EM-05 · Electrostatics field

Flux and Divergence

Flux is the quantity that says how much electric field passes through a surface, and it counts only the component perpendicular to the surface. You tilt the surface, watch the flux change as cosθ, and understand how the net flux through a closed surface leads to divergence.

Tilt the loop and see the flux

The field lines of a uniform electric field run side by side in one direction, and a loop sits among them. Tilting the loop with the slider changes the angle between the gold normal and the field, and the cyan ellipse, the loop's shadow cast along the field, shrinks with it. When the loop faces the field head-on, the most passes through; turned edge-on, the flux drops to 0.

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

Drag to rotate; the slider changes the tilt θ of the surface.

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 counts how much electric field passes through a surface. If you liken the field to flowing water, it corresponds to the amount of water that passes through a net. In a uniform field, setting the surface to face the field head-on lets the most through.

Only the perpendicular part counts

Tilting the surface reduces what passes through by cosθ, where θ is the angle between the surface normal and the field. When the surface lies parallel to the field (θ=90°), the field only grazes it and cannot pass through, so the flux is 0. In the end, flux is the dot product of the field vector and the area vector, written Φ = E·A = EA cosθ. The area vector A has the area as its magnitude and the normal as its direction, and the area of the cyan ellipse in the figure is exactly the effective area seen from the field.

ObserveΦ = EA cosθ

The more the surface tilts, the more the flux shrinks, following cosθ.

ChooseΦ = ? = EA cosθ

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

Curved surfaces, changing fields

When the surface is curved or the field varies from place to place, you cut the surface into small pieces dA, find the flux E·dA of each piece, and add them. That is the surface integral Φ = ∫E·dA. Just as in EM-04 you integrated a point-charge field along a rod, here you integrate the field over a surface.

Fill inΦ = ?

When the surface curves or the field varies, add it up with an integral.

Closed surfaces and divergence

When a surface wraps all the way around a region and closes, the outgoing flux minus the incoming flux is the net flux, written ∮E·dA. If the net flux is positive, inside the surface is a source that sends out field, that is, a net positive charge; if negative, a net negative charge that draws field in. If there is no net flux, the charges inside cancel or there are none. Divide this net flux by the enclosed volume and shrink the volume without limit, and the limit is the divergence, the amount of field springing out per unit volume at a point. Gauss's law in the next unit links exactly this net flux to the charge trapped inside.

On your ownΦ_net = ?

The net flux through a closed surface reveals the divergence.

Back to the first screen

On the first screen, the flux was largest with the loop facing the field head-on and became 0 when you turned it edge-on. What set the amount passing through was the cosine of the angle between the surface normal and the field, that is, the dot product, and the area of the cyan shadow ellipse matched that amount. Only as much field passed through as the surface casts in the field's direction. On a curved surface or in a varying field, this dot product is integrated over the surface, and on a closed surface the net flux reveals the charge inside.

Flux Φ says how much electric field passes through a surface. It counts only the component perpendicular to the surface, so Φ = E·A = EA cosθ, and on a curved surface or in a varying field Φ = ∫E·dA. The net flux ∮E·dA through a closed surface reveals the charge inside it, and that amount expressed per unit volume is the divergence.

What comes next

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