Steady Currents, the Continuity Equation and Ohm's Law

This chapter is the bridge between electrostatics and magnetism. Moving charge is a current, and steady (time-independent) currents are the sources of static magnetic fields, taken up in the next chapter. Before that, we need the language of current: the current density $\vec{J}$ , the local statement of charge conservation (the continuity equation), the microscopic content of Ohm's law, the idea of electromotive force that keeps a steady current flowing against resistance, and the boundary conditions and power relations that go with them.

The organising principle is charge conservation. Charge is never created or destroyed, only transported; the continuity equation $\nabla\cdot\vec{J} = -\partial\rho/\partial t$ is that fact written locally. For a steady current — the defining case of this chapter and of magnetostatics — nothing depends on time, so $\partial\rho/\partial t = 0$ and $\nabla\cdot\vec{J} = 0$ : steady current is divergence-free, its field lines form closed loops, and the same current crosses every section of a single wire. Everything else (Ohm's law, EMF, resistor networks) is built on that.

How to read this chapter. The two equations to carry forward are $\nabla\cdot\vec{J} = -\partial\rho/\partial t$ (always true) and, for the steady state, $\nabla\cdot\vec{J} = 0$ . Ohm's law in the form $\vec{J} = \sigma\vec{E}$ is a material property, not a law of nature — it holds for ordinary conductors in ordinary fields and fails for many materials. Keep "conductivity $\sigma$ " (this chapter) distinct from "surface charge $\sigma$ " (earlier chapters); context disambiguates, and this chapter uses $\sigma$ for conductivity throughout.

1. Current and Current Density

The current $I$ through a surface is the rate at which charge crosses it, $I = dQ/dt$ , in amperes (C/s). It is a scalar attached to a surface (with a chosen positive sense). The local quantity is the current density $\vec{J}$ , a vector field: $\vec{J}$ points in the direction of positive-charge flow, and its magnitude is the current per unit cross-sectional area perpendicular to the flow. The current through any surface $\mathcal{S}$ is the flux

I = \int_{\mathcal{S}} \vec{J}\cdot d\vec{a}

If a density $\rho$ of mobile charge moves with average (drift) velocity $\vec{v}$ , then

\vec{J} = \rho\,\vec{v}

For a metal with $n$ conduction electrons per unit volume, each of charge $-e$ drifting at $\vec{v}_d$ , this is $\vec{J} = -n e\,\vec{v}_d$ (and the current flows opposite to the electron motion). Drift speeds are tiny — millimetres per second in a household wire — even though signals propagate near the speed of light, because the field, not the electrons, carries the information.

2. The Continuity Equation

Charge conservation applied to a fixed volume $\mathcal{V}$ : the rate of decrease of enclosed charge equals the net current out through the boundary,

-\frac{d}{dt}\int_{\mathcal{V}}\rho\,d\tau = \oiint_{\partial\mathcal{V}}\vec{J}\cdot d\vec{a}

Applying the divergence theorem to the right side and shrinking $\mathcal{V}$ to a point gives the continuity equation:

\boxed{\;\nabla\cdot\vec{J} = -\frac{\partial\rho}{\partial t}\;}

This is one of the most fundamental statements in physics — more robust than any particular force law. In electrodynamics it is built into Maxwell's equations (it follows from taking the divergence of the Ampère–Maxwell law), which is why the displacement-current term is not optional.

For a steady current, $\partial\rho/\partial t = 0$ everywhere, so

\nabla\cdot\vec{J} = 0 \qquad \text{(steady state)}

$\vec{J}$ is then solenoidal: field lines of $\vec{J}$ never begin or end, currents flow in closed loops, and $\oint \vec{J}\cdot d\vec{a} = 0$ through any closed surface — the current into any region equals the current out. In a single unbranched wire, $I$ is the same at every cross-section.

3. Ohm's Law: the Microscopic Statement

Experimentally, in a wide class of materials the current density is proportional to the field driving it:

\boxed{\;\vec{J} = \sigma\vec{E}\;}

where $\sigma$ is the conductivity (S/m) and its reciprocal $\rho_{\text{res}} = 1/\sigma$ is the resistivity ( $\Omega\cdot$ m). This is Ohm's law in local form. It is a constitutive relation, valid for "ohmic" materials over a range of fields, not a universal law: semiconductors, plasmas, and metals at high field all deviate.

The Drude model gives it microscopic content. Free electrons are accelerated by $\vec{E}$ between random collisions that occur, on average, every $\tau$ (the mean free time) and which randomise the velocity. Between collisions, $m\dot{\vec{v}} = -e\vec{E}$ ; averaging, the drift velocity reaches a steady value $\vec{v}_d = -\dfrac{e\tau}{m}\vec{E}$ . Then

\vec{J} = -n e\,\vec{v}_d = \frac{n e^2\tau}{m}\,\vec{E} \quad\Rightarrow\quad \sigma = \frac{n e^2\tau}{m}

Conductivity is high when there are many carriers ( $n$ ), they are light ( $m$ ), and they travel far between collisions ( $\tau$ ). Rising temperature shortens $\tau$ (more lattice vibrations), so metals' resistivity increases with temperature.

4. Resistance and Ohm's Law for a Device

For a uniform conductor of length $L$ , cross-section $A$ , carrying current $I$ driven by a potential difference $V$ across its ends: the field inside is $E = V/L$ , the current density is $J = I/A$ , and $\vec{J} = \sigma\vec{E}$ gives $\dfrac{I}{A} = \sigma\dfrac{V}{L}$ , i.e.

V = IR, \qquad R = \frac{L}{\sigma A} = \frac{\rho_{\text{res}}\,L}{A}

the familiar "Ohm's law" for a resistor. $R$ is a geometric-plus-material property, in ohms. Note that inside a current-carrying wire $\vec{E}\ne 0$ — unlike the electrostatic conductor of Chapter 1, a resistor sustains an interior field, maintained by a slight surface-charge gradient along the wire.

5. Electromotive Force

A steady current in a resistive circuit dissipates energy, so something must continuously supply it. That "something" is a source of electromotive force (EMF), $\mathcal{E}$ — a battery, a generator, a solar cell. Inside the source, a non-electrostatic force per unit charge $\vec{f}_s$ (chemical, mechanical, photovoltaic) pushes charge from low to high potential. The EMF is the work done per unit charge by that source force around the circuit:

\mathcal{E} = \oint \vec{f}_s\cdot d\vec{\ell}

For an ideal source the entire loop integral comes from inside the source, and $\mathcal{E}$ equals the open-circuit terminal voltage. A real source has internal resistance $r$ : with a current $I$ flowing, the terminal voltage is $V = \mathcal{E} - Ir$ , and the load gets $\mathcal{E}$ minus the internal drop. Motional EMF — the $q\vec{v}\times\vec{B}$ force on charges in a conductor moving through a magnetic field — is the same idea and is developed fully in Chapter 7.

6. Power

The rate at which the field does work on the moving charge, per unit volume, is $\vec{J}\cdot\vec{E}$ . Integrated over a resistor:

P = \int \vec{J}\cdot\vec{E}\,d\tau = VI = I^2 R = \frac{V^2}{R}

For an ohmic conductor this power goes entirely into heat (Joule heating): the drift energy gained from the field between collisions is handed to the lattice at each collision. In a source delivering current, the source does work at rate $\mathcal{E}I$ ; of this, $I^2 r$ is lost internally and $VI = (\mathcal{E} - Ir)I$ reaches the load.

Worked example — the two current-carrying geometries. (a) A wire of resistivity $\rho_{\text{res}}$ , length $L$ , radius $a$ : $R = \rho_{\text{res}}L/\pi a^2$ . Doubling the radius quarters the resistance. (b) Current flowing radially between two concentric spherical shells (radii $a < b$ ) through a medium of conductivity $\sigma$ : by the steady condition the same $I$ crosses every sphere, so $J(r) = I/4\pi r^2$ and $E = J/\sigma$ ; integrating, $V = \dfrac{I}{4\pi\sigma}\left(\dfrac{1}{a} - \dfrac{1}{b}\right)$ , giving $R = \dfrac{1}{4\pi\sigma}\left(\dfrac{1}{a} - \dfrac{1}{b}\right)$ . The same $1/r^2$ that governed the electrostatic field of a point charge governs the current spreading from a small electrode.

7. Boundary Conditions and Charge Relaxation

At an interface between two conducting media in the steady state, $\nabla\cdot\vec{J} = 0$ forces the normal component of $\vec{J}$ to be continuous:

J_{1\perp} = J_{2\perp} \quad\Longleftrightarrow\quad \sigma_1 E_{1\perp} = \sigma_2 E_{2\perp}

Since $\varepsilon_1 E_{1\perp} \ne \varepsilon_2 E_{2\perp}$ in general (different $\sigma$ and $\varepsilon$ ), a free surface charge $\sigma_f = D_{1\perp} - D_{2\perp}$ (normal pointing from medium 2 into medium 1) accumulates at the interface — a real, measurable effect wherever two materials with different $\sigma/\varepsilon$ ratios carry current across their junction.

Charge relaxation. If a charge density $\rho$ is placed in the interior of an ohmic conductor, combine $\nabla\cdot\vec{J} = -\partial\rho/\partial t$ with $\vec{J} = \sigma\vec{E}$ and $\nabla\cdot\vec{E} = \rho/\varepsilon$ :

\frac{\partial\rho}{\partial t} = -\frac{\sigma}{\varepsilon}\,\rho \quad\Rightarrow\quad \rho(t) = \rho_0\,e^{-t/\tau_r}, \qquad \tau_r = \frac{\varepsilon}{\sigma}

Interior charge decays exponentially to the surface with time constant $\tau_r = \varepsilon/\sigma$ — about $10^{-19}$ s for copper, but seconds or longer for a poor conductor. This is why the electrostatic assumption "all charge on the surface" is justified for good conductors, and it quantifies how quickly a conductor reaches electrostatic equilibrium.

8. Common Pitfalls

Assuming $\vec{E} = 0$ inside a current-carrying conductor. That is the electrostatic result. A resistor with current flowing has $\vec{E} = \vec{J}/\sigma \ne 0$ inside, sustained by surface charge.
Treating $\nabla\cdot\vec{J} = 0$ as always true. It holds only in the steady state. In general $\nabla\cdot\vec{J} = -\partial\rho/\partial t$ ; a charging capacitor plate has $\nabla\cdot\vec{J}\ne 0$ at its surface.
Confusing conductivity $\sigma$ with surface charge $\sigma$ . Different quantities sharing a symbol; this chapter's $\sigma$ is conductivity. Some texts write $\rho$ for resistivity, which then collides with charge density — watch the context.
Thinking EMF is a force. It is work per unit charge (units: volts), an integral of a force per unit charge around the loop.
Using $V = \mathcal{E}$ for a battery under load. With current flowing, the terminal voltage is $\mathcal{E} - Ir$ ; only the open-circuit voltage equals $\mathcal{E}$ .
Forgetting the interface charge in steady currents. Continuity of $J_\perp$ (not $D_\perp$ ) is the steady-current condition; the mismatch generically deposits free charge at the junction.
Expecting instantaneous electrostatic equilibrium. Charge relaxes with $\tau_r = \varepsilon/\sigma$ ; for good conductors this is fantastically fast, but for insulators it can be macroscopically slow.

Summary

Current $I = \int\vec{J}\cdot d\vec{a}$ ; current density $\vec{J} = \rho\vec{v}$ , or $-ne\vec{v}_d$ in a metal.
Continuity equation: $\nabla\cdot\vec{J} = -\partial\rho/\partial t$ (charge conservation, local form). Steady state: $\nabla\cdot\vec{J} = 0$ , so $\vec{J}$ is solenoidal and $I$ is the same at every section of a wire.
Ohm's law (local): $\vec{J} = \sigma\vec{E}$ , a material relation. Drude model: $\sigma = ne^2\tau/m$ ; resistivity rises with $T$ because $\tau$ falls.
Resistance: $V = IR$ with $R = L/\sigma A = \rho_{\text{res}}L/A$ . Inside a current-carrying resistor $\vec{E}\ne 0$ .
EMF: $\mathcal{E} = \oint\vec{f}_s\cdot d\vec{\ell}$ , work per unit charge by the source. Real source: $V_{\text{terminal}} = \mathcal{E} - Ir$ .
Power: $P = \int\vec{J}\cdot\vec{E}\,d\tau = VI = I^2R = V^2/R$ ; Joule heating in an ohmic conductor.
Steady-current boundary condition: $J_\perp$ continuous ( $\sigma_1 E_{1\perp} = \sigma_2 E_{2\perp}$ ); a free charge builds up at an interface where $\sigma/\varepsilon$ differs.
Charge relaxation: interior charge decays as $e^{-t/\tau_r}$ with $\tau_r = \varepsilon/\sigma$ — the timescale for a conductor to reach electrostatic equilibrium.