the magnetic vector potential
The magnetic field has no sources — no magnetic charges for its field lines to begin or end on — and that single fact, div B = 0, has a powerful consequence: B can always be written as the curl of another field. That field is the magnetic vector potential A.
Because the divergence of any curl is identically zero, setting B = curl A automatically satisfies div B = 0. The vector potential also enters the electric field once things vary in time: combining it with the scalar potential gives E = -grad V - dA/dt, so E and B together descend from the pair (V, A). For a steady current in the Coulomb gauge, A obeys its own Poisson equation laplacian A = -mu_0 J, mirroring the electrostatic laplacian V = -rho/epsilon_0, and the field of a current follows by taking the curl.
A is not merely a calculational convenience. Like V it is gauge-dependent — you may add the gradient of any scalar, A goes to A + grad lambda, without changing B — so it seems less real than the field. Yet the Aharonov-Bohm effect shows a charged quantum particle responds to the line integral of A even in regions where B is exactly zero, and it is A, not B, that couples to charge in the Lagrangian and gets quantized in quantum electrodynamics. Relativity then merges V and A into the single four-potential.
An infinite solenoid confines all its magnetic field inside, with B = 0 everywhere outside. Yet the vector potential A outside is nonzero, circling the solenoid, and its loop integral equals the enclosed magnetic flux — exactly the quantity a charged particle detects in the Aharonov-Bohm effect, despite feeling no field.
B = curl A guarantees div B = 0; together with V it gives E = -grad V - dA/dt.
A is defined only up to the gradient of an arbitrary scalar (A goes to A + grad lambda), so it is not directly measurable — yet its loop integral (the flux) is physical, as the Aharonov-Bohm effect proves. Do not read too much reality into any one value of A; read it into gauge-invariant combinations.