Electrodynamics

Magnetostatics and the vector potential

Steady currents produce a divergence-free magnetic field; writing B = ∇×A automatically satisfies ∇·B = 0.

In magnetostatics, current density J is time independent. Ampère’s law fixes the curl of B, while the vector potential A is useful in boundary-value problems and quantum mechanics.

Interpretation and consequences

A is not unique: the gauge change A′ = A + ∇χ leaves B unchanged. In electrodynamics, scalar potential φ and A give E = −∇φ − ∂A/∂t.

The vector potential depends on gauge, but its circulation around a closed loop is fixed: $\oint A\cdot d\ell=\int B\cdot dS$. This is Stokes's theorem applied to $B=\nabla\times A$. For a wire loop, changing magnetic flux gives the induced emf $\mathcal E=-d\Phi_B/dt$, so the potential links field geometry directly to electromagnetic induction.

In Coulomb gauge, each component of $A$ obeys a Poisson equation. Its integral solution depends on the whole current distribution, not just a point; taking the curl then yields $B$. Another gauge may simplify a boundary condition or symmetry without changing observables. For time-dependent currents, this instantaneous integral must be replaced by a retarded potential to respect finite signal speed.

The integral vector-potential solution is nonlocal: currents at every source point contribute with weight $1/|r-r'|$. In a simply connected region, Coulomb gauge selects a convenient representative, but boundaries and topology can leave residual freedom. For a thin wire, the circulation around a loop is tied to magnetic flux through any spanning surface; the chosen surface must be consistent with the source region and boundary conditions.

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