Formula of Electric Flux: Φ = EA cos θ and Gauss's Law

The formula of electric flux through a flat surface in a uniform field is $\Phi_E = EA\cos\theta$ , with θ measured from the outward normal of the surface to the field. Think of it as a count of how many field lines pierce the surface. It is measured in newton square metre per coulomb (N m²/C), which is the same as volt metre (V m).

Flux becomes powerful when the surface is closed, like a balloon around some charges. Gauss's law then says the total outward flux depends only on the charge inside: $\Phi_E = \frac{q_{in}}{\varepsilon_0}$ . The shape and size of the closed surface do not matter. This page covers both forms. For the field itself, see the electric field formula, and for the rest of the chapter the electricity formulas hub.

Quick facts

Formula
$\Phi_E = EA\cos\theta = \vec{E} \cdot \vec{A}$
Closed surface
$\oint \vec{E} \cdot d\vec{A} = \frac{q_{in}}{\varepsilon_0}$
SI unit
N m²/C, equivalent to V m
Dimensional formula
[M¹ L³ T⁻³ I⁻¹]
Type
Scalar
Sign rule
Positive for field lines leaving a closed surface, negative for lines entering

Symbols in the formula of electric flux

Quantities used for Φ_E
Symbol	Meaning	SI unit
$\Phi_E$	Electric flux through the surface	N m²/C
$E$	Electric field strength at the surface	N/C or V/m
$A$	Area of the surface	m²
$\theta$	Angle between $\vec{E}$ and the normal	degree
$\varepsilon_0$	Electric constant; in SI it equals 8.854 × 10⁻¹² C²/(N m²)	C²/(N m²)

A surface facing the field head on (θ = 0°) collects the most flux, EA. Tilt it and the flux shrinks with cos θ. Turn it edge on (θ = 90°) and the flux is zero, since the lines slide past without crossing. If the field changes over the surface, break it into small patches and add $\vec{E} \cdot d\vec{A}$ for each.

Two ways to find flux

Open flat surface, uniform field: use $EA\cos\theta$ . Convert the area to m² and measure θ from the normal.
Closed surface of any shape: skip the geometry and use Gauss's law. Add up the charge inside and divide by ε₀. Charges outside give zero net flux.
Part of a symmetric closed surface: find the total with Gauss's law, then share it out by symmetry, for example one sixth per face for a charge at the center of a cube.

The calculator above handles the first case. Give the angle in degrees from the normal.

Electric flux problems solved

Example 1: a tilted sheet

A flat sheet of area 1500 cm² is placed in a uniform field of 500 N/C. The field makes 30° with the normal. Find the flux.

Convert: A = 1500 × 10⁻⁴ = 0.15 m².

$\Phi_E = (500)(0.15)\cos 30^\circ = 75 \times 0.866 = 65\ \text{N m}^2\text{/C}$

Example 2: Gauss's law

A closed surface surrounds a 3.0 μC charge. How much flux leaves it in total?

$\Phi_E = \frac{q}{\varepsilon_0} = \frac{3.0 \times 10^{-6}}{8.854 \times 10^{-12}} = 3.4 \times 10^{5}\ \text{N m}^2\text{/C}$

The answer is the same for a small sphere, a large cube or an odd shaped bag, as long as the charge is inside.

Example 3: one face of a cube (JEE style)

The same 3.0 μC charge sits at the center of a cube. Find the flux through one face.

By symmetry each of the 6 faces gets an equal share:

$\Phi_{face} = \frac{3.39 \times 10^{5}}{6} = 5.6 \times 10^{4}\ \text{N m}^2\text{/C}$

Unit and dimensional formula

Field in N/C times area in m² gives N m²/C. Because 1 N/C = 1 V/m, the same unit can be written V m. The V m form gives the quickest route to dimensions, since a volt is [M L² T⁻³ I⁻¹]:

$[\Phi_E] = [\text{volt}] \times [\text{length}] = [M L^3 T^{-3} I^{-1}]$

Each step, plus other quantities that match it, appears on the dimensional formula of electric flux page.

Electric flux next to magnetic flux

Comparing the two fluxes
Feature	Electric flux	Magnetic flux
Open surface	$EA\cos\theta$	$BA\cos\theta$
Unit	N m²/C	weber (Wb)
Closed surface	$q_{in}/\varepsilon_0$	always zero
Why	Field lines start and end on charges	Field lines form closed loops

A changing electric flux also produces a magnetic field, which is the idea behind the displacement current formula. The magnetic side is on the magnetic flux formula page.

Slips to watch for

Angle from the surface. When a problem quotes the tilt relative to the sheet itself, subtract that from 90°.
Counting outside charges. They change the field on the surface but not the net flux through a closed one.
Sign of the charge. A negative charge inside gives negative (inward) flux.
Units of area. 1500 cm² is 0.15 m², not 15 m².

A dipole fully inside a closed surface gives zero net flux, since its two charges add to zero.

Frequently asked questions

What is the formula of electric flux?

Through a flat surface in a uniform field, Φ_E = EA cos θ. Through a closed surface, Gauss's law gives Φ_E = q_in / ε₀.

Which unit is electric flux measured in?

Newton square metre per coulomb, N m²/C, which you may also see as V m. In base quantities this becomes [M¹ L³ T⁻³ I⁻¹].

Is electric flux a scalar or vector?

Scalar. Combining E and A through a dot product leaves a plain number, positive for outward flux and negative for inward flux.

Does electric flux depend on the size of the closed surface?

No. Gauss's law says the net flux through any closed surface depends only on the net charge it encloses, not on its size or shape.

What is the flux through a cube with a charge at its center?

The total is q/ε₀, and each of the six faces receives one sixth, q/(6ε₀).