The Biot–Savart Law

Moving charges feel magnetic fields, and they also make them. The Biot–Savart law tells us the field produced by each little piece of a current.

The Biot–Savart law#

A current element i dl⃗ produces at a point P, located by the vector r⃗ drawn from the element to P, a field

where c is the speed of light. The constant 1/ε₀c² is written μ₀, the permeability of vacuum, equal to 4π × 10⁻⁷ T m/A:

dB⃗ is perpendicular to the plane containing the element and P. To find its sense, lay your right palm along dl⃗ with the fingers curling toward r⃗: the thumb points along dl⃗ × r⃗. When the diagram’s plane contains both, the field points into the page (⊗) on one side of the wire and out of it (⊙) on the other.

A straight wire#

Take a straight wire MN carrying current i, and a point P at perpendicular distance d from it (O is the foot of the perpendicular). Every element gives a field in the same direction (into the page, say), so we can just add magnitudes. With l = −d cot θ, dl = d cosec²θ dθ and r = d cosec θ:

On the perpendicular bisector of a segment of length a:

For a very long wire θ₁ → 0 and θ₂ → π:

Right-hand thumb rule: point your right thumb along the current; your curled fingers show the direction of B⃗ round the wire. The field lines of a long straight wire are circles centred on it.