Electric Flux · 电通量
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| electric flux/ɪˈlektrɪk flʌks/ | 电通量 | diàn tōng liàng |
| area vector/ˈeərɪə ˈvektə/ | 面积矢量 | miàn jī shǐ liàng |
| normal/ˈnɔːml/ | 法线 | fǎ xiàn |
How many field lines pierce this net?
- Hold a net in a stream: the flow through it depends on its size and tilt.
- Do the same with an electric field and a surface.
- Electric flux 电通量 counts how much field pierces an area.
- It is the idea that makes Gauss's law possible in the next lesson.
有多少电场线穿过这张网?
- 把网放进水流里:穿过它的流量取决于网的大小和倾斜。
- 对电场和一个面做同样的事。
- 电通量计量有多少场穿过一块面积。
- 它就是让下一课高斯定律成为可能的想法。
Flux is a dot product
- Give the area an area vector 面积矢量 $\vec A$: length $=$ the area, direction $=$ the normal 法线.
- Flux is $\Phi = \vec E \cdot \vec A = EA\cos\theta$.
- $\theta$ is the angle between the field and the area's normal.
- The dot product automatically counts only the field that goes through.
通量是一个点积
- 给面积一个面积矢量 $\vec A$:长度 $=$ 面积,方向 $=$ 法线。
- 通量是 $\Phi = \vec E \cdot \vec A = EA\cos\theta$。
- $\theta$ 是场与面积法线之间的夹角。
- 点积自动只计量穿过的那部分场。
Electric flux through a flat area is: · 穿过平面的电通量为：
$\Phi = \vec E \cdot \vec A = EA\cos\theta$, with $\theta$ from the normal. · $\Phi = \vec E \cdot \vec A = EA\cos\theta$，其中$\theta$来自法向量。
The area vector points along the ____ to the surface. · 面积矢量指向表面的____方向。
The area vector is perpendicular to the surface — its normal. · 面积矢量垂直于表面——即其法向量。
The tilt decides everything
- Face-on ($\theta = 0$): all the field pierces it, $\Phi = EA$ (maximum).
- Edge-on ($\theta = 90^\circ$): the field skims past, $\Phi = 0$.
- In between, only the $\cos\theta$ part counts.
- Tilting a surface changes its flux without changing the field.
倾斜决定一切
- 正对($\theta = 0$):所有场都穿过它,$\Phi = EA$(最大)。
- 侧对($\theta = 90^\circ$):场擦过而过,$\Phi = 0$。
- 介于两者之间时,只有 $\cos\theta$ 那部分计入。
- 倾斜一个面在不改变场的情况下改变它的通量。
A field $E = 100\ \text{N/C}$ passes straight through ($\theta = 0$) an area of $2\ \text{m}^2$. Find the flux (N·m²/C). · 场$E = 100\ \text{N/C}$笔直穿过（$\theta = 0$）面积为$2\ \text{m}^2$的区域。求磁通量（N·m²/C）。
$\Phi = EA\cos 0 = 100 \times 2 \times 1 = 200$.
A field parallel to a surface gives zero flux through it. · 平行于表面的电场在该表面上的通量为零。
Parallel means $\theta = 90^\circ$, so $\cos\theta = 0$ and $\Phi = 0$. · 平行意味着$\theta = 90^\circ$，因此$\cos\theta = 0$和$\Phi = 0$。
Curved or changing surfaces need an integral
- If $E$ varies or the surface curves, split it into patches $d\vec A$.
- Each patch has flux $d\Phi = \vec E \cdot d\vec A$.
- Add them all: $\Phi = \displaystyle\int \vec E \cdot d\vec A$.
- Over a closed surface we write $\oint \vec E \cdot d\vec A$.
弯曲或变化的面需要积分
- 若 $E$ 变化或面弯曲,把它切成小片 $d\vec A$。
- 每片的通量为 $d\Phi = \vec E \cdot d\vec A$。
- 把它们全加起来:$\Phi = \displaystyle\int \vec E \cdot d\vec A$。
- 对闭合面我们写作 $\oint \vec E \cdot d\vec A$。
When is the flux greatest? · 何时通量最大？
Electric flux measures how many field lines cross a surface. Sort each case. · 电通量衡量穿过表面的电场线数量。对每种情况排序。
Select all · 所有 true statements about electric flux. · 选择关于电通量的所有正确陈述。
Flux is a dot product, peaks face-on, and integrates over curves. The angle always matters. · 通量是点积，正对时峰值，并在曲面上积分。角度始终重要。
Sign and units
- Flux out of a closed surface is positive; flux in is negative.
- Units are N·m²/C.
- A closed surface with no charge inside has zero net flux — as much goes in as out.
- That last fact is the seed of Gauss's law.
符号与单位
- 穿出闭合面的通量为正;穿入的为负。
- 单位是 N·m²/C。
- 内部没有电荷的闭合面净通量为零——进多少就出多少。
- 最后这个事实就是高斯定律的种子。
A closed surface has no charge inside. Its net flux is: · 封闭表面内部无电荷。其净通量为：
With no enclosed charge, as much flux enters as leaves — net zero. · 无包围电荷时，进入的通量等于离开的通量——净通量为零。
A field $E = 200\ \text{N/C}$ hits a $0.5\ \text{m}^2$ area tilted at $\theta = 60^\circ$.
- $\Phi = EA\cos\theta = 200 \times 0.5 \times \cos 60^\circ$.
- $\Phi = 200 \times 0.5 \times 0.5 = 50\ \text{N}\cdot\text{m}^2/\text{C}$.
一个场 $E = 200\ \text{N/C}$ 打到一块 $0.5\ \text{m}^2$、倾角 $\theta = 60^\circ$ 的面上。
- $\Phi = EA\cos\theta = 200 \times 0.5 \times \cos 60^\circ$。
- $\Phi = 200 \times 0.5 \times 0.5 = 50\ \text{N}\cdot\text{m}^2/\text{C}$。
The angle $\theta$ is measured from the normal, not from the surface itself. A field parallel to a surface has $\theta = 90^\circ$ and gives zero flux — a common sign-and-angle trap.
角 $\theta$ 是从法线量起的,不是从面本身。平行于面的场有 $\theta = 90^\circ$,给出零通量——一个常见的符号与角度陷阱。
Electric flux is the field piercing an area: $\Phi = \vec E \cdot \vec A = EA\cos\theta$, with $\theta$ from the area vector (normal). Face-on gives maximum flux, edge-on gives zero. For curved surfaces, $\Phi = \int \vec E \cdot d\vec A$.
电通量是穿过面积的场:$\Phi = \vec E \cdot \vec A = EA\cos\theta$,$\theta$ 从面积矢量(法线)量起。正对给出最大通量,侧对给出零。对弯曲面,$\Phi = \int \vec E \cdot d\vec A$。