Biot–Savart fields and current geometry

Let the source position be $\mathbf r'$ and the observation position be $\mathbf r$. The separation points from source to observer. Crossing the source current with this separation fixes the field direction; reversing either vector reverses the contribution. Current density has units of amperes per square metre. Integrating its contributions over a volume gives teslas, rather than a force or a potential. In a homogeneous linear medium a modified permeability can sometimes be used, but interfaces generally require solving the material boundary problem.

A stationary current distribution obeys $\nabla\cdot\mathbf J=0$: charge cannot accumulate anywhere. A finite straight segment therefore belongs to a larger closed circuit or to a specified feed arrangement. Its integral is a useful contribution, but claiming it is the entire field silently discards the return path. Superposition is vector addition at the same observation point. Separate contributions can cancel even when every wire carries a large current.

For a thin circular loop, choose an observation point on its symmetry axis. Opposite wire elements produce transverse components of equal magnitude and opposite direction. Their axial components reinforce. The distance to every source element is the same, but projecting the cross product supplies an extra factor of loop radius. This is why simply multiplying the magnitude of one element by the circumference gives an incorrect axial result.

At the centre the axial field scales inversely with loop radius for fixed current. Far along the axis it instead decreases as the inverse cube of distance, as expected for a localized magnetic dipole. These limits are powerful checks on the integration. The infinite-wire inverse-distance law describes a different geometry and cannot be used for a distant localized loop merely because both sources happen to be made of wire.

For sources varying on timescale T, let a be the largest relevant source-to-observer separation or apparatus dimension. A necessary near-zone condition is $a/(cT)\ll1$. A magnetoquasistatic treatment also requires displacement-current effects and radiation to be negligible at the requested accuracy. The current pattern must remain approximately divergence free; rapid charging of an interrupted wire is a warning that this approximation is inappropriate. Replacing a steady current by its value at the present time is not an exact solution of the full time-dependent Maxwell equations.

The thin-wire idealization has its own independent restriction: the observation distance should greatly exceed the conductor radius if its cross section is being ignored. Its apparent divergence on the filament does not predict an infinite field inside a real wire. Retardation, conductor thickness, and material response are separate issues. State which have been neglected before interpreting a close-range measurement as a failure of the underlying field equations.