Orbiting Charges | 轨道电荷

📚 Orbiting Charges | 轨道电荷

A charged particle moving through a magnetic field experiences a force that is always perpendicular to both its velocity and the field. When the velocity is perpendicular to a uniform magnetic field, this force provides the centripetal force needed for circular motion. The resulting ‘orbiting charge’ is central to many A-Level applications, including mass spectrometry and cyclotrons.

带电粒子在磁场中运动时会受到一个始终垂直于其速度与磁场的力。当速度垂直于匀强磁场时，该力提供圆周运动所需的向心力。由此形成的“轨道电荷”是许多 A-Level 应用的核心，包括质谱仪和回旋加速器。

1. What is an orbiting charge? | 什么是轨道运动电荷

In CIE A-Level physics, an orbiting charge usually refers to a charged particle moving in a circular path under the influence of a magnetic field. Since a magnetic force does no work, the particle’s speed stays constant but its direction changes continuously. This creates uniform circular motion.

在 CIE A-Level 物理中，轨道电荷通常指带电粒子在磁场作用下沿圆形路径运动。由于磁场力不做功，粒子的速率保持不变，但方向不断改变，从而形成匀速圆周运动。

A familiar microscopic example is an electron moving perpendicular to a uniform magnetic field. Although we cannot see the path directly, its circular track can be revealed in a cloud chamber or by the circular arcs in particle detectors.

一个常见的微观例子是电子垂直于匀强磁场运动。虽然我们无法直接看见路径，但可以在云室中或粒子探测器的圆形弧线中观察到它的圆周轨迹。

2. Magnetic force on a moving charge | 运动电荷受到的磁场力

The magnetic force on a charge q moving with velocity v in a magnetic field B is given by F = qvB sin θ, where θ is the angle between v and B. The direction is given by Fleming’s left-hand rule for positive charges; for electrons, the force direction is reversed.

电荷 q 以速度 v 在磁场 B 中运动时受到的磁场力为 F = qvB sin θ，其中 θ 是 v 与 B 之间的夹角。对正电荷，方向由左手定则判断；对电子，力的方向相反。

When v is parallel to B, θ = 0° or 180°, so sin θ = 0 and the force is zero. The particle continues in a straight line. When v is perpendicular to B, θ = 90°, so sin θ = 1 and the force has its maximum value F = qvB.

当 v 平行于 B 时，θ = 0° 或 180°，sin θ = 0，因此力为零，粒子沿直线运动。当 v 垂直于 B 时，θ = 90°，sin θ = 1，力达到最大值 F = qvB。

3. Circular motion in a uniform magnetic field | 匀强磁场中的圆周运动

If a charged particle enters a uniform magnetic field with its velocity exactly perpendicular to the field, the magnetic force is always perpendicular to the velocity. It changes the direction of motion but never the speed, so the particle follows a circular path at constant speed.

如果带电粒子以速度恰好垂直于磁场进入匀强磁场，磁场力始终垂直于速度。它只改变运动方向，不改变速率，因此粒子以恒定速率沿圆形路径运动。

The magnetic force acts as the centripetal force: F = qvB = mv²/r. This expression is the starting point for nearly all calculations involving orbiting charges.

磁场力充当向心力：F = qvB = mv²/r。这一表达式是几乎所有涉及轨道电荷计算的出发点。

4. Deriving the orbital radius | 推导轨道半径

Rearranging qvB = mv²/r gives the radius of the circular orbit: r = mv/(qB). The radius is directly proportional to the particle’s momentum mv and inversely proportional to its charge q and the magnetic flux density B.

将 qvB = mv²/r 整理可得圆周轨道半径：r = mv/(qB)。半径与粒子动量 mv 成正比，与电荷 q 和磁通密度 B 成反比。

For a fixed magnetic field, faster particles with greater momentum follow larger circles. For a fixed speed, a stronger magnetic field forces the particle into a tighter, smaller-radius orbit.

在磁场一定时，动量越大的高速粒子圆周半径越大。在速度一定时，磁场越强，粒子被约束在半径越小的轨道中。

5. Period and cyclotron frequency | 周期与回旋频率

The time T taken to complete one full circle is found from speed = circumference / period: v = 2πr/T. Substituting r = mv/(qB) gives T = 2πm/(qB). This period does not contain v, which is a remarkable result.

完成一整圈所需的时间 T 可由速度 = 周长/周期求得：v = 2πr/T。代入 r = mv/(qB) 可得 T = 2πm/(qB)。这个周期公式不含 v，这是一个非常重要的结果。

The cyclotron frequency f, the number of revolutions per second, is the reciprocal of the period: f = 1/T = qB/(2πm). It is sometimes called the cyclotron frequency because it is used in cyclotron accelerators.

回旋频率 f 是每秒转动的圈数，等于周期的倒数：f = 1/T = qB/(2πm)。它有时被称为回旋频率，因为它在回旋加速器中得到应用。

6. Independence of period from speed | 周期与速度无关

Because T = 2πm/(qB) and f = qB

Published by TutorHao | A-Level Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）