IB Physics: Motion in Electromagnetic Fields | IB物理：电磁场中的运动

📚 IB Physics: Motion in Electromagnetic Fields | IB物理：电磁场中的运动

When a charged particle encounters an electric or magnetic field, its motion transforms into a rich variety of patterns — parabolic trajectories in uniform electric fields, perfect circles in uniform magnetic fields, and helical spirals when fields and velocities are misaligned. This topic unifies Newton’s laws, circular motion, and field theory, and it is tested in both IB Physics SL and HL paper 2 exams with surprising frequency.

当带电粒子进入电场或磁场时，其运动会呈现出丰富多样的形态——在匀强电场中做抛物线运动、在匀强磁场中做完美圆周运动、当场与速度方向不共面时做螺旋运动。本主题将牛顿定律、圆周运动与场论完美统一，在 IB 物理 SL 和 HL 的 Paper 2 中考查频率极高。

Mastering this topic requires more than memorising formulas. You must visualise vector geometry: the direction of force, the orientation of velocity, and the resulting path curvature all depend on the right-hand rule, the sign of the charge, and the field configuration. In this article, we will methodically build the full picture, from first principles to exam-level problem solving.

掌握这一主题远不止背诵公式。你必须想象矢量几何：力的方向、速度的取向和路径的曲率都取决于右手定则、电荷正负以及场构型。在本文中，我们将从第一性原理出发，系统地构建完整图景，直至达到应试级别的问题解决能力。

1. The Lorentz Force | 洛伦兹力

The Lorentz force law is the master equation for the entire topic. For a charge q moving at velocity v through an electric field E and a magnetic field B, the total electromagnetic force is the vector sum of the electric and magnetic contributions:

洛伦兹力公式是整个主题的核心总方程。电荷 q 以速度 v 穿过电场 E 和磁场 B 时，所受到的电磁力是电场力与磁场力的矢量和：

F = qE + qv × B

The electric term qE acts parallel to the field direction and can either speed up or slow down the particle. The magnetic term qv × B is always perpendicular to both v and B; therefore it changes only the particle’s direction, never its kinetic energy.

电场项 qE 沿场方向作用，既可能加快也可能减慢粒子。磁场项 qv × B 始终垂直于 v 和 B，因此它只改变粒子的运动方向，从不改变其动能。

The cross product qv × B is critical. For a positive charge, the force direction follows the right-hand rule: point your fingers from v toward B, and your thumb gives the direction of v × B. For a negative charge, the force is exactly reversed. Neglecting this sign reversal is one of the most common exam errors.

叉积 qv × B 至关重要。对正电荷，力的方向遵循右手定则：四指从 v 弯向 B，大拇指指向 v × B 的方向。对负电荷，力的方向恰好相反。忽视这一符号反转是考试中最常见的失误之一。

2. Motion in a Uniform Electric Field | 匀强电场中的运动

Consider a charged particle entering a uniform electric field with its initial velocity perpendicular to the field direction. This is the classic parallel-plate capacitor configuration, where a particle enters a region between two charged plates. The field exerts a constant force, giving a constant acceleration entirely analogous to gravity on a projectile.

考虑带电粒子以垂直于场方向的初速度进入匀强电场。这是经典平行板电容器构型——粒子进入两带电板之间的区域。电场施加恒定力，产生恒定加速度，完全类似于重力场中的抛体运动。

Take E along the +y direction and the initial velocity v₀ along +x. The acceleration components are:

设 E 沿 +y 方向，初速度 v₀ 沿 +x 方向，则加速度分量为：

aₓ = 0, aᵧ = qE / m

Integrating with respect to time gives the position at any instant:

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