·           To understand the motion of a charged particle in a magnetic field

·           解带电粒子在磁场中的运动。

The region of the atmosphere existing above 80 kilometre from the Earth’s surface is called the ionosphere. It consists of charged particles and plays a crucial role in long range radio communication. The ultraviolet rays from the Sun ionize the air molecules in this region and produce ions and electrons. Now, let us study why these ions do not escape to outer space? And, how do they form the ionosphere?

距离地球表面80公里的大气区域称为电离层。电离层由带点粒子组成，在长波无线电通讯过程中发挥着关键性作用。从太阳射出的紫外线将该区域中的空气分子离子化，并产生离子和电子。现在我们来研究为何这些离子不会跑向外太空以及它们如何形成电离层。

We know that the equation of the magnetic force which acts on a charge q moving with a velocity  in a magnetic field  is

我们知道，施加在电荷量为q，移动速度为 的电荷上的磁力 的方程式为

 

Its magnitude can be written as

using the cross product. Here,  is the angle between the velocity  and the magnetic field .

磁力方程式也可以用如下向量方程式表述，其中的 是速度与磁场方向的夹角：

This equation tells us that if a charge moves in the magnetic field with its velocity  parallel to  that is q = 0°, the magnetic force acting on the charge is 0. The same is true if  and   are anti-parallel, that is q = 180°. In both the cases, the motion of a charge is not influenced by the magnetic field.

这个方程式告诉我们，如果一个电荷在磁场中运动的方向与磁场平行，也就是q = 0°，那么施加在这个电荷上的磁力为0。如果电荷运动方向与磁场逆平行，也就是q = 180°，那么施加在这个电荷上的磁力也是0。在这两种情况下，电荷的运动不受磁场影响。

Consider a freely moving positive charge Q entering the magnetic field such that is perpendicular to . For this case, q = 90° and hence, the magnitude of the magnetic force is maximum and is

如果一个自由运动的正电荷Q以速度 垂直进入磁场，这种情况下q = 90°，磁力最大；

To find the direction of on the positive charge, we use the right hand rule. Orient your first finger along the velocity and second finger along the magnetic field . The direction of the thumb gives the direction of the magnetic force .

我们使用右手定则来确定施加在正电荷上的磁力 的方向。让食指指向速度 的方向，中指指向磁场 的方向，则大拇指所指的方向就是磁力 的方向。

This force changes the path of the charge in the magnetic field.

磁力能改变电荷在磁场中的运动路径。

At point L, the force  is perpendicular to v. This force bends the trajectory of the charge and it reaches M. At this point, the force pulls the charge inward bending its path further and the charge reaches the point N. The charge is acted upon by a constant centripetal magnetic force at every point of its trajectory. As this force is always perpendicular to its velocity and hence the displacement, the work done by the magnetic force on the charge is 0. The energy of the charge remains constant and hence, its speed also remains constant. Therefore, the magnetic force merely changes the direction of the velocity of charge and not its magnitude. As a result, the charge performs a circular motion within the field. Since the charge is now held, we say that the magnetic field can trap the charged particles.

在L点，磁力 与速度v垂直。磁力使电荷的运动轨迹偏转并到达M点。在这点上，磁力将电荷往里拉，使其轨迹继续弯曲，电荷到达N点。在其运动轨迹的每一点上施加的向心磁力都是恒定的。由于磁力总是垂直于速度方向，因此由磁力产生的位移为0。电荷的能量是不变的，因此电荷的速度也是不变的，磁力只改变电荷的运动方向而不改变运动速度。所以，电荷在磁场中作圆周运动。由于电荷在磁场中运动，因此我们可说磁场能捕捉带电离子。

The above fact is true for the negative charge also; but, the direction of the magnetic force acting on it is opposite to that given by the thumb in the right hand rule.

For circular motion, the centripetal force is

以上这种情况对负电荷也适用。但对于负电荷，磁力的方向则与右手定则所指的方向相反。粒子做圆周运动时受到的向心力为：

Here, the centripetal force is provided by the magnetic force F_B. Hence,

其中，向心力由磁力F_B提供。因此：

Therefore, the radius (R) of its circular orbit is

所以，圆周轨道的半径(R)等于：

 

The experimental demonstration can be seen using a device called a fine beam tube. This is a vertically oriented spherical chamber containing an inert gas at a low pressure. There is an electron gun mounted horizontally inside it. Two coils, called Helmholtz coils, arranged as shown, produce a uniform magnetic field perpendicular to the axis of the electron gun when the current is passed through them. When the electron gun is switched on, one can see a bright ring inside the bulb. At a microscopic level, one can see that this is due to the presence of the magnetic field B. The magnetic force F_B forces the electrons into a circular motion as explained earlier. During their passage, the electrons interact with atomic electrons of gas and excite them, which in turn de-excite and emit radiation. A large number of such interactions along the circular path individually emit radiation that can be collectively seen as light in the form of a ring at the macroscopic level. Increasing the accelerating potential increases the speed of the electrons. As a result, the radius of the ring increases. Increasing the current in the coils results in an increase in the magnetic field. Consequently, the radius of the circular path decreases. These observations are in accordance with the derived formula  使用称为细束管的装置可观察试验展示。这是一个含有低压惰性气体竖直定位的球形室。室内有一个水平放置的电子枪。当电流通过两个如图所示设置的称为赫姆霍兹的线圈时，线圈产生与电子枪轴垂直的均一磁场。当电子枪开启时，在球内可见光环。在显微镜水平，可观察到光环是因为磁场B的存在。如先前的解释，磁力F_B迫使电子做圆周运动。运动过程中，这些电子用气体原子电子相互作用并使后者激发，接着气体原子电子去激发并发出辐射。在圆周轨迹内大量的如此相互作用均发出辐射，显微镜水平下同时观察这些辐射便产生环形光。增加加速电势可加快电子运动速度。结果，环半径加大。加大线圈电流可导致磁场加强。结果，圆周运动的半径减小。这些观察与导出公式一致。

Let us consider the case of a cathode ray tube (CRT). When no potential is applied to its deflecting systems, you can see a spot at the centre of its screen as the electron beam strikes the fluorescent screen. If you place two bar magnets facing their opposite poles, as shown, the point is deflected. This is because, each electron is influenced by the magnetic force due to the magnetic field and is deflected from its straight path. In a CRT, the bar magnets produce a weak magnetic field and the electrons have a very high velocity. Hence, the radius of curvature of the electrons is very large compared to the cross-section of the magnets. Consequently, the path traversed by electrons in the field is a circular arc of radius R. The electrons don't get trapped and can escape the field. They move further along the straight line to strike the screen and form a spot. 现在讨论阴极射线管。当无电势施加于其偏向系统时，一旦电子束轰击荧光屏，可见荧光屏中央出现一斑点。若如图所示向反极放置两个磁铁棒，斑点出现偏斜。这是因为由于磁场存在磁力使所有电子均受到影响故电子脱离原先的直线路径。在阴极射线管内，磁铁棒产生弱磁场且电子运动很快。因此，与横断面放置磁铁比较，电子运动弯曲的半径大。结果，电子在磁场中横贯穿越的路径为圆弧，半径为R。电子未被束缚，故可脱离磁场。这些电子沿直线继续运动变轰击荧光屏并形成斑点。

Now, fire a positive charge with its velocity vector making an angle ? with the magnetic field. The  can be decomposed into two perpendicular vector components. One is parallel to   the other is perpendicular to ,     . Since, the angle between  and  is zero, the parallel velocity component is not influenced by the magnetic force and contributes to its linear motion along . However, the component  being perpendicular to the magnetic field, is subject to a magnetic force causing circular motion of the charge in the plane perpendicular to . Hence, the motion of charge is the resultant of the linear and circular motion, which is a helical motion with a definite pitch. As the angle ? between velocity and magnetic field increases, the pitch of its helical path decreases. The pitch is given by the formula 现在，发出一个正电荷，使其运动矢量与磁场构成一定角度。 可分解为两个垂直的矢量组分。一个与 平行， 另一个与 垂直，     。因为 与 的角度为零，平行速度组分不受磁力影响，使其沿 做直线运动。然而，组分 与磁场垂直，故受制于磁力导致电荷垂直于 做平面环形运动。因此，电荷的运动为线性和环形运动的结果，以特定螺距做螺旋运动。随着速度与磁场角度的增加，其螺旋运动路径的螺距也增大。螺距可按下式计算

For its experimental proof, rotate the electron gun of the fine beam tube. You can see that the ring takes the form of a helix. This implies that the trajectory of the electrons is helical inside the bulb. 为得到证实，可旋转细束管的电子枪。将看到螺旋形式光环。这提示球内电子运动轨迹为螺旋状。

Let us now consider the question we first posed about the ionosphere. We know that the Earth has its own magnetic field. When air particles are ionized, ions and electrons are produced and the Earth's magnetic field does not let them escape from the atmosphere and traps them as explained earlier. The motion of +ve and -ve charges is opposite in direction and the charges follow a helical path around the magnetic field lines. This is how the ionosphere is formed. 现在讨论先前提出关于电离层的问题。我们已知地球有其自己的磁场。当气体颗粒电离时，可产生离子和电子但地球的磁场不允许其脱离大气层，而是以先前解释的方式被抑留。正电荷和负电荷的运动方向相反且，电荷以螺旋路径围绕磁场磁力线运动。这就是电离层形成的原因。

Now, can you guess why the picture of a conventional television becomes distorted when a magnet is kept near it?现在，你能否推测为什么当邻近有磁铁存在时老式电视的图像发生扭曲吗？