Electric Flux
Introduced| English |
|---|
| electric flux/ɪˈlektrɪk flʌks/ |
| area vector/ˈeərɪə ˈvektə/ |
| normal/ˈnɔːml/ |
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.
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.
Electric flux through a flat area is:
$\Phi = \vec E \cdot \vec A = EA\cos\theta$, with $\theta$ from the normal.
The area vector points along the ____ to the surface.
The area vector is perpendicular to the surface — its normal.
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.
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).
$\Phi = EA\cos 0 = 100 \times 2 \times 1 = 200$.
A field parallel to a surface gives zero flux through it.
Parallel means $\theta = 90^\circ$, so $\cos\theta = 0$ and $\Phi = 0$.
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 $\oint \vec E \cdot d\vec A$.
When is the flux greatest?
Electric flux measures how many field lines cross a surface. Sort each case.
Select all true statements about electric flux.
Flux is a dot product, peaks face-on, and integrates over curves. The angle always matters.
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.
A closed surface has no charge inside. Its net flux is:
With no enclosed charge, as much flux enters as leaves — net zero.
A field $E = 200\ \text{N/C}$ hits a $0.5\ \text{m}^2$ area tilted at $\theta = 60^\circ$.
- $\Phi = EA\cos\theta = 200 \times 0.5 \times \cos 60^\circ$.
- $\Phi = 200 \times 0.5 \times 0.5 = 50\ \text{N}\cdot\text{m}^2/\text{C}$.
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.
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, $\Phi = \int \vec E \cdot d\vec A$.