Electric Flux: Definition, Surface Integrals, Units, and Relation to Gauss's Law

Overview of electric flux: definition and computation via surface integrals and dot products, SI units and equivalents, and how flux through closed surfaces relates to enclosed charge via Gauss's law.

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: April 9, 2026

An electric field passing through a surface produces a quantity called electric flux, which measures how much of the field penetrates that surface. To describe this precisely, take a very small, infinitesimal patch of area and assume the field is essentially uniform over it. If the patch has a vector area dA and the field is the vector E, the amount of field crossing the patch is given by the dot product of those two vectors; see the entry on the dot product. A symbolic differential for the flux can be written as $d\Phi _{E}\,$ and the oriented patch is represented by $d\mathbf {A}$ .

Combining these gives the local expression for flux through the small patch: $d\Phi _{E}=\mathbf {E} \cdot d\mathbf {A}$

Flux over a surface

To find the total flux through a finite surface S, add (integrate) the contributions from each infinitesimal patch. This is the surface integral form: $\Phi _{E}=\int _{S}\mathbf {E} \cdot d\mathbf {A}$

Here S $S$ denotes the surface and dA is taken with the outward-pointing normal when a direction must be chosen.

Closed surfaces and Gauss's law

When the surface is closed (often called a Gaussian surface), the integral is written with a closed-contour symbol and relates directly to the net electric charge enclosed. In compact form: $\Phi _{E}=\oint _{S}\mathbf {E} \cdot d\mathbf {A} ={\frac {Q_{S}}{\epsilon _{0}}}$

In this relation Q_S is the total charge contained inside the surface, including any bound or free contribution, and ε₀ is the electric constant. This statement is the integral version of Gauss' law and appears as one of Maxwell's equations; in textbooks it is often described as the integral form of the law.

Charges located outside a closed surface do not change the net flux through that surface, though they can alter the field distribution on the surface itself. Gauss's law always holds, but solving for E from the law by hand is practical only when the field has a high degree of symmetry (for example, spherical or cylindrical situations). In more general geometries numerical methods and a computer are typically required.

Units

Electric flux is expressed in SI units as volt metres (V·m). Equivalent derived units are newton metres squared per coulomb (N·m²·C⁻¹). In base SI units this becomes kg·m³·s⁻³·A⁻¹.

Questions and Answers

Q: What is electric flux?

A: Electric flux is the dot product of an electric field, E, and a differential area on a surface, dA.

Q: How is electric flux calculated?

A: Electric flux can be calculated using the equation EdAcos(i), where E is the electric field and dA is an infinitesimal area on the surface across which E remains constant. The angle between E and dA is i.

Q: What does Gauss' Law for electric fields state?

A: Gauss' Law for electric fields states that for a closed Gaussian surface, the electric flux through it will equal to the net charge enclosed by it divided by the electrical constant (ε0). This relation holds true in all situations but can only be used to calculate when high degrees of symmetry exist in the electric field.

Q: What are some examples of symmetrical situations where Gauss' Law can be used to calculate?

A: Examples include spherical and cylindrical symmetry.

Q: What are SI units of electrical flux?

A: Electrical flux has SI units of volt metres (V m), or newton metres squared per coulomb (N m2 C−1). The SI base units of electrical flux are kg•m3•s−3•A−1.

Q: Does electrical flux depend on charges outside a closed surface?

A: No, electrical flux is not affected by charges that lie outside a closed surface; however, they may affect the net electric field within it.

Author

Creation date: November 13, 2022

Last updated: April 9, 2026