 Solution

For steady currents in vacuum, the Biot-Savart law gives the magnetic field by summing oriented current elements. With $\mathbf{R}=\mathbf{r}-{\mathbf{r}}^{\prime }$ from the source to the observation point,For a thin wire it becomes $\mathbf{B}=({\mu }_{0}I/4\pi )\int d{\mathbf{l}}^{\prime }\times \hat{\mathbf{R}}/R^2$. The cross product fixes the direction by the right-hand rule. Fields from different steady circuits add by superposition; the entire current path, including return wires where relevant, belongs in the calculation.

An infinite straight wire gives circular field lines of magnitude $\boxed{B={\mu }_{0}I/(2\pi s)}$ at perpendicular distance $s$. This follows by integrating the transverse component of each element along the wire. A circular loop of radius $a$ has cancelling transverse components on its axis, leavingMultiple coincident turns multiply the field by their number. Far from the loop this is the field of a magnetic dipole of moment $\mathbf{\mu }=I\pi a^2\hat{\mathbf{n}}$. Integrating over many loops gives the approximately uniform interior field $B\simeq {\mu }_{0}nI$ of a long solenoid, with $n$ turns per unit length. Finite coils have fringing fields, and appropriately spaced coil pairs can produce a more uniform measurement region.

The law is useful for predicting fields of wires and coils where direct symmetry is insufficient, designing electromagnets and estimating stray fields. It is a magnetostatic law: steady charge flow satisfies charge continuity. An instantaneous application to rapidly changing currents neglects propagation and radiation; such problems require the time-dependent Maxwell's equations and retarded fields. A quasistatic approximation can be valid when circuit dimensions are small compared with the light-travel distance during the variation timescale.