Unit D.2: Electric and Magnetic Fields单元 D.2：电场与磁场

Coulomb's law is an inverse-square law: doubling $r$ quarters the force. This shares its mathematical form with Newton's law of gravitation $F = \dfrac{G m_1 m_2}{r^2}$ — a parallel we exploit in D2.6.

库仑定律是平方反比律：$r$ 加倍，力变为四分之一。它与牛顿万有引力 $F = \dfrac{G m_1 m_2}{r^2}$ 形式相同——这一平行关系将在 D2.6 中利用。

For more than two charges, the net force on any one charge is the vector sum of the individual Coulomb forces (superposition). Resolve each pairwise force into components, add component-wise, then recombine. The constant can be written either as $k$ or via the permittivity of free space $\varepsilon_0 = 8.85 \times 10^{-12}\ \mathrm{C^2\,N^{-1}\,m^{-2}}$, since $k = \dfrac{1}{4\pi\varepsilon_0}$.

超过两个电荷时，某电荷所受合力是各对库仑力的矢量和（叠加原理）。把每对力分解为分量、按分量相加、再合成。常数可写为 $k$，也可用真空介电常数 $\varepsilon_0 = 8.85 \times 10^{-12}\ \mathrm{C^2\,N^{-1}\,m^{-2}}$，因为 $k = \dfrac{1}{4\pi\varepsilon_0}$。

At every point the field has a single, definite direction (the net force per unit charge there). If two field lines crossed, the field would have two directions at the crossing point — a contradiction. Hence lines never intersect.

每一点的场都有唯一确定的方向（该处单位电荷受到的合力）。若两条电场线相交，交点处的场将有两个方向——矛盾。因此电场线不相交。

For a dipole (equal $+$ and $-$ charges), lines leave the positive charge, curve through space, and enter the negative charge. Exactly midway on the perpendicular bisector the field is non-zero and points from $+$ toward $-$. Contrast with two like charges: there a null point (zero field) sits midway, and lines push apart rather than join.

对偶极子（等量 $+$ 与 $-$），电场线从正电荷发出、在空间弯曲、终止于负电荷。在中垂线正中点处场不为零，由 $+$ 指向 $-$。与两个同号电荷对比：那里中点是零场点，电场线互相排开而非相连。

From $E = V/d$, the field has units $\mathrm{V\,m^{-1}}$. From $E = F/q$, it has units $\mathrm{N\,C^{-1}}$. These are identical because $1\ \mathrm{V} = 1\ \mathrm{J\,C^{-1}}$ and $1\ \mathrm{J} = 1\ \mathrm{N\,m}$:

由 $E = V/d$，场的单位为 $\mathrm{V\,m^{-1}}$。由 $E = F/q$，单位为 $\mathrm{N\,C^{-1}}$。二者相同，因为 $1\ \mathrm{V} = 1\ \mathrm{J\,C^{-1}}$ 且 $1\ \mathrm{J} = 1\ \mathrm{N\,m}$：

A charged particle entering a uniform field perpendicular to the field lines follows a parabolic path, exactly like a projectile in gravity: constant velocity along the plates, constant acceleration $a = qE/m$ across them. That trajectory analysis is developed in D.3.

带电粒子垂直于电场线进入匀强场后走抛物线，与重力中的抛体完全类似：沿板方向匀速，垂直方向匀加速 $a = qE/m$。该轨迹分析在 D.3 展开。

Electric field strength is the negative gradient of potential: the field points "downhill" in potential, from high $V_e$ to low $V_e$. For a uniform field this reduces to the relation you already met:

电场强度是电势的负梯度：场指向电势"下坡"方向，由高 $V_e$ 指向低 $V_e$。对匀强场，这退化为你已学过的关系：

On a diagram, equipotentials are like contour lines on a map and field lines are the steepest-descent paths crossing them at right angles. Closely spaced equipotentials mean a large $\Delta V_e$ over a small $\Delta r$, hence a strong field. Note the sign and distance contrast with PE: potential $V_e \propto 1/r$ falls off more slowly than field $E \propto 1/r^2$.

在图上，等势面如地图上的等高线，电场线则是与之处处垂直的最陡下降路径。等势面越密表示小 $\Delta r$ 上有大 $\Delta V_e$，即场越强。注意与场的距离差别：电势 $V_e \propto 1/r$ 比场 $E \propto 1/r^2$ 衰减更慢。

Electric field lines start and end on charges (sources and sinks). Magnetic field lines never start or stop — they always form complete closed loops. The deep reason is that no isolated magnetic "charge" (a monopole) has ever been found: cut a bar magnet in half and you get two smaller magnets, each with its own N and S.

电场线起止于电荷（源与汇）。磁感线既不起也不止——总是构成完整闭合回路。根本原因是从未发现孤立的磁"荷"（磁单极子）：把条形磁铁切成两半会得到两块更小的磁铁，各自仍有 N、S 极。

Inside a solenoid, the loops from all the turns reinforce to give a nearly uniform interior field; outside, the return paths spread out and weaken, giving the solenoid the same external pattern as a bar magnet. This equivalence is why a current-carrying coil is an electromagnet, and reversing the current swaps its N and S ends.

在螺线管内部，各匝的回路相互增强，形成近乎匀强的内部场；外部，回路返回路径展开变弱，使螺线管外部图样与条形磁铁相同。正因这种等价，载流线圈即为电磁铁，反转电流即可互换其 N、S 端。

Between two protons the electrostatic repulsion exceeds the gravitational attraction by a factor of about $10^{36}$. Yet gravity governs planets, stars and galaxies. The resolution is sign: electric charge comes in $+$ and $-$, so bulk matter is almost perfectly neutral and its electric forces cancel over large scales. Mass has only one sign, so gravitational pulls always add up — never cancel. Over astronomical distances the relentless accumulation of a tiny force wins.

两个质子之间的静电斥力约为引力吸引的 $10^{36}$ 倍。然而引力主宰行星、恒星与星系。原因在于符号：电荷有 $+$、$-$ 两种，故宏观物质几乎完全中性，其电力在大尺度上相互抵消。质量只有一种符号，引力吸引总是叠加——从不抵消。在天文距离上，微小力的持续累积最终取胜。

This same sign argument explains why electric field lines can terminate (on the opposite charge) while gravitational field lines, having no "negative mass" to end on, only ever point inward toward mass.

同样的符号论证解释了：电场线可以终止（在相反电荷上），而引力场线因没有"负质量"可终止，只能指向质量向内。