Gauss's Law · 高斯定律
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| Gauss's law/ˈɡaʊsɪz lɔː/ | 高斯定律 | gāo sī dìng lǜ |
| permittivity/ˌpɜːmɪˈtɪvəti/ | 介电常数 | jiè diàn cháng shù |
| Gaussian surface/ˈɡaʊʃn ˈsɜːfɪs/ | 高斯面 | gāo sī miàn |
Turn a nasty integral into one line
- Finding $E$ from a big charged object by integrating is hard work.
- Gauss's law 高斯定律 offers a shortcut — when the shape is symmetric.
- It links the flux through a closed surface to the charge trapped inside.
- With the right surface, you read $E$ off in a single step.
把一个棘手的积分变成一行
- 通过积分从一个大带电物体求 $E$ 是件苦差事。
- 高斯定律提供了一条捷径——当形状对称时。
- 它把穿过闭合面的通量与被困在里面的电荷联系起来。
- 选对面,你一步就能读出 $E$。
The law itself
- For any closed surface: $\oint \vec E \cdot d\vec A = \dfrac{Q_{\text{enc}}}{\varepsilon_0}$.
- The left side is the total flux; the right is the enclosed charge over $\varepsilon_0$.
- $\varepsilon_0 = 8.85\times10^{-12}$ is the permittivity 介电常数 of free space.
- Only charge inside the surface counts.
定律本身
- 对任何闭合面:$\oint \vec E \cdot d\vec A = \dfrac{Q_{\text{enc}}}{\varepsilon_0}$。
- 左边是总通量;右边是封闭电荷除以 $\varepsilon_0$。
- $\varepsilon_0 = 8.85\times10^{-12}$ 是真空的介电常数。
- 只有面内部的电荷计入。
Flux from a charge · 电荷产生的通量
Watch the field lines from a charge spread out and cross any surface around it. · 观察电荷发出的电场线如何扩散并穿过周围任意表面。
In Gauss's law, the flux through a closed surface equals: · 在高斯定律中，穿过封闭表面的通量等于：
$\oint \vec E \cdot d\vec A = Q_{\text{enc}}/\varepsilon_0$.
Outside charge cancels
- A charge outside the surface sends as much flux in as out.
- So its net contribution to the closed-surface flux is zero.
- This is why only $Q_{\text{enc}}$ appears on the right.
- The field on the surface still feels outside charge — but the flux doesn't.
外部电荷相抵消
- 面外的电荷送进来的通量和送出去的一样多。
- 所以它对闭合面通量的净贡献为零。
- 这就是为什么右边只出现 $Q_{\text{enc}}$。
- 面上的场仍然感受到外部电荷——但通量不受影响。
A charge outside a closed surface adds zero net flux through it. · 封闭表面外的电荷对其净通量贡献为零。
Outside charge sends in as much flux as it takes out — net zero. · 外部电荷进入的通量等于离开的通量——净通量为零。
A surface encloses $+6\ \text{nC}$ and $-2\ \text{nC}$. What net charge (in nC) sets its flux? · 一个表面包围了$+6\ \text{nC}$和$-2\ \text{nC}$。设定其通量的净电荷是多少（单位：nC）？
Only enclosed charge counts: $6 + (-2) = 4\ \text{nC}$. · 仅计算包围电荷：$6 + (-2) = 4\ \text{nC}$。
Pick a surface that matches the symmetry
- Sphere of charge → use a spherical Gaussian surface 高斯面.
- Line of charge → use a coaxial cylinder.
- Plane of charge → use a box (pillbox) through it.
- Choose it so $E$ is constant and either along or across each patch.
选一个匹配对称的面
- 球形电荷 → 用球形高斯面。
- 线电荷 → 用同轴圆柱。
- 平面电荷 → 用穿过它的盒子(药盒面)。
- 选它使 $E$ 恒定,并且在每一片上要么沿着、要么横穿。
The imaginary closed surface you integrate over is called a ____ surface. · 你积分过的假想封闭表面称为____表面。
It is a Gaussian surface, chosen to match the symmetry. · 它是一个高斯面，根据对称性选定。
Select all · 所有 true statements about Gauss's law. · 选择关于高斯定律的所有正确陈述。
Always true, best with symmetry, driven by enclosed charge — over a closed surface. · 始终成立，在对称性下最佳，由包围电荷驱动——作用于封闭表面。
Powerful results, fast
- Outside a charged sphere: $E = \dfrac{kQ}{r^2}$ — exactly like a point charge.
- Around an infinite line: $E = \dfrac{\lambda}{2\pi\varepsilon_0 r}$.
- Near an infinite plane: $E = \dfrac{\sigma}{2\varepsilon_0}$ (constant).
- Each drops out of Gauss's law in a couple of lines.
结果强大,又快
- 带电球外部:$E = \dfrac{kQ}{r^2}$——完全像一个点电荷。
- 无限长线周围:$E = \dfrac{\lambda}{2\pi\varepsilon_0 r}$。
- 无限大平面附近:$E = \dfrac{\sigma}{2\varepsilon_0}$(恒定)。
- 每一个都从高斯定律中几行就得出。
Outside a uniformly charged sphere of charge $Q$, the field is: · 在电荷为$Q$的均匀带电球体外部，电场为：
Gauss's law gives $E = kQ/r^2$ outside — identical to a point charge. · 高斯定律给出球体外部的$E = kQ/r^2$——等同于点电荷。
Find $E$ a distance $r$ outside a sphere holding charge $Q$.
- A spherical Gaussian surface gives $\oint E\,dA = E(4\pi r^2)$.
- Set it equal to $Q/\varepsilon_0$: $E = \dfrac{Q}{4\pi\varepsilon_0 r^2} = \dfrac{kQ}{r^2}$.
求带电荷 $Q$ 的球外距离 $r$ 处的 $E$。
- 球形高斯面给出 $\oint E\,dA = E(4\pi r^2)$。
- 令它等于 $Q/\varepsilon_0$:$E = \dfrac{Q}{4\pi\varepsilon_0 r^2} = \dfrac{kQ}{r^2}$。
Gauss's law is always true, but only useful when symmetry makes $E$ constant on your surface. For a lopsided charge you can still write $\oint \vec E \cdot d\vec A = Q_{\text{enc}}/\varepsilon_0$, but you can't pull $E$ out of the integral.
高斯定律永远成立,但只有当对称使 $E$ 在你的面上恒定时才有用。对不对称的电荷你仍能写 $\oint \vec E \cdot d\vec A = Q_{\text{enc}}/\varepsilon_0$,但你无法把 $E$ 从积分里提出来。
Gauss's law says $\oint \vec E \cdot d\vec A = Q_{\text{enc}}/\varepsilon_0$ — flux through a closed surface depends only on the enclosed charge. Choose a Gaussian surface matching the symmetry (sphere, cylinder, plane) to read $E$ off in one step.
高斯定律说 $\oint \vec E \cdot d\vec A = Q_{\text{enc}}/\varepsilon_0$——穿过闭合面的通量只取决于所围电荷。选一个匹配对称的高斯面(球、圆柱、平面),一步读出 $E$。