KAHIBARO
Table of Contents
Seeing field through a surface
Electric flux is a way to describe how much electric field passes through a surface. It does not mean that something material is literally flowing like water. Instead, it is a mathematical measure of how strongly the electric field crosses an area.
If the electric field is strong and points straight through a surface, the flux is large. If the field lies along the surface instead of passing through it, the flux is zero. If the field crosses the surface in the opposite direction, the flux is negative.
Electric flux is important because it connects the geometry of a surface to the electric field around charges. This idea becomes especially powerful when studying Gauss's law, but here we focus only on what flux itself means.
Flux for a flat surface in a uniform field
Imagine a flat surface of area $A$ placed in a uniform electric field $\vec{E}$. The amount of flux depends on the angle between the field and the surface's perpendicular direction, called the area direction or surface normal.
The electric flux through the surface is
$$
\Phi_E = EA \cos\theta
$$
where $\theta$ is the angle between $\vec{E}$ and the outward normal to the surface.
If $\theta = 0^\circ$, the field goes straight through the surface and the flux is maximum:
$$
\Phi_E = EA
$$
If $\theta = 90^\circ$, the field is parallel to the surface and no field passes through it:
$$
\Phi_E = 0
$$
If $\theta = 180^\circ$, the field goes through in the opposite direction:
$$
\Phi_E = -EA
$$
Important formula for a flat surface in a uniform electric field:
$$
\Phi_E = EA\cos\theta
$$
Here, $\theta$ is measured between the electric field and the surface normal, not between the field and the surface itself.
The meaning of the surface normal
Every surface has two possible perpendicular directions. For open surfaces, one usually chooses one of them and keeps that choice consistently. For closed surfaces, the standard choice is the outward normal, pointing away from the enclosed region.
This chosen direction determines the sign of the flux. Field lines leaving a surface count as positive flux if the outward normal is used. Field lines entering count as negative flux.
Why the cosine appears
The cosine appears because only the part of the electric field perpendicular to the surface contributes to flux. If the field is tilted, we use only its perpendicular component:
$$
E_\perp = E\cos\theta
$$
Then the flux becomes
$$
\Phi_E = E_\perp A
$$
You can also think of it another way. Instead of changing the field, imagine shrinking the surface to the part that faces the field directly. That effective area is
$$
A_\perp = A\cos\theta
$$
So again,
$$
\Phi_E = EA_\perp
$$
Both viewpoints give the same result.
Vector form of electric flux
Using vectors, flux is written with the area vector $\vec{A}$. The area vector has magnitude equal to the area of the surface and direction along the chosen normal.
Then the flux is
$$
\Phi_E = \vec{E} \cdot \vec{A}
$$
This is a dot product, so it naturally gives
$$
\Phi_E = EA\cos\theta
$$
This vector form is compact and very useful.
Electric flux is a scalar quantity, even though it is built from vectors:
$$
\Phi_E = \vec{E}\cdot\vec{A}
$$
A dot product always gives a scalar.
Units of electric flux
The electric field has SI unit $\mathrm{N/C}$, and area has unit $\mathrm{m^2}$. Therefore the unit of electric flux is
$$
\mathrm{N \cdot m^2 / C}
$$
It may also be written in terms of volts as $\mathrm{V \cdot m}$, since electric field can also be expressed in $\mathrm{V/m}$.
| Quantity | Symbol | SI unit |
|---|---|---|
| Electric field | $E$ | $\mathrm{N/C}$ |
| Area | $A$ | $\mathrm{m^2}$ |
| Electric flux | $\Phi_E$ | $\mathrm{N \cdot m^2/C}$ |
Flux through curved surfaces
For a curved surface, the direction of the normal changes from point to point. Because of that, we cannot usually use one single angle for the whole surface. Instead, we divide the surface into very small pieces, each with area $dA$ and area vector $d\vec{A}$.
The small contribution to flux is
$$
d\Phi_E = \vec{E}\cdot d\vec{A}
$$
To find the total flux through the surface, we add all these small contributions:
$$
\Phi_E = \int \vec{E}\cdot d\vec{A}
$$
This is the general definition of electric flux.
General definition of electric flux through any surface:
$$
\Phi_E = \int \vec{E}\cdot d\vec{A}
$$
Use this when the surface is curved, or when the field changes from place to place.
Open surfaces and closed surfaces
An open surface is something like a sheet, a disk, or one side of a plane. A closed surface completely surrounds a volume, like a sphere, cube, or cylinder with end caps.
For an open surface, you choose one normal direction. For a closed surface, the normal is always taken outward.
The flux through a closed surface is often written with a circle on the integral sign:
$$
\Phi_E = \oint \vec{E}\cdot d\vec{A}
$$
This notation reminds us that the surface is closed.
Positive, negative, and zero flux
The sign of flux tells us the direction of crossing relative to the chosen normal.
| Situation | Flux sign |
|---|---|
| Field points along the chosen normal | Positive |
| Field points opposite the chosen normal | Negative |
| Field is parallel to the surface | Zero |
For a closed surface, positive flux means net field lines leaving the surface. Negative flux means net field lines entering the surface.
It is possible for some parts of a closed surface to have positive flux and other parts to have negative flux. The total flux is the sum of all parts.
A simple example
Suppose a uniform electric field of magnitude $200\ \mathrm{N/C}$ passes through a flat surface of area $0.50\ \mathrm{m^2}$. The angle between the field and the surface normal is $60^\circ$.
Then
$$
\Phi_E = EA\cos\theta
$$
so
$$
\Phi_E = (200)(0.50)\cos 60^\circ
$$
Since $\cos 60^\circ = 0.5$,
$$
\Phi_E = 50\ \mathrm{N \cdot m^2/C}
$$
The flux is positive if the field points in the same direction as the chosen normal.
If the same field were parallel to the surface, then $\theta = 90^\circ$ and the flux would be zero.
Geometric intuition
A larger area gives more flux because more field passes through the surface. A stronger field gives more flux because the crossing is more intense. A tilted surface gives less flux because less of the field goes straight through it.
This is why flux depends on three ideas at once, field strength, surface size, and orientation.
You can think of field lines as a picture only. The flux tells how many lines would cross the surface if the density of lines represented field strength. This is a visual aid, not a literal physical flow.
What flux does and does not mean
Electric flux measures the crossing of electric field through a surface. It is not the same as electric field itself. A surface can have zero net flux even if the electric field is not zero everywhere on it. For example, equal amounts of field may enter and leave different parts of a closed surface.
Also, flux is not a property of one point. It is a property of a field together with a chosen surface.
Do not confuse electric field and electric flux.
Electric field, $\vec{E}$, is defined at each point in space.
Electric flux, $\Phi_E$, is defined through a surface.
Preparing for the next idea
Electric flux becomes especially important when we ask how the total flux through a closed surface is related to the charge inside that surface. That relationship is the content of Gauss's law. Before using that law, it is essential to understand flux clearly as the surface measure
$$
\Phi_E = \int \vec{E}\cdot d\vec{A}
$$
and, for a simple flat surface in a uniform field,
$$
\Phi_E = EA\cos\theta
$$