Current Density

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.