Current, Current Density and Charge Conservation

The Flow of Charge

Imagine a river. The total amount of water passing a cross-section per second is the flow rate. In electricity, the total charge passing a cross-section per second is the current \(I\). However, just as the river might be wider in some places and narrower in others, the "density" of the flow varies. Current density \(\mathbf{J}\) is the electrical analogue of mass flux in fluid dynamics. It is a vector field defined such that its magnitude at a point is the amount of charge crossing a unit area perpendicular to the flow per unit time. If the flow is uniform and perpendicular to a surface, the total current is simply the current density multiplied by the area. If the flow is angled or non-uniform, we must integrate the component of \(\mathbf{J}\) normal to the surface over the entire area. This local description is essential because it allows us to analyze how current behaves inside conductors, not just at their boundaries.

Defining Current Density

Consider a small surface element \(d\mathbf{a}\) with an outward normal vector. The current \(dI\) passing through this element in time \(dt\) is the charge \(dq\) that crosses it. If the charge density is \(\rho\) and the velocity of the charges is \(\mathbf{v}\), the volume of charge crossing the surface in time \(dt\) is \(dV = (\mathbf{v} \cdot d\mathbf{a}) dt\). Thus, \(dq = \rho (\mathbf{v} \cdot d\mathbf{a}) dt\). Dividing by \(dt\), we find the differential current \(dI = \rho (\mathbf{v} \cdot d\mathbf{a})\). By definition, the current density \(\mathbf{J}\) is the current per unit area perpendicular to the flow, leading to the fundamental relation:

For a general surface \(S\), the total current \(I\) is the flux of \(\mathbf{J}\) through that surface:

This integral form is the bridge between the microscopic particle motion and the macroscopic circuit current. It highlights that current is not a property of a wire alone, but a property of the field of moving charges within the volume.

The Continuity Equation

Charge is a conserved quantity. It cannot appear or disappear; it can only move from one place to another. Consider a fixed volume \(V\) bounded by a closed surface \(S\). The rate of change of the total charge \(Q\) inside \(V\) must equal the net current flowing into the volume. The net current flowing *out* is the flux integral \(\oint_S \mathbf{J} \cdot d\mathbf{a}\). Therefore, the rate of change of charge inside is the negative of the outward flux:

Using the divergence theorem, we can convert the surface integral into a volume integral of the divergence of \(\mathbf{J}\):

Since this must hold for *any* arbitrary volume \(V\), the integrands must be equal at every point in space. This yields the local form of the continuity equation, which expresses local charge conservation:

This equation is a fundamental constraint on any physical solution in electromagnetism. It ensures that the fields we calculate respect the conservation of charge.

Ohm's Law in Microscopic Form

In many materials, such as metals, the current density is proportional to the electric field driving it. This is the microscopic form of Ohm's law. We define the conductivity \(\sigma\) (the reciprocal of resistivity \(\rho_{\text{res}}\)) such that:

Combining this with the continuity equation for a steady state (where \(\partial \rho / \partial t = 0\)), we get \(\nabla \cdot \mathbf{J} = 0\). Substituting Ohm's law, we find \(\nabla \cdot (\sigma \mathbf{E}) = 0\). If the material is homogeneous (\(\sigma\) is constant), this simplifies to \(\nabla \cdot \mathbf{E} = 0\). In a region with no free charge (\(\rho = 0\)), Gauss's law also states \(\nabla \cdot \mathbf{E} = 0\). Thus, in a homogeneous ohmic conductor with no net charge accumulation, the electric field is divergence-free. This consistency check reinforces the validity of linking the transport properties (conductivity) with the field equations.