadfly.jl.git) Gtk v0.16.4 Hexagons v0.2.0 IJulia v1.14.1+ [`C:\Users\Igor\.julia\dev\IJulia`] ImageMagick v0.7.1 Interact v0.9.0 LaTeXStrings v1. 0.3 Makie v0.9.0+ #master (https://github.com/JuliaPlots/Makie.jl.git) MeshIO v0.3.1 ORCA v0.2.0 Plotly v0.2.0 PlotlyJS v0.12.0+ #master (https://github .com/sglyon/PlotlyJS.jl.git) Plots v0.21.0 PyCall v1.18.5 PyPlot v2.6.3 Rsvg v0.2.2 StatPlots v0.8.1 UnicodePlots v0.3.1 WebIO v0.4.2 ZMQ v1.0.0

and let's start setting the problem

Movement of charged particles in an electromagnetic field

A charged particle with a charge moving in an EMF with speed is acted upon by the Lorentz force: . This formula is valid with a number of simplifications. Neglecting corrections for the theory of relativity, we assume the mass of the particle to be constant, so that the equation of motion has the form:

Let us direct the Y axis along the electric field, the Z axis along the magnetic field, and assume for simplicity that the initial velocity of the particle lies in the XY plane. In this case, the entire trajectory of the particle will also lie in this plane. The equations of motion will take the form:

Let's make it dimensionless: . Asterisks indicate dimensional quantities, and - the characteristic size of the physical system under consideration. We obtain a dimensionless system of equations of motion of a charged particle in a magnetic field:

Let's lower the order:

As the initial configuration of the model, we will choose: T, V/m, m/s. For numerical solution let's use the package DifferentialEquations:

Code and graphs

using DifferentialEquations, Plots pyplot() M = 9.11e-31 # kg q = 1.6e-19 # C C = 3e8 # m/s λ = 1e-3 # m function model solver(Bo = 2., Eo = 5e4, vel = 7e4) B = Bo*q*λ / (M*C) E = Eo*q*λ / (M*C*C) vel /= C A = syst(u,p,t) = A * u + # ODE system u0 = # start cond-ns tspan = (0.0, 6pi) # time period prob = ODEProblem(syst, u0, tspan) # problem to solve sol = solve(prob, Euler(), dt = 1e-4, save_idxs = , timeseries_steps = 1000) end Solut = modelsolver() plot(Solut)

Here the Euler method is used, for which the number of steps is specified. Also, not the entire solution of the system is stored in the answer matrix, but only the 1st and 2nd indices, that is, the x and y coordinates (we don’t need velocities).

X = for i in eachindex(Solut.u)] Y = for i in eachindex(Solut.u)] plot(X, Y, xaxis=("X"), background_color=RGB(0.1, 0.1, 0.1)) title !("Particle trajectory") yaxis!("Y") savefig("XY1.png")#save the graph to the project folder

Let's check the result. Let's introduce instead X new variable. Thus, a transition is made to a new coordinate system, moving relative to the original one at a speed u in the direction of the axis X:

If we select and denote , the system will be simplified:

The electric field has disappeared from the last equations, and they represent the equations of motion of a particle under the influence of a uniform magnetic field. Thus, the particle in the new coordinate system (x, y) should move in a circle. Since this new coordinate system itself moves relative to the original one with speed , the resulting motion of the particle will consist of uniform motion along the axis X and rotation around a circle in a plane XY. As is known, the trajectory resulting from the addition of such two movements is, in the general case, trochoid. In particular, if the initial speed is zero, the simplest case of motion of this kind is realized - by cycloid.
Let's make sure that the drift speed is really equal E/B. For this:

• let’s spoil the response matrix by replacing the first element (maximum) with an obviously smaller value
• let's find the number of the maximum element in the second column of the response matrix, which is plotted along the ordinate
• Let's calculate the dimensionless drift speed by dividing the abscissa value at the maximum by the corresponding time value

Y = -0.1 numax = argmax(Y) X / Solut.t

Out: 8.334546850446588e-5

B = 2*q*λ / (M*C) E = 5e4*q*λ / (M*C*C) E/B

Out: 8.333333333333332e-5
With an accuracy of seventh order!
For convenience, we will define a function that accepts model parameters and a graph signature, which will also serve as the file name png, created in the project folder (works in Juno/Atom and Jupyter). Unlike Gadfly, where the graphs were created in layers, and then were output by the function plot(), in Plots, to do in one frame different schedules, the first of them is created by the function plot(), and subsequent ones are added using plot!(). In Julia, the names of functions that change accepted objects usually end with an exclamation mark.

function plotter(ttle = "qwerty", Bo = 2, Eo = 4e4, vel = 7e4) Ans = modelsolver(Bo, Eo, vel) X = for i in eachindex(Ans.u)] Y = for i in eachindex( Ans.u)] plot!(X, Y) p = title!(ttle) savefig(p, ttle * ".png") end

At zero initial speed, as expected, we obtain cycloid:

plot() plotter("Zero start velocity", 2, 4e4, 7e4)

We obtain the trajectory of the particle when the induction and voltage are zero and when the sign of the charge changes. Let me remind you that the dot means sequential execution of the function with all elements of the array

Hidden away

plot() plotter.("B is zeroed E varies", 0, )

plot() plotter.("E is zero B varies", , 0)

q = -1.6e-19 # C plot() plotter.("Negative charge")

And let’s see how a change in initial velocity affects the trajectory of a particle:

plot() plotter.("Variation of speed", 2, 5e4, )

A little about Scilab

There is already enough information on Habré about Sailab, for example, so we will limit ourselves to links to Wikipedia and the home page.

On my own behalf, I’ll add about the availability of a convenient interface with checkboxes, buttons and graph output, and a rather interesting visual modeling tool, Xcos. The latter can be used, for example, to simulate a signal in electrical engineering:

Actually, our problem can be solved in Scilab:

Code and pictures

clear function du = syst(t, u, A, E) du = A * u + // ODE system endfunction function = model solver(Bo, Eo, vel) B = Bo*q*lambda / (M*C) E = Eo*q*lambda / (M*C*C) vel = vel / C u0 = // start cond-ns t0 = 0.0 tspan = t0:0.1:6*%pi // time period A = U = ode(" rk", u0, t0, tspan, list(syst, A, E)) endfunction M = 9.11e-31 // kg q = 1.6e-19 // C C = 3e8 // m/s lambda = 1e-3 / / m = modelsolver(2, 5e4, 7e4) plot(cron, Ans1) xtitle("Dimensionless coordinates and velocities","t","x, y, dx/dt, dy/dt"); legend("x", "y", "Ux", "Uy"); scf(1)//creating a new graphic window plot(Ans1(1, :), Ans1(2, :)) xtitle ("Particle trajectory","x","y"); xs2png(0,"graf1");// you can save graphs in different formats xs2jpg(1,"graf2");// however, it works every now and then

Information on the function for solving difurs ode. Basically this begs the question

Why do we need Julia?

... if there are already such wonderful things as Scilab, Octave and Numpy, Scipy?
I won’t say anything about the last two - I haven’t tried them. And in general, the question is complex, so let’s think offhand:

Scilab
On a hard drive it will take a little more than 500 MB, it starts quickly and the difuro calculation, graphics and everything else are immediately available. Good for beginners: excellent guide (mostly localized), there are many books in Russian. Internal errors have already been mentioned and, and since the product is very niche, the community is sluggish, and additional modules are very scarce.

Julia
As packages are added (especially any Python stuff a la Jupyter and Mathplotlib), it grows from 376 MB to quite more than six gigabytes. It doesn’t spare the RAM either: at the start it’s 132 MB and after you draw up graphs in Jupiter, it will easily reach 1 GB. If you work in Juno, then everything is almost like in Scilab: You can execute code directly in the interpreter, you can type in the built-in notepad and save as a file, there is a variable browser, a command log and online help. Personally, I am outraged by the absence of clear() , i.e. I ran the code, then started correcting and renaming it, but the old variables remained (there is no variable browser in Jupiter).

But all this is not critical. Scilab is quite suitable for the first couples; making a lab, a course, or calculating something in between is a very handy tool. Although there is also support for parallel computing and calling C/Fortran functions, it cannot be seriously used for anything. Large arrays plunge him into horror; to define multidimensional ones, one has to engage in all sorts of obscurantism, and calculations are beyond the scope classical problems They may well drop everything along with the operating system.

And after all these pains and disappointments, you can safely move on to Julia, to rake even here. We will continue to study, fortunately the community is very responsive, problems are resolved quickly, and Julia has many more interesting features that will turn the learning process into an exciting journey!

The sedimentation of solid and liquid particles suspended in gas under the influence of an electric field has advantages over other sedimentation methods. The effect of an electric field on a charged particle is determined by the magnitude of its electric charge. With electrodeposition, small particles manage to impart a significant electrical charge and, thanks to this, carry out the process of deposition of very small particles, which cannot be carried out under the influence of gravity or centrifugal force.

The principle of electrical purification of air (gases) from suspended particles is to charge the particles and then release them from the suspended medium under the influence of an electric field.

Physical entity electrodeposition consists in the fact that a gas flow containing suspended particles is pre-ionized, and the particles contained in the gas acquire an electrical charge. Charging of particles in the field of a corona discharge occurs under the influence of an electric field and due to ion diffusion. The maximum charge value of particles larger than 0.5 µm is proportional to the square of the particle diameter, and for particles smaller than 0.2 µm - to the particle diameter.

Under normal conditions, most gas molecules are neutral, i.e.

carries an electrical charge of one sign or another; Due to the action of various physical factors, a gas always contains a certain amount of electrical charge carriers. Such factors include strong heating, radioactive radiation, friction, bombardment of gas by fast-moving electrons or ions, etc.

Gas ionization is carried out in two ways:

1) on one's own, at a sufficiently high potential difference across the electrodes;

2) dependentO- as a result of exposure to radiation from radioactive substances, x-rays.

In industry, electrodeposition of suspended particles from gas is carried out in such a way that the gas flow is directed inside tubular (or between plate) positive electrodes, which are grounded (Fig. 2.6). Thin wire or rod electrodes, which are cathodes, are stretched inside the tubular electrodes.

If a certain voltage is created in the electric field between the electrodes, then the charge carriers, i.e., ions and electrons, receive significant acceleration, and when they collide with molecules, the latter are ionized. Ionization involves the ejection of one or more outer electrons from the orbit of a neutral molecule. As a result, a neutral molecule is converted into a positive ion and free electrons. This process is called impact ionization.

Rice. 2.6. Schemes of gas cleaning electrodes

When an ionized gas flow passes in an electric field between two electrodes, charged particles under the influence of an electric field move to the oppositely charged electrodes and settle on them.

The part of the interelectrode space adjacent to the corona electrode, in which impact ionization occurs, is called the corona region. The rest of the interelectrode space, i.e. between the corona and collecting electrodes, is called the external region.

A bluish-violet glow (corona) is observed around the corona electrode. Corona discharge is also accompanied by a quiet crackling sound. During a corona discharge, ozone and nitrogen oxides are released.

The ions and free electrons formed as a result of impact ionization also receive acceleration under the influence of the field and ionize new molecules. Thus, the process is of an avalanche nature. However, as you move away from the corona electrode, the electric field strength is no longer sufficient to maintain high speeds, and the process of impact ionization gradually fades.

Electric charge carriers, moving under the influence of an electric field, as well as as a result of Brownian motion, collide with dust particles suspended in a gas flow passing through an electrostatic precipitator and transfer an electric charge to them.

During ionization, both positive and negative ions are formed: positive ions remain near the “corona” at the cathode, and negative ions are directed at high speed to the anode, meeting and charging particles suspended in the gas on their way.

Most of the suspended particles passing in the interelectrode space receive a charge opposite to the sign of the collecting electrodes, move to these electrodes and are deposited on them. Some of the dust particles located in the sphere of action of the corona receive a charge opposite to the sign of the corona electrode and are deposited on this electrode.

If a potential difference (4...6) kV/cm is created on the electrodes, and a current density of (0.05...0.5) mA/m cathode length is provided, then the dusty gas, when passed between the electrodes, is almost completely freed from suspended particles.

Let us consider the main dependencies characterizing the electrical purification of gases (air) from dust particles.

The basic law of interaction of electric charges is Coulomb's law

expressed by the formula

F = k 1 (q 1 q 2 /r 2), (2.28)

Where q 1 , q 2 - magnitudes of interacting point charges; r– the distance between them; k 1 - proportionality coefficient ( k 1 > 0).

Point charges are understood as charges located on bodies of any shape, and the sizes of the bodies are small compared to the distance at which their action is felt.

Proportionality factor k 1 depends on the properties of the medium. This coefficient can be represented as a ratio of two coefficients

k 1 = k/ε (2.29)

Where k- coefficient; ε is a dimensionless quantity called the relative dielectric constant of the medium. For vacuum ε = 1.

Coulomb's law can also be expressed

Coefficient k in the SI system they accept k= 1/4 π.ε 0 ; here ε 0 is the electrical constant.

Let's substitute this value into formula (2.52.)

F = q 1 ∙q 2 /(4 π∙ε 0 ∙ε∙r 2), (2.31)

where ε 0 = 8.85∙10 -12 Cl 2 /(N.m 2).

To characterize the electric field, a physical quantity is used - field strength E. The intensity at any point in the electric field is the force with which this field acts on a single positive charge placed at this point.

Corona discharge occurs at a certain field strength. This value is called critical voltage and for negative polarity of the electrode can be determined by the empirical formula

Ecr= 3.04(β + 0.0311 √β / r)10 6 , (2.32)

Where r- radius of the corona electrode, m; β - gas density ratio in

operating conditions to gas density under standard conditions ( t= 20 0 C; R= 1.013∙10 5 Pa):

Here IN- barometric pressure, Pa; R r is the magnitude of rarefaction or absolute pressure of gases, Pa; t- gas temperature, °C.

Formula (2.54) is intended for air, but with some approximation it can also be applied to flue gases.

Field voltage at a distance x from the axis of the corona electrode:

Where U- voltage applied to the electrodes; R 1 and R 2 - radii of corona and precipitation electrodes.

Charge amount q(kA) acquired by a conductive spherical particle under the influence of an electric field is calculated using the formula:

q= 3∙π∙ d h 2 ∙ε ∙ E, (2.35)

where ε is the dielectric constant of the medium; d h - particle diameter; E- electric field strength of the corona discharge.

The amount of charge acquired by an electrically non-conducting particle:

where εch is the relative dielectric constant of the particle.

The maximum charge of particles with a diameter of more than 1 micron is determined by the formula

q prev =n e=0.19∙10 -9 r 2 E, (2.37)

Where n- number of elementary charges; e- the value of the elementary charge equal to 1.6∙10 -19 C; r- particle radius, m; E- electric field strength, V/m.

Formula (2.59.) is directly applicable if the dielectric constant of the dust substance is e equals 2.5. For many substances the value e significantly different: for gases e= 1; for plaster e= 4; for metal oxides e=12. ..18; for metals e= ∞.

If e≠2.5, then the value q pre, obtained by formula (2.38.), is multiplied by the correction, which is the ratio

D e =m/D e =2.5, (2.39)

Where De=m- meaning D= 1 + 2(ε - 1)/(ε + 2) at e= m; at ε = 2.5, D= 1.66; for ε = 1, D= 1.

In an electric precipitator, particle charging occurs very quickly: in less than a second, the particle charge approaches its limit value (Table 2.5).

Table 2.4

Ratio of particle charge versus charging time

The speed of movement of charged dust particles with a diameter of more than 1 micron in an electric field, m/s, can be determined by the formula

w h = 10 -11 E 2 r/μ 0 (2.40)

Where E- electric field strength, V/m; r- particle radius, m; μ 0 - dynamic viscosity of gas (air), Pa.s.

The speed of movement of charged dust particles with a diameter of less than 1 micron in an electrostatic field, m/s, can be determined by the formula

w h = 0.17.10 -11 E/μ 0(2.41)

The speed of movement of suspended particles that have received a charge depends on the size of the particles and the hydraulic resistance of the gaseous medium.

The rate of particle deposition in an electric field in a laminar mode of motion:

w h = n∙ e 0 ∙ E x /(3π d h ∙ μ 0) , (2.42)

Where n- the number of charges received by the particle; e 0 - the value of the elementary charge; μ 0 - coefficient of dynamic viscosity of the gas flow.

The deposition time can be found from the equation:

Where R- distance from the axis of the corona electrode to the surface of the collecting electrode; R 1 – radius of the corona electrode.

Magnitude w h changes with changes in value x.

The degree of cleaning efficiency in an electric precipitator can be determined by a formula obtained theoretically

η = 1 – exp(- w D f), (2.44)

Where w d - speed of movement (drift) of charged particles towards the collecting electrode, m/s; f- specific deposition surface, i.e. the surface of the precipitation electrodes per 1 m 3 /s of the gas (air) being purified, m 2.

Dust with low electrical conductivity causes a reverse corona phenomenon, which is accompanied by the formation of positively charged ions that partially neutralize the negative charge of the particles, as a result of which they lose the ability to move to the collecting electrode and be deposited. The conductivity of dust is influenced by the composition of the gas and dust. With increasing humidity of gases, the electrical resistivity of dust decreases. At high gas temperatures, the electrical strength of the interelectrode space decreases, which leads to deterioration in dust collection.