Dimensional Formula of Electric Flux: [M L³ T⁻³ I⁻¹]

The dimensional formula of electric flux is [M¹ L³ T⁻³ I⁻¹]. The power of length is three, higher than for almost any other quantity in Class 12 electrostatics, and that single detail is what most questions test.

Electric flux counts the electric field passing through a surface, Φ = EA cos θ. The field contributes [MLT⁻³I⁻¹] and the area adds two more powers of length. Gauss's law gives an independent check, because the net flux leaving any closed surface is q/ε₀, with q the charge trapped inside. For flux through planes, cubes and spheres, see the electric flux formula guide.

Quick facts

Dimensional formula
[M¹ L³ T⁻³ I⁻¹]
Defining formula
$\Phi_E = EA\cos\theta$
Gauss's law
$\Phi_E = \frac{q_{enc}}{\varepsilon_0}$
SI unit
N m² C⁻¹, equal to V m
Base units
kg m³ s⁻³ A⁻¹
Type
Scalar

Two ways to the dimensional formula of electric flux

Route 1: field times area.

$\Phi_E = EA\cos\theta$ , and cos θ has no dimension.
Field is force per charge: $[E] = \frac{[MLT^{-2}]}{[IT]} = [MLT^{-3}I^{-1}]$ .
Area: [L²]. So $[\Phi_E] = [MLT^{-3}I^{-1}][L^{2}] = [ML^{3}T^{-3}I^{-1}]$ .

Route 2: Gauss's law. Flux through a closed surface is $\frac{q}{\varepsilon_0}$ . Charge carries [IT] and permittivity carries [M⁻¹L⁻³T⁴I²]. Dividing flips every power of ε₀: $[IT] \times [ML^{3}T^{-4}I^{-2}]$ , which collects to [ML³T⁻³I⁻¹]. The routes agree. The first one leans on the electric field bracket; the second on the bracket of ε₀.

N m²/C, V m and the base units

From Φ = EA, the unit is (N/C)(m²), written N m² C⁻¹. Since a newton per coulomb equals a volt per metre, the same unit is (V/m)(m²), which tidies to V m. In base units both become kg m³ s⁻³ A⁻¹.

Example: a steady field of 4.0 × 10³ N/C crosses a flat card of area 25 cm² at right angles. Find the flux.

Area: 25 cm² = 25 × 10⁻⁴ m² = 2.5 × 10⁻³ m². The field is along the normal, so θ = 0 and cos θ = 1.

$\Phi_E = (4.0 \times 10^{3})(2.5 \times 10^{-3}) = 10\ \text{N m}^{2}\text{/C}$ .

Squaring 1 cm = 10⁻² m gives 10⁻⁴ m² for each square centimetre; using 10⁻² for an area is the classic slip.

Gauss's law as a dimension and number check

Example: a charge of 8.854 nC sits inside a closed box. How much flux leaves the box in total?

$\Phi_E = \frac{q}{\varepsilon_0} = \frac{8.854 \times 10^{-9}\ \text{C}}{8.854 \times 10^{-12}\ \text{C}^{2}\text{N}^{-1}\text{m}^{-2}} = 1.0 \times 10^{3}\ \text{N m}^{2}\text{/C}$ .

Unit check: C divided by C² N⁻¹ m⁻² leaves N m² C⁻¹, the flux unit. The size and shape of the box play no part.

Notice how the unit algebra mirrors the bracket algebra exactly. That is a quick way to catch an upside down ε₀ in an exam.

Electric flux beside nearby electrical brackets

How flux compares with field and potential
Quantity	Bracket	Link to flux
Electric field E	[M L T⁻³ I⁻¹]	Flux per area
Potential V	[M L² T⁻³ I⁻¹]	Flux per length
Electric flux Φₑ	[M L³ T⁻³ I⁻¹]	Field times area
Magnetic flux Φ_B	[M L² T⁻² I⁻¹]	Unrelated in dimensions

The first three form a ladder that climbs one power of L at each step, which is a neat memory aid. Magnetic flux, despite the similar name, has a different power of time and of length. The term ε₀ dΦₑ/dt in the displacement current has the bracket of current, a good test of the whole set; see displacement current.

Slips that cost marks on flux dimensions

Writing L². That is the potential bracket. Flux has L³.
Using the magnetic flux bracket. Weber and N m²/C are different units.
Multiplying by ε₀ instead of dividing. Gauss's law has ε₀ below the line.
Converting cm² with 10⁻². Area conversion needs 10⁻⁴.

To see the field, potential and flux brackets in a single table, open the dimensional formulas collection and search under electrostatics.

Frequently asked questions

Can you state the dimensional formula of electric flux in one line?

Yes: [M¹ L³ T⁻³ I⁻¹]. Take the electric field bracket [MLT⁻³I⁻¹] by area [L²], or divide charge [IT] by ε₀.

In which SI unit is electric flux measured?

Newton metre squared per coulomb, N m²/C. The volt metre, V m, is the same unit written another way.

Is electric flux a vector quantity?

No. It is a scalar, formed by a dot product of the field and the area vector. It can be positive, outward, or negative, inward.

Do electric flux and magnetic flux share dimensions?

No. Electric flux is [ML³T⁻³I⁻¹] while magnetic flux is [ML²T⁻²I⁻¹]. A single L and a single T separate them.

Does the size of a closed surface change the flux from a charge inside it?

It does not. Gauss's law ties the total flux only to the enclosed charge and ε₀. A bigger surface has a weaker field spread over a larger area, and the two effects cancel.