IB Physics: Motion of Charged Particles in Electric and Magnetic Fields | IB物理：带电粒子在电磁场中的运动

📚 IB Physics: Motion of Charged Particles in Electric and Magnetic Fields | IB物理：带电粒子在电磁场中的运动

The motion of charged particles in electric and magnetic fields is a cornerstone topic in IB Physics. It appears in both SL and HL examinations, often as multi-part questions combining kinematics, circular motion, and energy conservation. Mastering this topic requires a clear understanding of when each force acts and how it affects the trajectory.

带电粒子在电场和磁场中的运动是IB物理的核心内容，在SL和HL考试中均频繁出现，常以多部分综合题考查运动学、圆周运动和能量守恒。掌握这一主题的关键在于清楚每种力何时起作用，以及它们如何改变粒子的运动轨迹。

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1. The Lorentz Force | 洛伦兹力

A charged particle in an electromagnetic field experiences the Lorentz force, which is the vector sum of the electric and magnetic forces:

处于电磁场中的带电粒子受到洛伦兹力，它是电场力和磁场力的矢量和：

F = qE + qv × B

The electric force F = qE is parallel to the field direction and acts regardless of the particle’s motion. The magnetic force F = qv × B is given by a cross product: its magnitude is F = qvB sinθ, where θ is the angle between v and B, and its direction is given by the right-hand rule for positive charges.

电场力 F = qE 平行于场方向，与粒子的运动状态无关。磁场力 F = qv × B 由叉积定义：其大小为 F = qvB sinθ，其中 θ 为 v 与 B 的夹角，方向由右手定则确定（针对正电荷）。

Three key facts about the magnetic force:

关于磁场力的三个关键事实：

• It is always perpendicular to both v and B; hence it never does work.
• 它始终垂直于 v 和 B，因此永不做功。
• It changes only the direction of the velocity, not its magnitude.
• 它只改变速度的方向，不改变速度的大小。
• If v is parallel to B (θ = 0° or 180°), the magnetic force is zero.
• 若 v 平行于 B（θ = 0° 或 180°），磁场力为零。

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2. Motion Parallel to a Uniform Electric Field | 平行于匀强电场的运动

When a charged particle moves parallel or antiparallel to a uniform electric field, the electric force is constant and collinear with the velocity. This produces uniform acceleration.

当带电粒子平行或反平行于匀强电场运动时，电场力恒定且与速度共线，产生匀加速运动。

For a particle of charge q in a field E, the acceleration is:

对于电荷量为 q、处于电场 E 中的粒子，其加速度为：

a = qE/m

This is analogous to projectile motion under gravity. If the particle starts from rest, its final speed after moving through a potential difference V is found from the work-energy theorem:

这与重力作用下的抛体运动类似。若粒子从静止出发，经过电势差 V 后的末速度可由功能定理求出：

qV = ½mv²

Since the electrostatic force is conservative, this result is independent of the path taken. The speed depends only on the potential difference and the charge-to-mass ratio q/m.

由于静电力是保守力，该结果与路径无关。末速度仅取决于电势差和荷质比 q/m。

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3. Motion Perpendicular to a Uniform Electric Field | 垂直于匀强电场的运动

When a particle enters a uniform electric field with its initial velocity perpendicular to the field, its motion resembles projectile motion under gravity.

当粒子以垂直于电场方向的初速度进入匀强电场时，其运动类似于重力场中的抛体运动。

Taking the x-axis along the initial velocity and the y-axis along the field:

以初速度方向为 x 轴、电场方向为 y 轴：

• x-direction: constant velocity, x = v₀t
• x 方向：匀速直线运动，x = v₀t
• y-direction: uniform acceleration, y = ½at² = ½(qE/m)t²
• y 方向：匀加速运动，y = ½at² = ½(qE/m)t²

Eliminating t gives a parabolic trajectory:

消去 t 可得抛物线轨迹：

y = (qE/(2mv₀²))x²

The particle emerges from a field region of length L having been deflected by an angle φ where:

粒子穿出长度为 L 的场区域时，偏转角 φ 满足：

tan φ = v_y/v₀ = qEL/(mv₀²)

This deflection principle is used in cathode ray tubes and in the deflection plates of oscilloscopes.

这一偏转原理被应用于阴极射线管和示波器的偏转板中。

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4. Motion in a Uniform Magnetic Field | 匀强磁场中的运动

When a charged particle enters a uniform magnetic field with velocity perpendicular to the field, the magnetic force acts as a

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