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Knowledge Analysis of Charged Particle Motion in Uniform Electromagnetic Field Based on Maxwell Equation

Publié en ligne: 15 Jul 2022
Volume & Edition: AHEAD OF PRINT
Pages: -
Reçu: 15 Apr 2022
Accepté: 05 Jun 2022
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Magazine
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2 fois par an
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Anglais
Introduction

There is an extremely close relationship between electricity and magnetism, broadly speaking, electromagnetism includes electricity and magnetism, but in a narrow sense, electromagnetism is a discipline that describes the relationship between electricity and magnetism. It mainly includes knowledge of electromagnetic fields, electromagnetic waves, related charges and the dynamics of charged objects [1]. In this discipline, an electromagnetic field is a physical field produced by charged objects, the charged object in it is subjected to the force of the electromagnetic field on it, in turn, electromagnetic fields are also affected by charged objects (charges or currents) [2]. The electromagnetic field theory is developed by human beings in the understanding of natural laws and production practice, and is closely related to modern technology and actual production and life. Maxwell's equations are a set of partial differential equations that can be used to describe the relationship between electric and magnetic fields and charge density and current density. It is the theoretical basis of modern electromagnetism. The rich physics contained in Maxwell's equations has been widely used in all aspects of people's life and production, including communications, medical, scientific research and military technology and many other fields. The laws of motion of charged particles in electromagnetic fields are of great significance in many research fields of physics and science and technology [3]. In terms of applications, mass spectrometers, oscilloscope tubes, electron microscopes, television picture tubes, magnetic focusing, particle accelerators, etc. are all closely related to it. As far as basic research is concerned, most of people's understanding of atomic nuclei and elementary particles comes from the study of the collision process between them, and the collision process of charged particles is closely related to their motion laws in the electromagnetic field [4]. For the research of space physics and astrophysics, most of the research objects are plasma, there are also various magnetic fields (such as the earth's magnetic field, the sun's magnetic field, the stellar magnetic field, the galaxy's magnetic field, etc.). In response to this research question, Lu J briefly described the practical application of electromagnetic fields in science and technology, take the motion of charged particles in an electromagnetic field as an example, the close connection between technology applications and electromagnetic fields will be shown through two typical examples, students are required to conduct theoretical research on the test questions at the same time, pay more attention to the use of topic-related knowledge and the basic knowledge learned to solve practical problems [5]. Gjata O to solve the motion of charged particles in compound fields, it believes that in the mutual relationship of teaching, guiding students in a timely manner, in order to enable students to learn to organize the required knowledge, that is, the integration of knowledge, in order to establish a correct spatial model, classification for processing [6]. Ibarra-Sierra V G on the movement of charged particles in the electromagnetic field, combined with the analysis of the real questions in the college entrance examination paper, discuss some motion properties and motion laws of charged particles in the electromagnetic field [7]. Aquino G preliminarily explored the teaching of the moving part of charged particles in an electric field, it involves the related motion state of charged particles in the electric field, classifies the research objects, solves problems with the help of mechanics knowledge, and expounds the use of electric field control to change the motion state of charged particles [8]. Borah B K tried to use the idea of motion synthesis and decomposition through the real college entrance examination questions and their original questions, in order to deal with the complex curved motion of charged particles in the electromagnetic field, it provides some solutions for related problems [9]. Ikebe Y found that the electromagnetic field knowledge is distributed in the important test sites of the college entrance examination over the years, there is a more important position, among which the relevant concepts of electromagnetic field and the kinetic energy theorem, Newton's law of motion, momentum and other mechanics and electricity related knowledge are often skillfully combined in the comprehensive examination questions of Ulidian, and tend to apply relevant mathematical tools to solve physical problems [10].

Based on the current research, the author aims to explore the motion state of charged particles in a uniform electromagnetic field based on Maxwell's equation. The author proposes an analysis of the properties of charged particles based on Maxwell's equations and the electromagnetic field, first describe Maxwell's equations and the analysis of the electromagnetic field characteristics, list the important knowledge points about the motion of charged particles in an electric field, and determine the center of the circle where the charged particles move in a circular motion, simulate the motion of charged particles in a uniform orthogonal electromagnetic field. Obtained: the motion trajectory of the charged particle obliquely incident electric field is a parabola; The motion trajectory of the charged particle in a uniform orthogonal electromagnetic field is a cycloid; This provides a reference for the further study of the motion law of charged particles in other situations.

Maxwell's equations and analysis of electromagnetic field characteristics
Derivation of Maxwell's equations

Based on flux integral and circulation integral, namely equations (1)(2), if the vector field is taken as the physical quantity F={D,B,E,H} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over F} = \left\{ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over D} ,\,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over B} ,\,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over E} ,\,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over H} } \right\} describing the electromagnetic field, and the Gauss theorem and Stokes theorem are combined, the Maxwell equations can be obtained as follows: ψD¯=S¯DDS=GuassTheoremVDDV=VρdV {\psi _{\bar D}} = \oint {_{\bar S}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over D} \cdot D\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S} \mathop = \limits^{GuassTheorem} \int {_V\nabla \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over D} DV = \int {_V\rho dV} } } ψB=SBDS=GuassTheoremVBDV=0 {\psi _{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over B} }} = \oint {_{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S} }\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over B} \cdot D\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S} \mathop = \limits^{GuassTheorem} \int {_V\nabla \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over B} DV = 0} } ΓE=lEDl=StocksTheoremS×EDS=SBtdS {\Gamma _{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over E} }} = \oint {_{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over l} }\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over E} \cdot D\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over l} \mathop = \limits^{StocksTheorem} \int {_{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S} }\nabla \times \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over E} \cdot D\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S} = } - \int {_{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S} }{{\partial \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over B} } \over {\partial t}} \cdot d\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S} } } ΓH=lHDl=StocksTheoremS×HDS=S(JC+Dt)dS {\Gamma _{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over H} }} = \oint {_{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over l} }\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over H} \cdot D\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over l} \mathop = \limits^{StocksTheorem} \int {_{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S} }\nabla \times \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over H} \cdot D\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S} = } \int {_{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S} }\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over J} }_C}+{{\partial \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over D} } \over {\partial t}}} \right) \cdot d\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S} } }

This is the integral of Maxwell's equations. The differential formula of Maxwell's equations can be directly obtained from the integral equation as: d=ρ \nabla \cdot d = \rho b=0 \nabla \cdot \vec b = 0 ×E=BT \nabla \times \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over E} = - {{\partial \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over B} } \over {\partial T}} ×H=Jc+DT \nabla \times \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over H} = {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over J} _c} + {{\partial \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over D} } \over {\partial T}}

For the time-varying electromagnetic field in which the electric and magnetic fields change with time, let the electric field E(t) \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over E} \left( t \right) be: E(t)=EMCos(wt+ϕ0)a^N=Re[EMejwt] \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over E} \left( t \right) = {E_M}Cos\left( {wt + {\phi _0}} \right){\hat a_N} = {\mathop{\rm Re}\nolimits} \left[ {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over E} }_M}{e^{jwt}}} \right] Where E˙M=EMejϕ0a^N {{\dot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E}}}_{M}}={{E}_{M}}{{e}^{j{{\phi }_{0}}}}{{\hat{a}}_{N}} is the complex amplitude and ân is the unit vector of the electric field, then the first derivative of E(t) \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over E} \left( t \right) with respect to time can be expressed as: E(t)T=Re[jwE˙MejWt] \frac{\partial \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E}\left( t \right)}{\partial T}=\operatorname{Re}\left[ jw{{{\dot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E}}}}_{M}}{{e}^{jWt}} \right]

Similarly, for the magnetic field H(T) \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over H} \left( T \right) and its first derivative can be expressed as: H(T)=Re[H˙mejwt] \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H}\left( T \right)=\operatorname{Re}\left[ {{{\dot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H}}}}_{m}}{{e}^{jwt}} \right] H(t)T=Re[jwH˙mejwt] \frac{\partial \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H}\left( t \right)}{\partial T}=\operatorname{Re}\left[ jw{{{\dot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H}}}}_{m}}{{e}^{jwt}} \right]

Equations (13), (14), (15), (16) are Maxwell's equations in complex form. D˙M=ρ \nabla \cdot {{\dot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {D}}}_{M}}=\rho B˙M=0 \nabla \cdot {{\dot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B}}}_{M}}=0 ×E˙M=jwB˙M \nabla \times {{\dot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E}}}_{M}}=-jw{{\dot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B}}}_{M}} ×H˙=J˙M+jwD˙M \nabla \times \dot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H}}={{\dot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {J}}}_{M}}+jw\dot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {D}}M

Motion of charged particles in compound fields (when electric and magnetic fields coexist)

It is generally difficult and comprehensive to examine the motion of charged particles in a compound field [11]. In recent years, in the physics questions of the college entrance examination, there are generally two kinds of “composite fields” and the problem of space staggering, in the motion space of many charged particles, if there is an electric field E in addition to the magnetic field B, the charged particles are simultaneously affected by the electric field force and the Lorentz force, and the resultant force F is: F=qe+qv×b F = qe + qv \times b Here, in order to expand students' understanding of particle motion in the compound field, the following two common types are used to discuss the motion trajectory of particles in this compound field [12].

The charged particle is incident at any angle θ with the x-axis at an initial velocity of size υ0, in the x direction, the particles are not affected by the Lorentz force, only the electric field force, do a uniformly accelerated linear motion with υ0 cos θ, with x=υ0tcosθ+12Eqmt2 x = {\upsilon _0}t\cos \theta + {1 \over 2}{{Eq} \over m}{t^2} at the same time, the radius of its circle is R=mυ0sinθqB R = {{m{\upsilon _0}\sin \theta } \over {qB}} , at this time, the trajectory of the particle is also a spiral, when there is no electric field, the motion of particles is of equal pitch, and the helix here is of variable pitch, that is, the equation of motion of particles with increasing pitch is: {x=υ0tcosθ+12Eqmt2y=mυ0sinθqBsin(qBmt)z=Mυ0sinθQB[1cos(QBMT)] \left\{ {\matrix{ {x = {\upsilon _0}t\cos \theta + {1 \over 2}{{Eq} \over m}{t^2}} \hfill \cr {y = {{m{\upsilon _0}\sin \theta } \over {qB}}\sin \left( {{{qB} \over m}t} \right)} \hfill \cr {z = - {{M{\upsilon _0}\sin \theta } \over {QB}}\left[ {1 - \cos \left( {{{QB} \over M}T} \right)} \right]} \hfill \cr } } \right.

E and B are perpendicular to each other

The electric field E is along the y direction, the magnetic field B is along the x direction, and the charged particles enter this space with a certain initial velocity U0, taking the incident point as the origin of the coordinates, the charged particles move under the action of electrostatic force and Lorentz force, the motion trajectory in the orthogonal electromagnetic field is a cycloid (cycloid), and its motion equation can be obtained by solving the differential equation: {x=12qEcosθmt2y=mEsinθqB2[cos(qBmt)1]z=MEsinθqB2sin(QBMT)qEsinθMT \left\{ {\matrix{ {x = {1 \over 2}{{qE\,\cos \theta } \over m}{t^2}} \hfill \cr {y = - {{mE\sin \theta } \over {q{B^2}}}\left[ {\cos \left( {{{qB} \over m}t} \right) - 1} \right]} \hfill \cr {z = {{ME\sin \theta } \over {q{B^2}}}\sin \left( {{{QB} \over M}T} \right) - {{qE\sin \theta } \over M}T} \hfill \cr } } \right.

Results and Analysis
Important knowledge points about the motion of charged particles in an electric field

In the electric field, the speed and direction of the charged particles may change at any time, due to the electrostatic force acting on it in the electric field, so there is acceleration. For microscopic particles like electrons and protons, the universal gravitational force (gravity) is usually much smaller than the electrostatic force, because their mass is very small, so although they are also subject to the universal gravitational force (gravity), but usually only the influence of electrostatic force is considered and gravitational force (gravity) can be ignored. Under modern laboratory conditions, with the help of scientific and technological equipment, changing or controlling the electric field to affect the movement of electric particles, charged particles are accelerated or deflected by various modes of electric field, the two most basic conditions [13]. The motion of the charged particle is a uniform speed curve motion, because the motion situation and force method of this kind of motion can be analogous to the flat-projection motion, we call this kind of motion similar to the flat-projection motion. This seemingly complex motion is often equivalent to the uniform linear motion of charged particles in the horizontal direction, these two sub-motions of uniformly accelerated linear motion with zero initial velocity in the vertical direction are processed [14].

The motion of a charged particle in the same electric field is similar to the motion of a free fall when it is at rest, and the motion of the particle is the same if the initial velocity is not zero. and FIG. The motion in an electric field is similar to the motion in a gravitational field. However, although the particles in the gravitational field and the uniform electric field are both subjected to constant force, the nature and expression of the force are indeed different. Gravity is proportional to the mass of an object. Objects of different masses, as long as they are in the same place, have the same value g of the acceleration of gravity. Particles with the same charge but different masses have the same electric field force, however, the accelerations generated in the electric field are different, and some particles have different charges and masses, however, the accelerations that may be generated in the electric field are the same.

Important knowledge points about the motion of charged particles in a uniform magnetic field

The motion of charged particles in a uniform magnetic field is directly derived from the Lorentz force theory, which is difficult for most ordinary students. You can first use the Lorentz force demonstrator to allow students to observe the linear motion of charged particles, and explain the observation of the demonstration device to allow students to understand the electric field, excitation coil, and the direction of uniform magnetic field. After the excitation coil is energized, a combined magnetic field from the inside to the outside is generated in the glass bubble along the direction of the center line of the two coils. Then keep the speed of the electron beam unchanged, change the magnitude and direction of the magnetic induction intensity or keep the magnitude and direction of the magnetic induction unchanged, change the speed of the electron beam, and observe the change of the electron beam track respectively. Before students observe, teachers should explain the generation, direction and direction of electron beam movement of the uniform magnetic field. If students have a good foundation, they can also ask students to observe the experiment before the experiment, the deflection of the electron beam direction is determined according to the knowledge of the Lorentz force, and the prediction of the trajectory shape of the electron beam is discussed and then compared with the experimental observations.

Under normal circumstances, it is difficult for students to understand why the electrons will make a uniform circular motion at a constant speed after being vertically injected into a uniform magnetic field. When teaching, students can discuss in groups. Through cooperative learning and teacher sorting, the following points can be clarified:

Since the Lorentz force is always perpendicular to the velocity direction of the moving particle, the Lorentz force cannot do work on the moving charged particle, so the kinetic energy of the moving particle remains unchanged.

Since the magnitude of the Lorentz force is constant and its direction is perpendicular to the moving direction of the charged particle from beginning to end, it just provides the centripetal force required for the circular motion of the moving particle.

Since the Lorentz force on the particle provides the centripetal force required for its uniform circular motion, the motion radius and motion period of the charged particle can be deduced, discuss in detail which necessary factors are related to r and T, and make certain comparisons with experimental phenomena, so that students can realize that the period of moving particles has nothing to do with their speed.

In the research of physics, there are special devices called “bubble chamber” and “cloud chamber” for the study of particles, through the movement of particles in these devices, to show their tracks. If a uniform magnetic field is applied in these “cloud chambers” and “bubble chambers”, it can be seen through these devices that the motion trajectory of the charged particles is a circle. The photo below is the track of the moving charged particles in the bubble chamber, it can be seen that some particle tracks are spiral, the reason is that its energy is continuously reduced in the process of movement, resulting in a decrease in speed and speed, so it is spiral.

Determination of the center of the circular motion of charged particles

Because there are many types of finding the center of the circle, each type corresponds to a different method, so there are many ways to determine the center of particle trajectories. Even if some students have relatively solid mathematical knowledge, however, due to the lack of understanding of the system conditions and the method of finding the center of the circle, students often find it difficult to find the center of the circle in the process of solving the problem. After induction and summary, there are five conditions for determining the center of the circle, namely the incident point, the incident direction, the exit point, the exit direction and the orbit radius of the particle motion. And as long as the three conditions are known, the position of the center of the circle can be determined.

The incident point, incident direction and exit point are known, that is, the vertical line of the incident velocity is firstly drawn through the incident point, then make the perpendicular line connecting the incident point, then the intersection of the two straight lines is the center of the circular motion of the charged particle.

Knowing the incident point, exit point and exit direction, then make the vertical line of the exit velocity through the exit point, then the intersection of these two straight lines is the center of the circle where the charged particles do circular motion.

Knowing the incident direction, the exit direction and the orbit radius, the intersection of the line connecting the incident direction and the exit radius direction can be made, make an angle bisector through the intersection point, and then use the radius to find the center of the charged particle motion.

Knowing the incident point, incident direction and outgoing direction, that is, through the incident point, the vertical line perpendicular to the incident velocity and the intersection of the angle bisector of the angle between the two velocities is the center of the charged particle motion.

Knowing the exit point, exit direction and entrance direction, the intersection of the vertical line perpendicular to the exit velocity and the bisector of the angle between the two velocities is the center of the charged particle motion.

Knowing the incident point, incident direction and radius, the vertical line of the incident velocity can be made through the incident point, and the point where the length of the incident point along the vertical line is the radius is the center of the charged particle motion.

Knowing the exit point, exit direction and radius, the vertical line of the exit velocity can be drawn through the exit point, and the length of the exit point along the vertical line is the center of the charged particle motion.

The exit point and the radius, the mid-perpendicular line of the outgoing point can be made, and then the center of the charged particle motion can be found on the mid-perpendicular line according to the radius.

Knowing the incident point, exit point and exit direction, you can first draw the incident point and the exit point to make the mid-perpendicular line connecting the two points, and then make the exit velocity perpendicular through the exit point, then the two straight lines are the intersection is the center of the circular motion of the charged particles.

Motion of the charged particles in a uniform electromagnetic field

Let B=Bk \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over B} = B\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over k} , E=Ej \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over E} = E\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over j} and the particle move at υ0 along the x-axis from the origin at time t=0. B and E are constant vectors and are orthogonal.

The component form of the equation of motion is: {x¨=ωLy˙y¨=QEMωLx˙z¨=0 \left\{ {\matrix{ {\ddot x = {\omega _L}\dot y} \hfill \cr {\ddot y = {{QE} \over M}-{\omega _L}\dot x} \hfill \cr {\ddot z = 0} \hfill \cr } } \right.

Differentiate equation (20) and make substitutions, we get: {x+ωL2x˙=QEmωLy+ωL2Y=0z¨=0 \left\{ {\matrix{ \ldots \hfill \cr {x + \omega _L^2\dot x = {{QE} \over m}{\omega _L}} \hfill \cr \ldots \hfill \cr {y + \omega _L^2Y = 0} \hfill \cr \ldots \hfill \cr {\ddot z = 0} \hfill \cr } } \right.

Using the initial conditions: When t=0, there are: {x=Y=z=0x˙=υ0y˙=z˙=0 \left\{ {\matrix{ {x = Y = z = 0} \hfill \cr {\dot x = {\upsilon _0}} \hfill \cr {\dot y = \dot z = 0} \hfill \cr } } \right.

Integrating Equation (22), the trajectory equation of the charged particle is obtained as: {x=υ0υEωLsinωLt+υEtY=υ0υEωL(1cosωLT)z=0 \left\{ {\matrix{ {x = {{{\upsilon _0} - {\upsilon _E}} \over {{\omega _L}}}\sin \,{\omega _L}t + {\upsilon _E}t} \hfill \cr {Y = - {{{\upsilon _0} - {\upsilon _E}} \over {{\omega _L}}}\left( {1 - \cos \,\,{\omega _L}T} \right)} \hfill \cr {z = 0} \hfill \cr } } \right. Make: {A=υ0υEωL=mqB(υ0EB)B=υEωL=MEQB2 \left\{ {\matrix{ {A = - {{{\upsilon _0} - {\upsilon _E}} \over {{\omega _L}}} = - {m \over {qB}}\left( {{\upsilon _0} - {E \over B}} \right)} \hfill \cr {B = {{{\upsilon _E}} \over {{\omega _L}}} = {{ME} \over {Q{B^2}}}} \hfill \cr } } \right.

Then the trajectory equation (24) of the charged particle can be transformed into: {xbωLtasinωLty=a(1cosωLt)z=0y=a(1cosωLt){(XbωLt)2+(ya)2=a2Z=0 \matrix{ {\left\{ {\matrix{ {xb{\omega _L}t - a\,\sin \,{\omega _L}t} \hfill \cr {y = a\left( {1 - \cos {\omega _L}t} \right)} \hfill \cr {z = 0} \hfill \cr } } \right.} \hfill \cr {y = a\left( {1 - \cos {\omega _L}t} \right) \Rightarrow } \hfill \cr {\left\{ {\matrix{ {{{\left( {X - b{\omega _L}t} \right)}^2} + {{\left( {y - a} \right)}^2} = {a^2}} \hfill \cr {Z = 0} \hfill \cr } } \right.} \hfill \cr }

Numerical simulation of formula (24) is carried out according to different initial velocities and magnitudes of the electromagnetic field, and the simulation diagrams in Figures 1, 2, 3, and 4.

It can be seen from Figures 1, 2, 3, and 4 that if υ0 < E / B is present, the trajectory of the charged particle is a short-amplitude cycloid, in Figure 1; If υ0 < E / B, the trajectory of the charged particle is a long cycloid, as shown in Figure 2; If υ0 < E / B, the charged particles move in a straight at a uniform speed along the x-axis, as shown in Figures 3 and 4.

Figure 1

Short-amplitude cycloid of charged particles in a uniform orthogonal electromagnetic field

Figure 2

Long-amplitude cycloid of charged particles in a uniform orthogonal electromagnetic field

Figure 3

The state of uniform linear orthogonal electromagnetic field 1

Figure 4

The state of uniform linear orthogonal electromagnetic field 2

Conclusion

Maxwell's equations are the core theoretical basis of modern electromagnetism, which has a wide range of applications in many fields such as modern communication, industrial production and military. A deep understanding of the physics implied by Maxwell's equations is of great practical significance to the research work related to electromagnetism. On the basis of briefly introducing some basic concepts and laws, based on flux integral and circulation integral, Maxwell's equations are deduced and the characteristics of electromagnetic fields contained in them are analyzed accordingly, help to improve students' deep understanding of Maxwell's equations and the characteristics of electromagnetic fields. This paper analyzes the motion law of charged particles in the electromagnetic field, and simulates the motion of charged particles in the electromagnetic field, obtained: This provides a reference for the further study of the motion law of charged particles in other situations. In addition, the author has only simulated some basic motion situations involving middle school physics, for more complex motion situations, how to completely simulate the motion laws of particles needs to be discussed and studied in the follow-up work.

Figure 1

Short-amplitude cycloid of charged particles in a uniform orthogonal electromagnetic field
Short-amplitude cycloid of charged particles in a uniform orthogonal electromagnetic field

Figure 2

Long-amplitude cycloid of charged particles in a uniform orthogonal electromagnetic field
Long-amplitude cycloid of charged particles in a uniform orthogonal electromagnetic field

Figure 3

The state of uniform linear orthogonal electromagnetic field 1
The state of uniform linear orthogonal electromagnetic field 1

Figure 4

The state of uniform linear orthogonal electromagnetic field 2
The state of uniform linear orthogonal electromagnetic field 2

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