Circular Motion of Charged Particles in Magnetic Fields | 带电粒子在磁场中的圆周运动

📚 Circular Motion of Charged Particles in Magnetic Fields | 带电粒子在磁场中的圆周运动

When a charged particle enters a uniform magnetic field perpendicular to its velocity, the magnetic force acts as a centripetal force, causing the particle to move in a circular path. This is one of the most important applications of combining electromagnetism with circular motion in A-Level Physics.

当带电粒子垂直于匀强磁场方向进入磁场时，磁场力充当向心力，使粒子做匀速圆周运动。这是 A-Level 物理中电磁学与圆周运动结合的重要考点。

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1. The Lorentz Force | 洛伦兹力

A charged particle moving in a magnetic field experiences a force known as the Lorentz force. For a particle with charge q moving with velocity v perpendicular to a magnetic field B, the magnitude of the force is:

带电粒子在磁场中运动时会受到洛伦兹力的作用。当电荷量为 q 的粒子以速度 v 垂直于磁感应强度 B 的方向运动时，力的大小为：

F = Bqv

This force is always perpendicular to both the velocity and the magnetic field. If the velocity is not perpendicular to the field, only the perpendicular component v sin θ contributes to the force.

这个力始终同时垂直于速度和磁场方向。如果速度与磁场不垂直，只有垂直分量 v sin θ 对力有贡献。

F = Bqv sin θ

When θ = 90°, sin θ = 1, and the force is maximum. The direction of the force can be determined using Fleming’s Left-Hand Rule.

当 θ = 90° 时，sin θ = 1，力达到最大值。力的方向可用弗莱明左手定则判断。

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2. Deriving the Radius of Circular Motion | 推导圆周运动半径

Since the magnetic force is always perpendicular to the velocity, it does no work on the particle. It only changes the direction of motion, not the speed. For uniform circular motion, the centripetal force F = mv²/r must equal the magnetic force:

由于洛伦兹力始终垂直于速度方向，它对粒子不做功，只改变运动方向而不改变速度大小。对于匀速圆周运动，向心力 F = mv²/r 必须等于洛伦兹力：

Bqv = mv²/r

Rearranging this equation gives the radius of the circular path:

整理该方程即可得到圆周运动的半径：

r = mv / (Bq)

This result tells us that the radius is proportional to the momentum of the particle and inversely proportional to both the magnetic field strength and the charge.

这一结果表明，半径与粒子的动量成正比，与磁感应强度和电荷量成反比。

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3. Angular Velocity and Period | 角速度与周期

The angular velocity ω is defined as v/r. Substituting the expression for r:

角速度 ω 定义为 v/r。将 r 的表达式代入：

ω = v/r = v × Bq / (mv) = Bq / m

The period T is the time for one complete revolution. Since ω = 2π/T, we have:

周期 T 是完成一周运动所需的时间。由 ω = 2π/T，可得：

T = 2πm / (Bq)

This is a very important result: the period and angular velocity are independent of the speed of the particle and the radius of the orbit. They depend only on the magnetic field strength, the charge, and the mass of the particle.

这是一个非常重要的结论：周期和角速度与粒子的速度及轨道半径无关，仅取决于磁感应强度、电荷量和粒子的质量。

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4. Fleming’s Left-Hand Rule | 弗莱明左手定则

To determine the direction of the magnetic force on a moving positive charge, use Fleming’s Left-Hand Rule:

要判断运动正电荷所受磁场力的方向，可使用弗莱明左手定则：

• First finger (Index): Points in the direction of the magnetic field (from N to S).

  食指：指向磁场方向（从 N 极到 S 极）。

• Second finger (Middle): Points in the direction of conventional current (motion of positive charge).

  中指：指向电流方向（正电荷运动方向）。

• Thumb: Points in the direction of the force (motion).

  大拇指：指向力的方向（运动方向）。

For a negative charge, the direction of the force is opposite to that given by the rule. Alternatively, point the middle finger in the direction of electron motion and the thumb still gives the force direction.

对于负电荷，力的方向与定则给出的方向相反。另一种方法是将中指指向电子运动方向，大拇指仍然指向力的方向。

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