AP® Physics C: Electricity and Magnetism · Unit 8 · Topic 8.5
Electric Flux Through a Surface
Electric flux measures how much electric field pokes through a surface, not just how strong the field is. It equals E times A times cos(theta), where theta is the angle between the field and the surface's normal vector.
Picture a wire hoop held in a steady wind. Face it straight into the wind and every gust blows through the opening. Tilt the hoop and some wind now slides along the plane of the hoop instead of passing through it. The hoop's area never changed, but the amount of wind crossing it did. Electric flux works the same way: it is not just the field strength times the area, it is how much of the field actually pierces the surface.
To capture this, every surface gets a normal vector, an arrow perpendicular to it with length equal to the surface's area. Flux is the dot product of the field and that area vector: Phi = E dot A = E A cos(theta), with theta measured between E and the normal. When the surface faces the field head-on, theta = 0, cos(theta) = 1, and flux is maximum. When the surface is edge-on to the field, theta = 90 degrees, cos(theta) = 0, and no field lines cross it at all, no matter how large the surface is. Turning the loop from face-on to edge-on smoothly dials the flux from E A down to zero, following that cosine curve.
This idea of counting only the perpendicular component is what makes flux additive and well behaved for curved or closed surfaces: you chop the surface into tiny flat patches, take E dot dA on each, and add them up. That sum is exactly the setup Gauss's law needs. Flux through a surface is the quantity Gauss's law will shortly relate to the charge enclosed, so getting comfortable with the cosine and the normal vector here is what makes the next topic's shortcut make sense.
Key terms
- Electric flux
- A measure of how many electric field lines pass through a given surface, Phi = E A cos(theta).
- Normal vector
- A vector perpendicular to a surface whose length represents the surface's area.
- Dot product
- A way to multiply two vectors that keeps only the component of one along the direction of the other.
- Theta
- The angle between the electric field vector and the surface's normal vector, not the surface itself.
Watch it explained
- How it worksHow does the dot product of E and dA count only the field poking through the surface?
- Worked exampleFind the flux through a 0.2 m by 0.2 m square tilted 60 degrees in a 500 N/C field.
- Common mistakeWhy isn't flux just the field strength multiplied by the area of the surface?
- Going furtherWhat happens to the flux through a loop as you slowly turn it from face-on to edge-on?
Check yourself
End of topic.