Divergence, Curl, and Ampère's Law

The Magnetostatic Bridge

This chapter works in the magnetostatic limit: currents are steady, charge density is time-independent, and the electric field is not changing with time ($\partial \mathbf{E}/\partial t=0$). Under these assumptions the displacement-current term is absent, so the magnetic laws take their magnetostatic forms.

The Biot-Savart law is a field-construction formula: given a known steady current distribution, it adds the contributions of its current elements to calculate $\mathbf{B}$ directly. That is general, but the integral can be laborious. For highly symmetric currents, Ampère's law can find the same field more efficiently by using its circulation instead.

The circulation $\oint \mathbf{B}\cdot d\mathbf{l}$ is the total tangential component of $\mathbf{B}$ around an oriented closed loop. A current through the loop's surface makes the magnetic field tend to circle that current. In magnetostatics, Ampère's integral law is

\oint_{\partial S} \mathbf{B}\cdot d\mathbf{l} = \mu_0 I_{\text{enc}} = \mu_0\int_S \mathbf{J}\cdot d\mathbf{a}.

Stokes' theorem says that this loop circulation equals the surface integral of $\nabla\times\mathbf{B}$. Because the relation holds for every suitable surface $S$, its local form is $\nabla\times\mathbf{B}=\mu_0\mathbf{J}$: curl measures the local tendency of $\mathbf{B}$ to circulate around the current density.

A separate magnetic field law is the absence of magnetic sources or sinks. Classical electromagnetism has no isolated magnetic charges, so magnetic field lines do not begin or end at a point. The divergence theorem connects its two forms:

\oint_{\partial V}\mathbf{B}\cdot d\mathbf{a} = \int_V (\nabla\cdot\mathbf{B})\,d\tau = 0 \quad\Longleftrightarrow\quad \nabla\cdot\mathbf{B}=0.

Thus Biot-Savart constructs $\mathbf{B}$ from steady currents; Ampère's law constrains its circulation, especially when symmetry helps; $\nabla\times\mathbf{B}=\mu_0\mathbf{J}$ is the local circulation law; and $\nabla\cdot\mathbf{B}=0$ is the local no-sources-or-sinks law. These are consistent descriptions of magnetostatics, not four statements that all follow trivially from Biot-Savart.

Later, when electric fields vary with time, $\nabla\times\mathbf{B}=\mu_0\mathbf{J}$ is incomplete and becomes the Ampère-Maxwell law.

1. The Divergence of B

The behavior of magnetic fields regarding "sources" and "sinks" is described by the divergence. Unlike electric fields, which diverge from positive charges, magnetic fields are solenoidal.

\underbrace{\nabla \cdot \mathbf{B} = 0}_{\text{Differential Form}} \quad \longleftrightarrow \quad \underbrace{\oint_{\mathcal{S}} \mathbf{B} \cdot d\mathbf{a} = 0}_{\text{Integral Form}}

Physical Interpretation:

No Magnetic Monopoles: There is no magnetic equivalent to electric charge ($q$). You cannot isolate a "North" pole.
Closed Loops: Magnetic field lines have no beginning or end; they form continuous closed loops.
Flux Conservation: The net magnetic flux entering any closed volume must exactly equal the flux leaving it.

2. The Curl of B (Ampère's Law)

While the divergence of $\mathbf{B}$ is zero, the curl is non-zero wherever current flows. This describes the rotational or "circulating" nature of the magnetic field.

\underbrace{\nabla \times \mathbf{B} = \mu_0 \mathbf{J}}_{\text{Differential Form}} \quad \longleftrightarrow \quad \underbrace{\oint_{\mathcal{P}} \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}}}_{\text{Integral Form}}

Physical Interpretation:

Circulation: Electric currents ($\mathbf{J}$ or $I$) act as vortices that cause the magnetic field to curl around them.
Calculation Tool: The integral form (Ampère's Law) allows us to calculate $\mathbf{B}$ easily when the system possesses high symmetry (cylindrical, planar, or solenoidal), allowing us to pull $|\mathbf{B}|$ out of the integral.

Deep Dive: Applications of Ampere's Law

3. Comparison: Magnetostatics vs. Electrostatics

The theories of static electric and magnetic fields run parallel to each other, yet they describe distinct physical phenomena.

Similarities

Superposition Principle: Both fields satisfy the superposition principle. The total field is the vector sum of fields produced by individual source elements.
Inverse Square Laws: Both Coulomb's Law and the Biot-Savart Law involve a $1/r^2$ dependence for the field contribution from a point source (or current element).
Long Range: Both forces can act over infinite distances.

Differences and Parallels

\begin{aligned} &\textbf{Electrostatics} & &\textbf{Magnetostatics} \\ \hline \text{Source} \quad & \rho \text{ (Scalar Charge)} & & \mathbf{J} \text{ (Vector Current)} \\ \text{Field Law} \quad & \nabla \cdot \mathbf{E} = \rho/\epsilon_0 \text{ (Gauss)} & & \nabla \times \mathbf{B} = \mu_0 \mathbf{J} \text{ (Ampère)} \\ \text{Null Property} \quad & \nabla \times \mathbf{E} = 0 \text{ (Conservative)} & & \nabla \cdot \mathbf{B} = 0 \text{ (Solenoidal)} \\ \text{Potentials} \quad & \mathbf{E} = -\nabla V \text{ (Scalar } V) & & \mathbf{B} = \nabla \times \mathbf{A} \text{ (Vector } \mathbf{A}) \\ \text{Constants} \quad & \epsilon_0 \approx 8.85 \times 10^{-12} \text{ F/m} & & \mu_0 = 4\pi \times 10^{-7} \text{ N/A}^2 \\ \end{aligned}

Constants: $\epsilon_0$ vs $\mu_0$

$\epsilon_0$ (Permittivity of free space) describes how easy it is to establish an electric field in a vacuum. $\mu_0$ (Permeability of free space) describes the ability of a material to support the formation of a magnetic field. They are linked by the speed of light: $c = 1/\sqrt{\epsilon_0 \mu_0}$.

Potentials: $V$ vs $\mathbf{A}$

Because $\nabla \times \mathbf{E} = 0$, we can define a Scalar Potential $V$ such that $\mathbf{E} = -\nabla V$. This relates to potential energy.
Because $\nabla \cdot \mathbf{B} = 0$, we cannot use a scalar potential. Instead, we define a Vector Potential $\mathbf{A}$ such that $\mathbf{B} = \nabla \times \mathbf{A}$. $\mathbf{A}$ does not have a simple potential energy interpretation but simplifies calculations in quantum mechanics and electrodynamics.

Behavior in Matter

Electrostatics: Matter responds via Polarization ($\mathbf{P}$). Electric dipoles align with the field, creating a bound charge $\sigma_b$ that typically reduces the total field inside the material (Dielectrics).

Magnetostatics: Matter responds via Magnetization ($\mathbf{M}$). Magnetic dipoles align, creating a bound current $\mathbf{K}_b$. Depending on the material, this can slightly reduce the field (Diamagnetic), slightly increase it (Paramagnetic), or massively increase it (Ferromagnetic).