KAHIBARO
Table of Contents
Local view of electric current
When electric current flows through a wire, we often describe it by the total current $I$, measured in amperes. That tells us how much charge passes through the wire each second. But sometimes we need a more detailed, local description. We want to know how the current is distributed inside the material, not just the total amount. This leads to the idea of current density.
Current density tells us how much current passes through a unit area inside a conductor. It is especially useful when the conductor is thick, has a nonuniform shape, or when the flow of charge is not the same everywhere.
Definition of current density
Current density is written as $\vec{J}$. It is a vector quantity. Its direction is the direction in which positive charge would flow.
If a current $I$ passes uniformly and perpendicularly through a cross-sectional area $A$, then the magnitude of the current density is
$$
J = \frac{I}{A}
$$
The SI unit of current density is
$$
\mathrm{A/m^2}
$$
This means amperes per square meter.
Important definition:
$$
J = \frac{I}{A}
$$
only when the current is uniformly distributed over the area and flows perpendicular to that area.
Current density is a vector, so in full form we write $\vec{J}$.
Why current density is a vector
Current itself is treated as a scalar in simple circuit problems, but inside matter the flow of charge has a direction in space. Because of that, current density must be a vector.
If positive charges move to the right, then $\vec{J}$ points to the right. If electrons move to the right, then $\vec{J}$ points to the left, because conventional current is defined in the direction of positive charge flow.
This is an important point in metals, where the actual moving charges are electrons.
Relation between current and area
Suppose a wire has cross-sectional area $A$. If the current is spread evenly across the whole cross section, then the current density is the same everywhere in that section.
A larger area carrying the same current has a smaller current density. A smaller area carrying the same current has a larger current density.
This is why narrow parts of a conductor can heat more strongly. The same total current is forced through less area, so the current density increases.
| Quantity | Symbol | Meaning | Unit |
|---|---|---|---|
| Current | $I$ | Total charge flow per second | A |
| Area | $A$ | Cross-sectional area | $\mathrm{m^2}$ |
| Current density | $J$ | Current per unit area | $\mathrm{A/m^2}$ |
Current through a surface
In a more general case, current may not be uniform, and the surface may not be perpendicular to the flow. Then we use the more complete relation
$$
I = \int \vec{J} \cdot d\vec{A}
$$
Here, $d\vec{A}$ is a small area vector. Its magnitude is the small area $dA$, and its direction is perpendicular to the surface.
The dot product means that only the component of $\vec{J}$ perpendicular to the surface contributes to the current through that surface.
If $\theta$ is the angle between $\vec{J}$ and the area vector, then for uniform current density over a flat surface,
$$
I = JA\cos\theta
$$
Special cases are easy to understand. If the flow is perpendicular to the surface, then $\theta = 0$ and
$$
I = JA
$$
If the flow is parallel to the surface, then $\theta = 90^\circ$ and
$$
I = 0
$$
because no charge passes through the surface.
General current formula:
$$
I = \int \vec{J} \cdot d\vec{A}
$$
For uniform flow through a flat surface:
$$
I = JA\cos\theta
$$
Only the component of $\vec{J}$ normal to the surface contributes.
Physical meaning
Current density gives a microscopic picture of conduction. It tells us where charge is flowing strongly and where it is flowing weakly. In a uniform straight wire, $\vec{J}$ is usually parallel to the wire. In more complex objects, such as broad metal plates or irregular conductors, $\vec{J}$ can vary from point to point.
A high current density means a large amount of charge crosses each square meter every second. A low current density means less charge crosses each square meter every second.
Uniform wire example
Imagine a cylindrical wire carrying current $I$ uniformly.
If the wire radius is $r$, then the cross-sectional area is
$$
A = \pi r^2
$$
So the current density is
$$
J = \frac{I}{\pi r^2}
$$
This shows that if the radius becomes smaller, the current density becomes larger.
Example calculation
Suppose a wire carries a current of $3.0\ \mathrm{A}$ and has cross-sectional area $2.0 \times 10^{-6}\ \mathrm{m^2}$.
Then
$$
J = \frac{I}{A} = \frac{3.0}{2.0 \times 10^{-6}} = 1.5 \times 10^6\ \mathrm{A/m^2}
$$
So the current density is
$$
1.5 \times 10^6\ \mathrm{A/m^2}
$$
Current density and moving charges
Current density is closely related to the motion of charge carriers in a material. If the material contains $n$ charge carriers per unit volume, each carrier has charge $q$, and they move with drift velocity $\vec{v}_d$, then
$$
\vec{J} = nq\vec{v}_d
$$
This equation connects the large-scale current we measure in a circuit with the microscopic motion of charges inside the material.
For electrons, $q = -e$, so the electron drift velocity points opposite to the current density vector.
Microscopic form of current density:
$$
\vec{J} = nq\vec{v}_d
$$
where $n$ is number of charge carriers per unit volume, $q$ is charge of each carrier, and $\vec{v}_d$ is drift velocity.
Interpreting the microscopic formula
The formula $\vec{J} = nq\vec{v}_d$ makes physical sense.
If there are more mobile charges in each cubic meter, then the current density is larger. That is the role of $n$.
If each charge is larger in magnitude, then the current density is larger. That is the role of $q$.
If the charges drift faster, then more charge crosses a given area each second. That is the role of $\vec{v}_d$.
This formula is very important because it links the microscopic world of atoms and electrons to the macroscopic current in wires.
Common situations
In many elementary problems, the current density is assumed uniform. In that case,
$$
J = \frac{I}{A}
$$
is enough.
But in real conductors, current density can vary due to shape changes, material changes, or nonuniform electric fields. Then the integral form is needed.
| Situation | Useful formula |
|---|---|
| Uniform current through flat cross section | $J = I/A$ |
| General current through a surface | $I = \int \vec{J} \cdot d\vec{A}$ |
| Microscopic relation | $\vec{J} = nq\vec{v}_d$ |
Direction and sign conventions
Because $\vec{J}$ points in the direction of positive charge flow, it follows the same direction as the electric field in many conducting materials. In metals, however, the electrons move opposite to this direction.
This difference often confuses beginners. The easiest rule is to remember that conventional current and current density are defined as if positive charges move.
Direction rule:
$\vec{J}$ points in the direction of conventional current, not necessarily in the direction of electron motion.
Summary
Current density is the current per unit area and gives a local description of charge flow inside a material. It is a vector quantity with SI unit $\mathrm{A/m^2}$. For uniform flow through area $A$,
$$
J = \frac{I}{A}
$$
In the general case,
$$
I = \int \vec{J} \cdot d\vec{A}
$$
and at the microscopic level,
$$
\vec{J} = nq\vec{v}_d
$$
These relations make current density one of the key ideas for understanding how electric current behaves inside conductors.