= Biot-Savart law
{c}
{title2=$d\mathbf B=\frac{\mu_0I}{4\pi}\frac{d\boldsymbol\ell'\times\widehat{\mathbf R}}{R^2}$}
{wiki}
For a steady <current density> $J$, the <magnetic field> is
$$
B(r)=\frac{\mu_0}{4\pi}\int
\frac{J(r')\times(r-r')}{|r-r'|^3}\,d^3r'.
$$
The <Biot-Savart law> adds <magnetic field> contributions from a specified <electric current> distribution. The <vectors> $\mathbf r'$ and $\mathbf r$ locate source and observer; $\mathbf r-\mathbf r'$ points toward the observer. Here $\mu_0$ is <vacuum permeability>. For a thin wire carrying <electric current> $I$, the <volume integral> becomes a <line integral>:
$$
\mathbf B(\mathbf r)=\frac{\mu_0 I}{4\pi}
\int_C\frac{d\boldsymbol\ell'\times(\mathbf r-\mathbf r')}
{|\mathbf r-\mathbf r'|^3},\qquad
d\mathbf B=\frac{\mu_0}{4\pi}\frac{I\,d\boldsymbol\ell'\times\widehat{\mathbf R}}{R^2},
\quad \mathbf R=\mathbf r-\mathbf r'.
$$
Here $d\boldsymbol\ell'$ follows positive <conventional current>; $\widehat{\mathbf R}$ is the <unit vector> toward the observer. With fixed path orientation, reversing the <electric current> replaces $I$ by $-I$. The <cross product> and <right-hand rule> fix the direction. Reversing the source-to-observer displacement would give the wrong sign.<The Feynman Lectures on Physics, Volume II by Richard Feynman and others>[II.14–6, equations (14.37)–(14.38); II.14–7, equations (14.42)–(14.43), including the cross-product order]{ref}
The setting is <magnetostatics>: time-independent <current density> and <charge density>, in <vacuum> or with the vacuum approximation for air. <Electric charge conservation> then gives $\nabla\cdot\mathbf J=0$. A finite segment is part of a complete <electrical circuit>; its endpoints cannot continuously accumulate <electric charge> in a steady state. The instantaneous integral above does not describe switching transients or <electromagnetic radiation>. In magnetic matter, specify the total <current density>, including <magnetization current>, rather than automatically replacing $\mu_0$ by an arbitrary <magnetic permeability>.<The Feynman Lectures on Physics, Volume II by Richard Feynman and others>[II.13–4, equations (13.12)–(13.13), steady-current and closed-circuit discussion; II.13–5, internal currents in magnetic matter]{ref}
The <superposition principle> makes the <Biot-Savart law> useful even when the complete <electrical circuit> lacks the <symmetry> needed to evaluate <Ampère's circuital law> directly. Integrate a straight piece, a curved piece, and the return path separately, then add their <magnetic fields> as <vectors>. The beauty is that a local oriented <current element> produces both the circulating <magnetic field of a finite straight wire> and the reinforcing <on-axis magnetic field of a circular current loop>.
An actual laboratory procedure is documented in WPI Physics Department's https://users.wpi.edu/~kwu/Teaching/PH2651/biot-savart-law_MField_2017.pdf[Magnetic Field Mapping], §§1.2–1.2.3 and the single-coil checklist on p. 9. Preparation locates the active area of a <Hall effect> sensor using a small magnet. A field meter reads the component normal to that area. The wire apparatus is a wall-mounted square loop selectable between one and five conductors. The coil connects to a power supply and <ammeter>; its diameter and turn count are recorded. Align the sensor and scan distance. The control is a current-off background reading at each position, subtracted from the current-on reading. The inference compares corrected <magnetic field> with position and the predicted geometry dependence. Active-area displacement, sensor tilt, background variation, and finite coil width limit that comparison. These are instructions, not reported measurements.<Magnetic Field Mapping by WPI Physics Department>[pp. 2–5, §§1.2–1.2.3: preparation, sensor orientation, wire background subtraction and single-coil apparatus; p. 9, single-coil checklist]{ref}
A further proposed control is to reverse $I$ at fixed sensor position and orientation. If the background and meter offset stay unchanged, the odd part isolates the <magnetic field> from the <electric current>:
$$
\frac{B_{\rm read}(+I)-B_{\rm read}(-I)}2=B_{\rm current}.
$$
This is a deduction from the <superposition principle>, not a reported WPI measurement. The measured component, sensor position, and complete return path must still match the model.
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