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

The dimensional formula of magnetic field is [M¹ L⁰ T⁻² I⁻¹], usually written [MT⁻²I⁻¹] or [MT⁻²A⁻¹]. Here the magnetic field means B, also called magnetic flux density or magnetic induction, measured in tesla.

The surprise for most students is that length cancels out completely. The force on a moving charge brings one power of L, and the velocity in the formula brings another power of L that divides it away. To see how B is calculated for wires, loops and solenoids, use the magnetic field formula guide, which has solved numbers for each case.

Quick facts

Dimensional formula
[M¹ L⁰ T⁻² I⁻¹]
Defining formula
$B = \frac{F}{qv\sin\theta}$
SI unit
tesla (T) = N A⁻¹ m⁻¹ = Wb/m²
CGS unit
gauss: 1 T = 10⁴ G
Type
Vector
Also called
Magnetic flux density, magnetic induction

From F = qvB to the dimensional formula of magnetic field

Use the magnetic part of the Lorentz force: a charge q travelling at speed v, at angle θ to the field lines, feels a sideways push.

Force on the charge: $F = qvB\sin\theta$ , so $B = \frac{F}{qv\sin\theta}$ .
sin θ is a ratio of lengths and has no dimensions.
Put in the brackets: $[F] = [MLT^{-2}]$ , $[q] = [IT]$ , $[v] = [LT^{-1}]$ .
Denominator: $[IT][LT^{-1}] = [IL]$ , because the time powers cancel.
Result: $[B] = \frac{[MLT^{-2}]}{[IL]} = [MT^{-2}I^{-1}]$ .

Notice the order of cancelling. Time cancels in the denominator first, then length cancels between top and bottom. If you get a leftover L, recheck the velocity bracket.

What the bracket says about the tesla

Reading [M T⁻² I⁻¹] as base units gives kg s⁻² A⁻¹. That looks strange, so it helps to rebuild it from more familiar units. From F = BIL, a field of one tesla exerts one newton on each metre of wire carrying one ampere: 1 T = 1 N A⁻¹ m⁻¹. Expanding the newton into kg m s⁻², the metre cancels and you are back at kg s⁻² A⁻¹.

The tesla is large. A fridge magnet is around 5 × 10⁻³ T, and the field of the Earth at its surface is roughly 5 × 10⁻⁵ T. The older CGS system uses the gauss instead, with 1 T = 10⁴ G, so the Earth's field is about 0.5 G.

Checks you can run with [MT⁻²I⁻¹]

Example: confirm that $F = BIL$ gives a force.

$[B][I][L] = [MT^{-2}I^{-1}][I][L] = [MLT^{-2}]$ . Current cancels and length appears once, which is the bracket of force.

Example: a light wave has E = 450 N/C. What is B, and do the dimensions of E = cB agree?

Value: $B = \frac{E}{c} = \frac{450}{3.00 \times 10^{8}} = 1.5 \times 10^{-6}\ \text{T}$ .

Dimensions: $[c][B] = [LT^{-1}][MT^{-2}I^{-1}] = [MLT^{-3}I^{-1}]$ , which is the bracket of electric field. The equation is consistent.

For the field near a long wire,

B = \frac{\mu_0 I}{2\pi r}

, the bracket of μ₀ is [MLT⁻²I⁻²]; multiply by [I] and divide by [L] to land on [MT⁻²I⁻¹] again.

B, H and their relatives compared

Magnetic quantities that students mix up
Quantity	Bracket	SI unit
Magnetic field B (flux density)	[M T⁻² I⁻¹]	tesla
Magnetic field intensity H	[L⁻¹ I]	A/m
Magnetic flux Φ	[M L² T⁻² I⁻¹]	weber
Permeability μ₀	[M L T⁻² I⁻²]	H/m

B and H are linked by $B = \mu_0 H$ in vacuum, and the table confirms it: [MLT⁻²I⁻²] times [L⁻¹I] gives [MT⁻²I⁻¹]. Flux is B times an area, so it simply gains L². Magnetic flux density and B are the same quantity under two names, so they share one bracket.

Errors examiners see every year

Leaving an L in the answer. The correct bracket has L⁰. Most wrong answers carry L¹ from forgetting the velocity.
Giving the bracket of H when B is asked. Read whether the question says flux density or intensity.
Writing I⁺¹. Current is in the denominator of B = F/(IL), so its power is minus 1.
Using Q in an SI answer. Convert charge to IT before you simplify.

Use the dimensional formulas table to see B beside E, Φ and μ₀ in one view.

Frequently asked questions

What is the dimensional formula of magnetic field B?

It is [M¹ L⁰ T⁻² I⁻¹], usually written [MT⁻²I⁻¹]. It comes from B = F/(qv), where force over charge times velocity leaves mass over time squared and current.

What is the dimension formula of magnetic field in terms of A?

Writing the ampere symbol in place of I, it reads [M T⁻² A⁻¹]. The meaning is identical, since A here is just the label for the current dimension.

Are B, magnetic induction and flux density one quantity?

For school and entrance exam purposes they are. All three names label the vector field measured in tesla, so they share a single bracket.

How is a tesla related to a gauss?

One tesla equals ten thousand gauss. The gauss belongs to the older CGS system. Treat 1 T = 10⁴ G as a fixed conversion factor to learn, and convert to tesla before using any SI formula.

Why does the magnetic field bracket have no length?

Force carries one L and the velocity in qv carries one L. Dividing the first by the second removes length entirely, which is why B has L⁰.