Dimensional Formula of Magnetic Flux: [M L² T⁻² I⁻¹]

The dimensional formula of magnetic flux is [M¹ L² T⁻² I⁻¹], or [ML²T⁻²A⁻¹] if you prefer the ampere symbol. Flux describes the amount of field passing across a surface, so it is field multiplied by area.

Since the magnetic field B carries [MT⁻²I⁻¹] and area carries [L²], the flux simply picks up two powers of length. Its SI unit is the weber. For coils, angles and solved numbers, the magnetic flux formula guide goes further; here the focus is on dimensions and what they let you check.

Quick facts

Dimensional formula
[M¹ L² T⁻² I⁻¹]
Defining formula
$\Phi = BA\cos\theta$
SI unit
weber (Wb) = T m² = V s
In base units
kg m² s⁻² A⁻¹
Type
Scalar, can be positive or negative
Linked laws
Faraday's law, Lenz's law, Φ = LI

Working out the dimensional formula of magnetic flux

Flux through a flat area A in a uniform field: $\Phi = BA\cos\theta$ , with θ the angle that the surface normal makes with the field.
cos θ is a ratio, so it adds nothing.
Field: from $F = qvB$ , $[B] = [MT^{-2}I^{-1}]$ . Area: $[A] = [L^{2}]$ .
Multiply: $[\Phi] = [MT^{-2}I^{-1}][L^{2}] = [ML^{2}T^{-2}I^{-1}]$ .

If you are not sure where [MT⁻²I⁻¹] comes from, the dimensions of B are worked out step by step separately. Keep the angle in mind for numbers, but remember that it never changes the bracket.

Weber, tesla metre squared and volt second

Translated into base units, the bracket becomes kg m² s⁻² A⁻¹. Three friendlier forms are all equal to it:

T m², straight from Φ = BA.
V s, from Faraday's law, because an emf is flux per unit time.
H A, from Φ = LI for a coil with inductance L.

Example: a square loop of side 20 cm sits in a 0.50 T field. The normal to the loop makes 60° with the field. Find Φ.

Area: $A = (0.20\ \text{m})^{2} = 0.040\ \text{m}^{2}$ . Convert cm to m before squaring.

$\Phi = (0.50)(0.040)\cos 60^\circ = 0.50 \times 0.040 \times 0.5 = 0.010\ \text{Wb}$ .

Using the bracket to test Faraday's law

Example: show that $\varepsilon = -N\frac{\Delta\Phi}{\Delta t}$ gives a voltage, then find the emf when 200 turns see the flux above fall to zero in 0.010 s.

Dimensions: N is a count, so $\frac{[ML^{2}T^{-2}I^{-1}]}{[T]} = [ML^{2}T^{-3}I^{-1}]$ , exactly what potential difference carries. The law is consistent.

Numbers: $|\varepsilon| = 200 \times \frac{0.010\ \text{Wb}}{0.010\ \text{s}} = 200\ \text{V}$ .

The minus sign is Lenz's law and carries direction only. Dimensions ignore it, so quote the size and explain the direction in words.

The match with the voltage bracket is exactly why one weber per second is one volt.

Where the same dimensions appear

Quantities carrying [M L² T⁻² I⁻¹]
Quantity	Formula	Why it matches
Magnetic flux	$BA\cos\theta$	Definition
Flux linkage	$N\Phi$	Number of turns has no dimension
Inductance times current	$LI$	Defines self inductance
Emf times time	$\varepsilon\,\Delta t$	Rearranged Faraday's law

A related check: dividing flux by current gives [ML²T⁻²I⁻²], the bracket of self inductance. Dividing flux by area takes you back to B. Electric flux is a completely different quantity, with [ML³T⁻³I⁻¹], so never borrow one for the other.

Errors that lose marks on flux

Giving the bracket of B. Flux has two more powers of L than the field.
Including the angle as a dimension. cos θ is a pure number.
Writing I to the power plus 1. B already has I⁻¹ and area adds no current, so flux keeps I⁻¹.
Squaring centimetres wrongly. 20 cm is 0.20 m, so its square is 0.040 m², not 0.40 m².

All electromagnetic induction brackets are collected on the dimensional formulas hub, so flux, inductance and emf can be compared side by side.

Frequently asked questions

Which dimensional formula belongs to magnetic flux?

It is [M¹ L² T⁻² I⁻¹]. Multiply the magnetic field bracket [MT⁻²I⁻¹] by the area bracket [L²].

Why is one weber equal to one volt second?

Faraday's law says emf equals rate of change of flux. So flux equals emf multiplied by time, and the unit volt times second must be the weber. Either one expands to kg m² s⁻² A⁻¹.

Is magnetic flux a scalar or vector quantity?

Magnetic flux is a scalar. It comes from a dot product of the field vector and the area vector, so the result can be negative yet never points anywhere.

How do magnetic flux and flux density differ?

B tells you the flux crossing each square metre, [MT⁻²I⁻¹], measured in tesla. Φ is the total through the whole surface, [ML²T⁻²I⁻¹], measured in weber.

Does flux linkage NΦ have different dimensions from flux?

No. N counts turns of wire, a pure number, so flux linkage keeps the bracket of flux and is also measured in weber, often written weber turns.