Electrodynamics/Biot-Savart Law

Biot-Savart Law

The Biot-Savart law relates the magnetic B field to the distances and strengths of magnets in the field. In many respects, it's very similar to Coulomb's law, both in form and concept.

We can state the Biot-Savart Law as:

$d𝐁=K_m\frac{Id𝐥\times \widehat{𝐫}}{r^2}$

Where:

$K_m=\frac{{\mu }_0}{4\pi }$, where ${\mu }_0$ is the magnetic constant

$I$ is the current, measured in amperes

$d𝐥$ is the differential length vector of the current element

$\widehat{𝐫}$ is the unit displacement vector from the current element to the field point and

$r$ is the distance from the current element to the field point

Forms

General

In the magnetostatic approximation, the magnetic field can be determined if the current density j is known:

$𝐁=K_m\int \frac{𝐣\times \widehat{𝐫}}{r^2}dv$

where

$\widehat{𝐫}=\frac{𝐫}{r}$ is the unit vector in the direction of r.

$dv$ = is the differential unit of volume.

Constant uniform current

In the special case of a constant, uniform current I, the magnetic field B is

$𝐁=K_{m}I\int \frac{d𝐥\times \widehat{𝐫}}{r^2}$

Point charge at constant velocity

In the special case of a charged point particle $q$ moving at a constant velocity $𝐯$, then the equation above reduces to a magnetic field of the form:

$𝐁=K_m\frac{q𝐯\times \widehat{𝐫}}{r^2}$

Microscopic Scale

On the microscopic scale, the Biot-Savart law becomes,

$𝐇=ϵ𝐯\times 𝐄$

and hence,

$𝐁=𝐯\times \frac{1}{c^2}𝐄$