Định luật Biot-Savard

Định luật Biot-Savard định nghỉa tương quan giửa Dòng điện và Từ trường . Định luật Biot-Savart Law cho rằng:

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

Với

$K_m=\frac{{\mu }_0}{4\pi }$, 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

Phương trình trên chỉ đúng với dòng điện ổn

Mật độ dòng điện

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

Với

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

$dv$ = is the differential unit of volume.

Dòng điện ổn

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

Điện tích điểm với vận tốc không đổi

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 approximately of the form:

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

This formula, however, is wrong. This is because a point charge moving in a straight line does not constitute a steady current nor a constant charge distribution, both of which are essential to magnetostatics. In particular, in this case, there is a changing electric field and an induced magnetic field that must be added to correct the above formula. Even though the equation is not precisely correct, it is a very good approximation (unreasonably good, in fact).

Cực nhỏ

$𝐇=ϵ𝐯\times 𝐄$

and hence,

$𝐁=𝐯\times \frac{1}{c^2}𝐄$