Current Density Formula: J = I/A, Unit A/m² and Examples

The current density formula is $J = \frac{I}{A}$ , which means current density equals the current divided by the area it flows through at right angles. It tells you how crowded the flow of charge is at a point in a conductor, measured in amperes per square metre.

Two wires can carry the same 5 A, yet the thin one runs hotter because its charge is packed into a smaller section. Current density captures that. It is a vector pointing along the flow of positive charge, while current itself is treated as a scalar. You meet it in Class 12 current electricity alongside drift velocity and the microscopic form of Ohm's law.

Quick facts

Formula
$J = \frac{I}{A}$
SI unit
ampere per square metre (A/m²)
Dimensional formula
[M⁰ L⁻² T⁰ I¹]
Vector or scalar
Vector, along the direction of conventional current
Electron form
$J = nev_d$
Ohm's law form
$\vec{J} = \sigma \vec{E}$
Tilted area
$I = JA\cos\theta$

Reading the current density formula

Terms in J = I / A
Symbol	Stands for	Unit
$J$	Current density, the current per unit normal area	A/m²
$I$	Current through the surface	A
$A$	Size of the surface the charge crosses, taken square on to the flow	m²

If the surface is tilted so that its normal makes an angle θ with the flow, only the component $A\cos\theta$ faces the current. The general relation is then $I = \vec{J}\cdot\vec{A} = JA\cos\theta$ . For a straight wire cut square across, θ = 0 and the simple form holds.

Three ways to write it

Forms of current density
Form	When to use it
$J = \frac{I}{A}$	You know the current and the wire size
$J = nev_d$	You know the carrier density and the drift speed
$J = \sigma E = \frac{E}{\rho}$	You know the field inside the material and its conductivity σ or resistivity ρ

The second form comes from $I = neAv_d$ divided by A. The third is the point form of Ohm's law. Put $E = V/L$ and $J = I/A$ into it and you recover $V = I\frac{\rho L}{A}$ , the familiar resistance relation.

Worked examples

Example 1: a thin lead

A wire with cross section 1.5 mm² carries 3.0 A. Find J.

A = 1.5 × 10⁻⁶ m².

$J = \frac{3.0}{1.5\times10^{-6}} = 2.0\times10^{6}\ \text{A/m}^2$

Example 2: a round wire given by diameter

A copper conductor of diameter 2.0 mm carries 5.0 A. Find the current density.

Half of 2.0 mm gives r = 1.0 × 10⁻³ m, so A = π(1.0 × 10⁻³)² = 3.14 × 10⁻⁶ m².

$J = \frac{5.0}{3.14\times10^{-6}} \approx 1.6\times10^{6}\ \text{A/m}^2$

Example 3: from the field

Inside a copper wire the electric field is 0.020 V/m. Taking σ = 6.0 × 10⁷ S/m, find J.

$J = \sigma E = (6.0\times10^{7})(0.020) = 1.2\times10^{6}\ \text{A/m}^2$

This is the same order as Examples 1 and 2, which is typical for household wiring.

Units and dimensions of current density

Amperes divided by square metres give A/m². Current is a base quantity with symbol I, and area has dimensions L². Writing mass and time with zero powers gives [M⁰ L⁻² T⁰ I¹]. The dimensional formula of current density page shows the full derivation and a check of $J = \sigma E$ .

Exam answers are often in A/cm² or A/mm². Since 1 m² = 10⁴ cm², a density of 2.0 × 10⁶ A/m² equals 200 A/cm², or 2.0 A/mm².

Why current density matters in practice

Wiring is rated by how much current density the copper can carry before it overheats. A fuse wire is made thin on purpose: its high J heats it quickly so it melts first. Along a wire that narrows, the current I is the same everywhere in a series path, but J rises at the narrow neck, and so does the drift speed.

Current is the same at every cross section of a single unbranched wire. Current density is not, unless the area is the same.

Common mistakes to avoid

Calling J a scalar. It has a direction at each point, even though current I is added like a scalar.
Dividing by the surface area of the wire's outside instead of its cross section.
Using the diameter in πr². Always halve it first.
Mixing σ and ρ. Conductivity multiplies E; resistivity divides it.

The full electricity formula list and the current formula page explain how charge flow is defined in the first place.

Frequently asked questions

What is the formula for current density?

J = I / A, the current divided by the cross section area at right angles to it. In terms of electrons it is J = nev_d.

In what unit is current density measured?

Ampere per square metre, A/m², which reduces to [M⁰ L⁻² T⁰ I¹] in base dimensions.

Is current density a vector or a scalar?

It is a vector. It points in the direction positive charge would flow at that point. Current, its integral over an area, is a scalar.

What is the relation between current density and electric field?

J = σE, where σ is the conductivity. This is Ohm's law written for a point inside the material.

Does current density change along a wire of varying thickness?

Yes. The current is the same throughout, so J is larger where the wire is thinner and smaller where it is thicker.