Unit D.3: Motion in Electromagnetic Fields单元 D.3：电磁场中的运动

A current is a stream of charge carriers, each with charge $q$ drifting at speed $v_d$. In a wire of length $L$ and cross-section carrying $n$ carriers per unit volume, the current is $I = nAqv_d$. Each carrier feels $qv_dB\sin\theta$; summing over all carriers in the length $L$ gives the wire force.

电流是一束载流子，每个带电荷 $q$、以漂移速率 $v_d$ 运动。长度 $L$、单位体积载流子数为 $n$ 的导线中，电流 $I = nAqv_d$。每个载流子受力 $qv_dB\sin\theta$；对长度 $L$ 内全部载流子求和得到导线所受力。

So the macroscopic wire law and the microscopic single-charge law $F = qvB\sin\theta$ are the same physics seen at two scales. This is why D3.1 and D3.2 share one hand rule.

所以宏观导线公式与微观单电荷公式 $F = qvB\sin\theta$ 是同一物理在两个尺度上的表现。这正是 D3.1 与 D3.2 共用同一手定则的原因。

The complete force on a charge in combined fields is $\vec{F} = q(\vec{E} + \vec{v}\times\vec{B})$. The magnetic part $q\vec{v}\times\vec{B}$ is, by the cross product, perpendicular to $\vec{v}$. Power delivered is $P = \vec{F}\cdot\vec{v}$; for the magnetic term this is $q(\vec{v}\times\vec{B})\cdot\vec{v} = 0$ because a vector perpendicular to $\vec{v}$ has zero dot product with $\vec{v}$.

电荷在复合场中受到的完整力为 $\vec{F} = q(\vec{E} + \vec{v}\times\vec{B})$。其磁分量 $q\vec{v}\times\vec{B}$ 由叉积可知垂直于 $\vec{v}$。功率 $P = \vec{F}\cdot\vec{v}$；磁项给出 $q(\vec{v}\times\vec{B})\cdot\vec{v} = 0$，因为垂直于 $\vec{v}$ 的矢量与 $\vec{v}$ 的标量积为零。

Hence the kinetic energy $\tfrac12 mv^2$ is constant under a purely magnetic force — the speed is fixed and only the direction turns. Electric fields, by contrast, can do work and change kinetic energy (D3.4).

因此在纯磁力作用下动能 $\tfrac12 mv^2$ 守恒——速率不变，只有方向转动。相反，电场可以做功并改变动能（D3.4）。

Start from $r = mv/(qB)$ and the circumference relation $T = 2\pi r / v$. Substitute $r$:

从 $r = mv/(qB)$ 与周长关系 $T = 2\pi r / v$ 出发，代入 $r$：

The speed $v$ cancels, so $T$ depends only on the mass-to-charge ratio $m/q$ and the field $B$. A faster particle covers a longer circumference at a proportionally higher speed, so the time per revolution is unchanged. This is exactly why a cyclotron can use a fixed-frequency accelerating voltage as the particle spirals outward.

速率 $v$ 被消去，故 $T$ 仅依赖荷质比 $m/q$ 与磁场 $B$。快粒子在按比例更高的速率下走更长的周长，故每圈用时不变。这正是回旋加速器能在粒子向外螺旋时使用固定频率加速电压的原因。

A charge enters horizontally at $v_0$ along $x$, between plates of length $\ell$ with field $E = V/d$ pointing along $y$. The acceleration is $a = qE/m = qV/(md)$, purely vertical. The horizontal motion is unaccelerated, so the time inside is $t = \ell / v_0$. The vertical deflection on exit is

电荷以 $v_0$ 沿 $x$ 水平进入长度为 $\ell$ 的两板之间，场 $E = V/d$ 沿 $y$ 方向。加速度 $a = qE/m = qV/(md)$，纯竖直。水平运动无加速，故板内时间 $t = \ell / v_0$。出口处竖直偏转为

This is precisely the projectile decomposition from kinematics, with $qE/m$ playing the role of $g$. Increasing $V$ or $\ell$, or lowering $v_0$, increases the deflection — exactly how a cathode-ray tube steers its beam.

这正是运动学中的抛体分解，其中 $qE/m$ 扮演 $g$ 的角色。增大 $V$ 或 $\ell$、或减小 $v_0$ 都使偏转增大——这正是阴极射线管偏转电子束的方式。

Wire 1 (current up the page) produces a field that, at the location of wire 2 to its right, points into the page (right-hand grip rule). Wire 2 also carries current up the page. Apply Fleming's left-hand rule to wire 2: current up, field into the page, force points to the left — back toward wire 1. By symmetry wire 1 is pushed toward wire 2. Hence same-direction currents attract.

导线 1（电流向上）在其右侧导线 2 处产生指向纸面内的磁场（右手握螺旋定则）。导线 2 也载向上的电流。对导线 2 用弗莱明左手定则：电流向上、磁场入纸面，力指向左——朝向导线 1。由对称性，导线 1 被推向导线 2。故同向电流相吸。

Reverse one current and every direction flips, so the forces point apart: opposite-direction currents repel. This force-per-length expression historically defined the ampere.

把一个电流反向，所有方向随之翻转，故力相互背离：反向电流相斥。这一单位长度受力表达式历史上曾用于定义安培。

A mass spectrometer chains two of this unit's ideas. First, a velocity selector passes only ions with $v = E/B$, guaranteeing every surviving ion has the same known speed. Those ions then enter a second region of pure magnetic field $B'$, where they follow circular arcs of radius

质谱仪串联了本单元两个思想。首先，速度选择器只放行 $v = E/B$ 的离子，保证每个通过的离子都有相同的已知速率。这些离子随后进入第二个纯磁场区域 $B'$，沿半径为下式的圆弧运动

Because $v$, $q$ and $B'$ are fixed and known, $r$ is directly proportional to mass $m$. Heavier ions land farther out on the detector, so position maps directly onto mass. The velocity selector (D3.6) and the gyroradius formula (D3.3) work together to make the measurement clean.

由于 $v$、$q$ 与 $B'$ 固定且已知，$r$ 与质量 $m$ 成正比。较重的离子落在探测器更外侧，故位置直接对应质量。速度选择器（D3.6）与回旋半径公式（D3.3）协同使测量变得清晰。