Current Density, the Continuity Equation & Ohm's Law in Point Form
From lumped circuits to fields: how charge flows steadily through conductors
You already know V=IR from circuit theory. But what happens inside the wire at the level of fields and charges? The answer connects everything you learned about electrostatics to the world of moving charge — and it starts with a single vector field.
From Electrostatics to Steady Currents
Having spent two modules on electrostatics — Coulomb's law, Gauss's law, electric potential, conductors, capacitors, dielectrics, and boundary conditions — you have a thorough understanding of what happens when charges sit still. But charges rarely sit still forever. When an electric field exists inside a conductor, free charges move, and that motion is what we call electric current. In this lesson, we take the first step from electrostatics into steady currents, bridging the lumped-element world of circuit theory (V=IR) with the field-theoretic description that will carry us through the rest of this module and into magnetostatics.
Current Density as a Vector Field
The total current I through a surface tells you how much charge crosses that surface per unit time. But I is an aggregate quantity — it tells you nothing about where, or in what direction, the charge is flowing at a specific point. For a field description, we need current density J, a vector field with units of A/m². At any point, J points in the direction of positive charge flow, and its magnitude equals the current per unit cross-sectional area flowing normal to that area.
For uniform current distributed evenly across a flat cross-section of area A, the relationship reduces to a scalar equation. When the flow is non-uniform, you must integrate J over the surface to recover the total current: I = ∫∫J·dS.
Direction Matters
Because J is a vector field, it can vary in both magnitude and direction from point to point. In a simple round wire carrying DC, J is uniform and points along the wire axis. In a grounded conductor carrying current away from a buried electrode, J fans outward radially and decreases with distance. The vector nature of J becomes essential when we write the continuity equation, which is a statement about how J flows through space.
- Use the uniform-current relation J = I/A.
- Substitute: J = 5 A / (2×10⁻⁶ m²).
- Compute: J = 2.5×10⁶ A/m² = 2.5 MA/m².
The value 2.5 MA/m² is entirely typical for household wiring — copper conductors routinely carry current densities of this order. A 'megaamp per square meter' sounds enormous compared to the 5 A you read on an ammeter, but it reflects how little cross-sectional area a thin wire actually has.
The Continuity Equation
Now we ask a deeper question: can charge pile up or deplete at a point in space? Intuitively, if more current flows into a tiny volume than flows out, the charge density inside that volume must be increasing. The continuity equation makes this precise.
Consider a small closed volume V bounded by surface S. The net current flowing out through S is ∫∫J·dS. By definition, that outflow must reduce the total charge inside: ∫∫J·dS = −d/dt∫∫∫ρdV. Applying the divergence theorem to the left side and noting that the volume is arbitrary, we obtain the differential (point-form) continuity equation.
Steady Currents Are Divergence-Free
This equation is a local statement of charge conservation. For steady currents — the focus of this module — nothing changes with time. Charge may be moving, but the charge density at every point is constant. That means ∂ρ/∂t = 0, and the continuity equation simplifies to ∇·J = 0.
This is the same 'what goes in must come out' principle you intuit from Kirchhoff's current law (KCL) in circuit theory. KCL is just the integral form of ∇·J = 0 applied at a circuit node. The field version holds at every point in space, not just at wire junctions.
Ohm's Law in Point Form
The last piece we need is a relationship between the current density J and the electric field E that drives it. Consider a cylindrical conductor of length L, cross-section A, and conductivity σ. A voltage V applied across its ends produces a uniform field E = V/L inside. The current is I = V/R, and for a uniform conductor R = L/(σA), so I = VσA/L = σEA. Dividing by area gives J = I/A = σE. This is Ohm's law in point form — it holds at every point inside the material, regardless of geometry.
Conductivity σ (units S/m) is a material property. Good conductors like copper have σ ≈ 5.8×10⁷ S/m; insulators have σ near zero. The point-form law J = σE works for any field distribution — uniform or non-uniform — as long as the material is ohmic (linear).
- Rearrange Ohm's law in point form: E = J/σ.
- Substitute: E = (1.16×10⁶) / (5.8×10⁷).
- Compute: E = 0.02 V/m.
Why Wires Look Like Equipotentials
The result — E = 0.02 V/m — is astonishingly small. It takes only two hundredths of a volt per meter of electric field to drive 5 A through a copper wire of this cross-section. Copper's enormous conductivity means that even tiny fields produce substantial currents. This is why, in circuit analysis, we often treat wires as equipotential: the field inside is negligible, and so the voltage drop along a short wire is practically zero.
Check your understanding
- Current density J (A/m²) is a vector field giving current per unit cross-sectional area; for uniform flow through a flat area, J = I/A.
- The continuity equation ∇·J = −∂ρ/∂t expresses local charge conservation: any net outflow of J from a region must deplete the charge density there.
- For steady currents, ∂ρ/∂t = 0, so ∇·J = 0 — current density is solenoidal, meaning what flows in must flow out with no accumulation.
- Ohm's law in point form, J = σE, relates current density to the local electric field via conductivity σ, and is the microscopic counterpart of V = IR.