EM-05 · Electrostatics field

Flux and Divergence

How much of the electric field passes through a surface (the flux) counts only the perpendicular component. Tilt the surface to watch the flux change as cosθ, and see how the net flux through a closed surface leads to divergence.

Tilt the loop to watch the flux

In a uniform electric field, tilt the loop. It catches the most when facing the field head-on and nothing when turned edge-on. What sets how much passes through?

Surface tilt θ (angle between normal and field)θ = 35°

Drag to orbit. The slider tilts the surface θ.

The flux Φ through the surface

Φ = E·A = EA cosθ · counts only the component perpendicular to the surface

∮ E·dA · net flux through a closed surface reveals the divergence

Flux ratio · 82%

Slanted

Flux, the field through a surface

Flux measures how much of the electric field passes through a given surface. If the field were a flow of water, flux would be the amount passing through a net. In a uniform field, a surface placed face-on to the field catches the most.

Only the perpendicular part counts

Tilt the surface and the throughput drops as cosθ, where θ is the angle between the surface normal and the field. When the surface lies along the field (θ=90°), the field only grazes it and nothing passes, so the flux is zero. That is why flux is the dot product of the field vector and the area vector: Φ = E·A = EA cosθ. The area vector A has magnitude equal to the area and direction along the normal.

ObserveΦ = EA cosθ

The more the surface tilts, the flux drops as cosθ.

ChooseΦ = ? = EA cosθ

Flux is the dot product of the field and the area vector.

Curved surfaces, changing fields

When the surface curves or the field varies from place to place, chop the surface into tiny pieces dA and add each piece’s flux E·dA. That is the surface integral Φ = ∫E·dA. Just as EM-04 integrated the field of point charges, here we integrate the field over a surface.

Fill inΦ = ?

For a curved surface or varying field, add by integrating.

Closed surfaces and divergence

When the surface closes around a region, the difference between flux in and flux out is the net flux, written ∮E·dA. A positive net flux means a source inside (positive charge) pumping field out; a negative one means a sink (negative charge). Dividing this net flux by the volume and shrinking it to a point gives the divergence — the field welling up per unit volume. The Gauss law in the next unit ties exactly this net flux to the enclosed charge.

On your ownΦ_net = ?

The net flux through a closed surface reveals the divergence.

Back to the first screen

On the first screen, flux was largest with the loop face-on and fell to zero edge-on. What set the throughput was the cosine of the angle between the normal and the field — the dot product. Only as much field passes as the projection of the area vector A onto E allows. For curved surfaces or varying fields you integrate this dot product over the surface, and for a closed surface the net flux reveals the charge inside.

The flux Φ is how much of the electric field passes through a surface. It counts only the component perpendicular to the surface, so Φ = E·A = EA cosθ, and for curved surfaces or varying fields Φ = ∫E·dA. The net flux ∮E·dA through a closed surface reveals the charge enclosed, and dividing it by unit volume gives the divergence.

What comes next

The net flux through a closed surface is proportional to the charge inside. The Gauss law in the next unit (EM-06) pins this down as Φ = Q/ε₀, and the divergence theorem (EM-07) links the surface integral to the volume integral — the integral and differential forms. With good symmetry, this law gives the field without integrating.