Synthesis: The Static Field Equations and the Road to Maxwell
~6 min read · Tier toggle changes depth (T1 plain · T2 standard · T3 rigour)
You should be comfortable with vector calculus identities, specifically the divergence and curl operators, and have a working knowledge of the Biot–Savart law and Coulomb's law from previous lectures.
Why this matters
This lecture unifies the separate electrostatic and magnetostatic results into a coherent field theory.
Throughout PU-102, we have treated electric and magnetic phenomena as distinct force laws. By expressing these forces in terms of local field quantities, we reveal the underlying symmetry of nature. This synthesis provides the complete set of static field equations, which serve as the foundation for the full Maxwell equations and the prediction of electromagnetic waves in PU-204 (Electromagnetism II).
The Field Concept in Plain Language
Imagine a charge sitting in space. It does not just exert a force on another charge at a distance; it modifies the space around it. This modification is the electric field. Similarly, a steady current modifies the space around it with a magnetic field. The "field" is a physical entity that carries information about the source. In electrostatics, the field lines start on positive charges and end on negative charges. In magnetostatics, the field lines form closed loops; they never start or stop. This difference is the key to understanding the four fundamental static equations we will derive today. We are moving from "action at a distance" to a local interaction mediated by fields.
Electrostatics: Sources and Conservation
For static charges, the electric field is determined by two fundamental properties. First, charges act as sources or sinks for the field. This is described by Gauss's law. In its differential form, the divergence of the electric field is proportional to the charge density. This tells us that if you place a small box in space, the net flux of the electric field out of the box is determined by the net charge inside.
Second, the electrostatic field is conservative. This means the work done moving a charge around a closed loop is zero. Mathematically, the curl of the electric field is zero. This property allows us to define a scalar potential \(\varphi\) such that \(\mathbf{E} = -\nabla\varphi\). The negative sign indicates that the field points in the direction of decreasing potential.
Magnetostatics: Solenoidal Fields
Magnetic fields behave differently. There are no magnetic monopoles; isolated north or south poles do not exist. Consequently, the net magnetic flux through any closed surface is always zero. In differential form, this is the statement that the divergence of the magnetic field is zero. This is a profound difference from the electric field, which has non-zero divergence in the presence of charge.
However, magnetic fields are generated by moving charges, i.e., currents. The relationship between the current density \(\mathbf{J}\) and the magnetic field is given by Ampère's law. In its differential form, the curl of the magnetic field is proportional to the current density. This indicates that magnetic field lines circulate around currents. Unlike electric field lines, which can begin and end, magnetic field lines must form closed loops or extend to infinity.
The Symmetry of the Static Equations
We can now summarize the four static field equations. Two are "source" equations (divergence equations) and two are "circulation" equations (curl equations). The electric field has a source (charge) but no circulation (in statics). The magnetic field circulates around currents but has no divergence (no monopoles). This symmetry is elegant but incomplete. It assumes that charges and currents are static. If charges move to create a changing electric field, or if currents change to create a changing magnetic field, these equations are no longer sufficient. The "road to Maxwell" adds Faraday's induction term, \(\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t\), and Maxwell's displacement current, \(\mu_0\varepsilon_0\,\partial\mathbf{E}/\partial t\), to Ampère's law — the subject of PU-204.
The derivations
Differential Gauss's Law via the Divergence Theorem
From the integral form of Gauss's law, \(\oint_{\partial V}\mathbf{E}\cdot d\mathbf{A}=\dfrac{Q_{\text{enc}}}{\varepsilon_0}\), valid for every closed surface, the divergence theorem yields the local (differential) law \(\nabla\cdot\mathbf{E}=\dfrac{\rho}{\varepsilon_0}\), holding pointwise wherever \(\mathbf{E}\) is continuously differentiable and \(\rho\) is the volume charge density.
Irrotationality of the Electrostatic Field
For any static charge distribution, the electrostatic field \(\vec{E}(\vec{r})\) satisfies \(\nabla \times \vec{E} = \vec{0}\) everywhere off the source points. Equivalently, the field is a gradient, \(\vec{E} = -\nabla \varphi\), and its line integral around any closed loop vanishes, \(\oint_{C} \vec{E}\cdot d\vec{\ell} = 0\). The electrostatic field is therefore conservative.
Absence of Magnetic Monopoles
The magnetostatic field produced by any bounded, steady current distribution through the Biot–Savart law is divergence-free everywhere: \( \nabla \cdot \mathbf{B} = 0 \) identically. Equivalently, the net magnetic flux through every closed surface vanishes, so magnetic field lines never begin or end, and no isolated magnetic charge (monopole) can act as a source of \( \mathbf{B} \).
Ampère's Circuital Law from Biot–Savart
For a steady (magnetostatic) current distribution with divergence-free current density \(\vec{J}\), the field produced by the Biot–Savart law satisfies \(\nabla\times\vec{B}=\mu_0\vec{J}\) at every point, and equivalently \(\oint_{C}\vec{B}\cdot d\vec{l}=\mu_0 I_{\text{enc}}\) for any surface \(S\) spanning the closed loop \(C=\partial S\), with \(I_{\text{enc}}=\int_S\vec{J}\cdot d\vec{a}\). Ampère's circuital law is thus obtained as a theorem, not a postulate.
A common misconception is that \(\nabla \cdot \mathbf{B} = 0\) implies \(\mathbf{B} = 0\). This is false; it only means there are no sources or sinks. A uniform magnetic field has zero divergence but is non-zero. Similarly, \(\nabla \times \mathbf{E} = 0\) does not mean \(\mathbf{E} = 0\); it means the field is conservative. Confusing zero divergence/curl with zero field is a critical error in vector calculus interpretation. C
Worked example
Problem. Consider an infinite line charge with linear charge density \(\lambda = 5.0 \times 10^{-9} \ \mathrm{C/m}\) along the \(z\)-axis. Calculate the magnitude of the electric field at a radial distance \(r = 0.10 \ \mathrm{m}\) from the axis. Use \(\varepsilon_0 = 8.85 \times 10^{-12} \ \mathrm{F/m}\).
- Identify the symmetry: Cylindrical symmetry implies \(\mathbf{E}\) is radial and depends only on \(r\).
- Apply Gauss's law in integral form: \(\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}\).
- Choose a Gaussian cylinder of radius \(r\) and length \(L\). The flux through the curved surface is \(E(2\pi r L)\). The flux through the end caps is zero.
- The enclosed charge is \(Q_{\text{enc}} = \lambda L\).
- Equate and solve for \(E\): \(E(2\pi r L) = \frac{\lambda L}{\varepsilon_0} \implies E = \frac{\lambda}{2\pi \varepsilon_0 r}\).
- Substitute values: \(E = \frac{5.0 \times 10^{-9}}{2\pi (8.85 \times 10^{-12}) (0.10)}\).
Units check. \(\frac{\mathrm{C/m}}{\mathrm{F/m} \cdot \mathrm{m}} = \frac{\mathrm{C}}{\mathrm{F} \cdot \mathrm{m}} = \frac{\mathrm{C}}{(\mathrm{C/V}) \cdot \mathrm{m}} = \frac{\mathrm{V}}{\mathrm{m}} = \mathrm{N/C}\). The units are consistent.
Three quick tests
- What is the divergence of the magnetic field \(\mathbf{B}\) everywhere in space?
Answer
Zero. \(\nabla \cdot \mathbf{B} = 0\). - If \(\nabla \times \mathbf{E} = 0\), what can we say about the line integral of \(\mathbf{E}\) around a closed loop?
Answer
It is zero. The field is conservative. - Which static equation relates the curl of \(\mathbf{B}\) to the current density \(\mathbf{J}\)?
Answer
Ampère's law: \(\nabla \times \mathbf{B} = \mu_0 \mathbf{J}\).
Problem set for PU-102 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.
Derivations cited: D-041 · D-042 · D-051 · D-052 · Threads: force, energy, fields