Electric Flux

Electric flux is simply a measure of how much electric field passes through a surface, the way you might measure how much wind blows through a hoop. The word flux is Latin for "flow." Nothing actually flows in an electric field, but field lines threading a surface behave a lot like a flow, so the name stuck. It's good for counting: the next lesson shows that the flux out of a closed surface tells you exactly how much charge is inside. The formula $\Phi_E = EA\cos\theta$ looks technical, but it's three pieces: $E$ (how strong the field is), $A$ (how big the surface is) and $\cos\theta$ (how squarely the field hits it, 1 head-on and 0 edge-on). It's just "strength times area, adjusted for tilt"; bear with it.

Picture it. Hold that hoop in a steady wind. Face it into the wind and the most air streams through; tilt it and less gets through; turn it edge-on and nothing does. Electric flux is the same bookkeeping for the electric field: it counts how much $\mathbf{E}$ passes through a surface.

Uniform field, flat surface. Give the surface an area vector $\mathbf{A}$, whose length is the area $A$ and which points along the surface's normal. Then $\Phi_E = \mathbf{E}\cdot\mathbf{A} = EA\cos\theta$, in N·m²/C. There are two ways to see the cosine, and they agree:

• only the field's normal component, $E\cos\theta$, goes through; the part along the surface just skims it;
• or keep all of $E$ but use the surface's shadow on a plane perpendicular to the field, of area $A\cos\theta$: every field line through the surface also crosses its shadow.

Flux is a signed number. Past $\theta = 90^\circ$ the field crosses backward and the flux is negative, so you always have to say which way the normal points. For a closed surface (a sphere, a box, a can) the rule is outward: flux leaving counts as positive and flux entering as negative.

Curved surfaces, changing fields. Chop the surface into small patches $\Delta\mathbf{A}_i = \hat{\mathbf{n}}_i\,\Delta A_i$, each small enough to be nearly flat with $\mathbf{E}$ nearly constant on it, and add: $\Phi_E \approx \sum\mathbf{E}_i\cdot\Delta\mathbf{A}_i$. In the limit that's the flux integral $\iint_S\mathbf{E}\cdot d\mathbf{A}$ from Flux Integrals: parametrize the surface, use $\mathbf{r}_u\times\mathbf{r}_v\,du\,dv$ for $d\mathbf{A}$, and integrate.

The calculation that matters most. A point charge $q$ sits at the center of a sphere of radius $R$. Its field, from The Electric Field, points straight out along $\hat{\mathbf{n}}$ with the same strength $kq/R^2$ everywhere on the sphere, so

Notice that the radius canceled. Think about why: a bigger sphere has more area, $4\pi R^2$, but the field on it is weaker by exactly $1/R^2$. In the field-line picture, every line leaving $q$ crosses every sphere around it once, so all those spheres catch the same flux. That's the inverse-square law in disguise, and the next lesson turns it into Gauss's Law.

A common mistake. It's tempting to use the angle between $\mathbf{E}$ and the surface, because that's the angle you can see in a drawing. It's the wrong one: $\theta$ is measured from the normal, and the surface angle gives you $\sin\theta$ where you need $\cos\theta$. Two smaller slips to watch for: flux entering a closed surface counts as negative, and flux is a plain number, not a vector.

Flux is how engineers count what passes through an opening. The power a solar panel collects scales with the flux of sunlight through it, $\cos\theta$ and all, which is why panels track the sun. Capacitive sensors read the flux of a field ending on an electrode, and the charge it induces, $\varepsilon_0$ times that flux. And every Gauss's-law calculation in high-voltage design (cables, bushings, insulators) starts by adding up $\mathbf{E}\cdot d\mathbf{A}$ over a carefully chosen surface.