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

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: (1) ψD¯=∮S¯D⇀⋅DS⇀=GuassTheorem∫V∇⋅D⇀DV=∫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} } } (2) ψB⇀=∮S⇀B⇀⋅DS⇀=GuassTheorem∫V∇⋅B⇀DV=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} } (3) ΓE⇀=∮l⇀E⇀⋅Dl⇀=StocksTheorem∫S⇀∇×E⇀⋅DS⇀=−∫S⇀∂B⇀∂t⋅dS⇀ {\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} } } (4) ΓH⇀=∮l⇀H⇀⋅Dl⇀=StocksTheorem∫S⇀∇×H⇀⋅DS⇀=∫S⇀(J⇀C+∂D⇀∂t)⋅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: (5) ∇⋅d=ρ \nabla \cdot d = \rho (6) ∇⋅b→=0 \nabla \cdot \vec b = 0 (7) ∇×E⇀=−∂B⇀∂T \nabla \times \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over E} = - {{\partial \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over B} } \over {\partial T}} (8) ∇×H⇀=J⇀c+∂D⇀∂T \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: (9) E⇀(t)=EMCos(wt+ϕ0)a^N=Re[E⇀Mejwt] \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 â_n 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: (10) ∂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: (11) 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] (12) ∂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. (13) ∇⋅D⇀˙M=ρ \nabla \cdot {{\dot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {D}}}_{M}}=\rho (14) ∇⋅B⇀˙M=0 \nabla \cdot {{\dot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B}}}_{M}}=0 (15) ∇×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}} (16) ∇×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

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: (17) 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]. (1)

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: (18) {x=υ0tcosθ+12Eqmt2y=mυ0sinθqBsin(qBmt)z=−Mυ0sinθQB[1−cos(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.

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.

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: (1)

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.

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.

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.

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: (20) {x¨=ωLy˙y¨=QEM−ωLx˙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: (21) {…x+ωL2x˙=QEmωL…y+ωL2Y=0…z¨=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: (22) {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: (23) {x=υ0−υEωLsin ωLt+υEtY=−υ0−υEωL(1−cos ωLT)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: (24) {A=−υ0−υEωL=−mqB(υ0−EB)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: (25) {xbωLt−a sin ωLty=a(1−cosωLt)z=0y=a(1−cosωLt)⇒{(X−bωLt)2+(y−a)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

Figure 2

Figure 3

Figure 4