Current Density Dimensional Formula: [M⁰ L⁻² T⁰ I¹]

The current density dimensional formula, with every power shown, reads [M⁰ L⁻² T⁰ I¹]; the compact version is [L⁻²I] or [L⁻²A]. Current density J tells you how much current flows through each square metre of a conductor's cross section.

It comes straight from J = I/A, current divided by area. Current is a base quantity [I] and area is [L²], so the bracket holds no mass and no time at all. Unlike total current, J has a direction at every point inside the wire, the way positive charge would travel there. The current density formula guide covers its links to drift speed and conductivity in detail.

Quick facts

Dimensional formula
[M⁰ L⁻² T⁰ I¹], or [L⁻²I]
Defining formula
$J = \frac{I}{A}$
SI unit
ampere per square metre (A/m²)
Other forms
$J = nev_d$ , $J = \sigma E$
Type
Vector
Conversion
1 A/cm² = 10⁴ A/m²

Step by step: current density dimensional formula

Definition: $J = \frac{I}{A}$ , where A is the cross section cut straight across the wire.
Current is a base quantity: [I].
Area: [L²].
Divide: $[J] = [IL^{-2}]$ , which in full is [M⁰ L⁻² T⁰ I¹].

No M and no T survive, and that is what most students get wrong. They expect a time power because current is "charge per time", but in SI the ampere is itself a base unit, so no T appears. If you start from charge instead, $J = \frac{Q}{At}$ gives $\frac{[IT]}{[L^{2}][T]}$ , and the time cancels anyway. The current bracket is the starting point.

Testing J = nev_d and J = σE

Example: check the drift relation $J = nev_d$ .

Number density n: electrons per volume, [L⁻³]. Charge e: [IT]. Drift speed: [LT⁻¹].

$[L^{-3}][IT][LT^{-1}] = [L^{-3+1}T^{1-1}I] = [L^{-2}I]$ . It matches.

Example: check the microscopic Ohm's law, $J = \sigma E$ .

Conductivity is the inverse of resistivity. Resistivity carries [ML³T⁻³I⁻²], so σ carries [M⁻¹L⁻³T³I²].

Electric field: [MLT⁻³I⁻¹]. Multiply: the M powers give 0, the L powers −3 + 1 = −2, the T powers 0 and the I powers 2 − 1 = 1. Result: [L⁻²I]. It matches as well.

Resistivity itself is explained on the resistivity formula guide.

A numerical example with drift speed

Example: a copper wire with cross section 2.5 mm² carries 10 A. Find J, then the drift speed, taking n = 8.5 × 10²⁸ electrons per m³.

Area: 2.5 mm² = 2.5 × 10⁻⁶ m², since 1 mm² = 10⁻⁶ m².

$J = \frac{10}{2.5 \times 10^{-6}} = 4.0 \times 10^{6}\ \text{A/m}^{2}$ .

First the product: $ne = 1.36 \times 10^{10}\ \text{C/m}^{3}$ . Then $v_d = \frac{J}{ne} = \frac{4.0 \times 10^{6}}{1.36 \times 10^{10}} = 2.9 \times 10^{-4}\ \text{m/s}$ .

Under a third of a millimetre each second: electrons drift very slowly even when a large current flows. The drift velocity formula explains why the bulb still lights at once.

Convert A/cm² by multiplying by 10⁴, because a square metre holds 10⁴ square centimetres.

Current density compared with similar brackets

Current per something
Quantity	Bracket	Unit
Current I	[I]	A
Current density J	[L⁻² I]	A/m²
Magnetizing field H	[L⁻¹ I]	A/m
Surface charge density σ	[L⁻² T I]	C/m²

Current density and surface charge density both divide by area, but one divides a current and the other a charge, so they differ by one power of time. H divides current by a length only. Reading the denominator tells you which is which.

Current density bracket slips

Adding T⁻¹. The ampere is a base unit; current density has no time power.
Writing L⁻¹. That is H. Current density divides by area, L².
Treating J as a scalar. Current density has a direction at each point, unlike total current.
Converting mm² with 10⁻³. Area needs 10⁻⁶.

For electrical brackets side by side, go to the dimensional formulas table and filter by current electricity.

Frequently asked questions

How do you write the dimensional formula of current density?

It is [M⁰ L⁻² T⁰ I¹], or [L⁻²I]. Current density is current divided by area, J = I/A.

Which unit does current density take in SI?

Ampere per square metre, A/m². Older problems sometimes give A/cm², which is 10⁴ times larger.

Does current density have a direction?

It does, so J is a vector. At every point it aims where positive charge would move, whereas the total current through a cross section is a scalar.

How is current density related to drift velocity?

J = nev_d, where n is the number of free electrons per cubic metre and e is the electron charge. A bigger J at fixed n means a faster drift.

Why does current density have no time dimension?

The ampere is a base unit in SI, which leaves current with the bare bracket [I]. Dividing by area adds L⁻² but never brings in time.