Biot-Savart law

Also called Biot Savart law

The rule giving the magnetic field contributed by a single short segment of current-carrying wire. Adding up those contributions along the whole wire gives the field, whatever the shape of the wire.

d\vec{B} = \frac{\mu_0}{4\pi}\frac{I\, d\vec{\ell} \times \hat{r}}{r^2}

AP Physics C: Electricity and Magnetism, topic 12.3, and printed on that course's equation sheet. Read the pieces: an inverse square in the distance, a cross product fixing the direction, and a current element $I\, d\vec{\ell}$ standing in for the source.

It is the general tool. Ampere's law is faster wherever the geometry is symmetric, and silent everywhere else. Biot-Savart makes no symmetry demand, so it handles a finite straight segment, an arc, or a point off the axis of a loop, at the price of an integral you have to set up yourself.

The cross product means the field around a small segment is tangent to circles centred on the wire, with no component toward it, away from it, or parallel to it. It also means a point lying along the line of the segment gets nothing: $d\vec{\ell}$ and $\hat{r}$ are then parallel and the cross product vanishes.

The CED prints one derived result, the field at the centre of a circular loop:

B_{\text{centre of loop}} = \frac{\mu_0 I}{2R}

Its boundary statement limits quantitative work to a few cases, naming a point on the perpendicular bisector of a straight conductor, a point on the central axis of a circular loop, and the centre of a segment of a circular loop. Nothing here appears in AP Physics 2, which computes only the long straight wire.

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