Current Density

Current density is one of many characteristics that can be used to describe an electric circuit. On a macroscopic scale, we can determine the current that is flowing through wires: it gives us the bigger picture of the flow of charges as a group. However, the amount of current flowing through a wire is often dependent on the shape and size of the wire. If we want a more general understanding of the current in a circuit, which can be applied to different wire dimensions, its useful to determine the current density. That's where the current density is practical, as it accounts for the exact dimensions of the wire, providing us with useful information about every charge that passes through it. In this article, we'll explain exactly what current density is, and derive various equations used to calculate it.

Volume Current Density

To explain volume current density, we first must understand electric current.

It's a macroscopic quantity that describes the behavior of charge made up of many electrons, rather than considering them individually. Mathematically, it can be expressed as

\[I=\frac{\mathrm{d}q}{\mathrm{d}t},\]

where \(I\) is current and \(q\) is the charge.

However, if we want to know how much current passes through a wire independent of the size of the cross-sectional area, we use current density.

Current density is often referred to as volume current density. Let's take a look at how a current density arises.

Current Density and Electric Field

Let's imagine a long cylindrical wire with a cross-sectional area \(A\) pictured in Figure 1 below.

Fig. 1 - Potential difference in a conductive wire produces a parallel electric field.

Creating a potential difference between the two ends of this wire will result in an electric field inside the wire. The direction of the field lines will be parallel to the walls of the cylinder; the same as the direction of the electric current.

Let's look at all the relevant equations regarding current density on its own and in relation to the electric field.

Current Density Formula

In its simplest forms, the current density can be expressed as

\[J=\frac{I}{A}.\]

This equation can be rearranged to obtain the formula for current in a uniform electric field:

\[I=JA,\]

while in a non-uniform electric field, it'll be

\[I= \oint \vec{J}\,\mathrm{d}\vec{A}.\]

The electric field in the wire from Figure 1 is proportional to the resistivity of the conductor and the current density. That means that the equation for current density can be written as

\[\vec{J}=\frac{\vec{E}}{\rho},\]

where \(\vec{J}\) is the current density measured in amperes per meter squared \(\left ( \frac{\mathrm{A}}{\mathrm{m}^2}\right )\), \(\vec{E}\) is the electric field with the units of volts per meter \(\left ( \frac{\mathrm{V}}{\mathrm{m}}\right )\), and \(\rho\) is the resistivity measured in Ohm-meters \(\left ( \frac{\mathrm{\Omega}}{\mathrm{m}}\right )\).

Finally,

\[\vec{J}=nq\vec{v}_\mathrm{d}\]

describes current density using the number of charge carriers \(n\) and drift velocity \(v_\mathrm{d}\). This shows the microscopic nature of current density.

Current Density And Magnetic Field

First, let's define a magnetic field.

In the case of the conductive wire, the magnetic field is produced by the moving charges and is proportional to the current.

Fig. 2 - A conductive wire with a magnetic field.

We can distinguish three separate cases of magnetic fields when it comes to conductive wires, as pictured in Figure 2 above.

Inside the wire, the magnetic field is equal to

\[B=\frac{\mu_0 J r}{2},\]

where we can reexpress the current density as

\[J=\frac{I}{A}= \frac{I}{\pi R^2},\]

so the final expression becomes

\[B=\frac{\mu_0 r I}{2\pi R^2}.\]

As we get further away from the center of the wire and reach its surface, the equation transforms into

\[B=\frac{\mu_0 I}{2\pi R}.\]

Finally, on the outside, the magnetic field is equal to

\[B=\frac{\mu_0 I}{2\pi r'}.\]

Here, \(\mu_0\) is the vacuum permeability equal to \(4\pi\times10^{-7} \, \frac{\mathrm{T} \, \mathrm{m}}{\mathrm{A}}\).

Exchange Current Density

A common concept in electrochemistry is the exchange current density.

These two opposing currents are called anodic and cathodic. As a result, the net current density is zero, as the anodic and cathodic currents cancel each other out. These values must be obtained experimentally, as it's the main way to quantify the performance of electrodes.

Fig. 3 - Exchange current density can be used to quantify the properties of electrodes.