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:

$d\mathbf {B} =K_{m}{\frac {Id\mathbf {l} \times \mathbf {\hat {r}} }{r^{2}}}$

Với

$K_{m}={\frac {\mu _{0}}{4\pi }}\,$, where $\mu _{0}$ is the magnetic constant

$I\mathbf {}$ is the current, measured in amperes

$d\mathbf {l}$ is the differential length vector of the current element

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

$r\mathbf {}$ is the distance from the current element to the field point

Phương trình trên chỉ đúng với dòng điện ổn

Mật độ dòng điện

$\mathbf {B} =K_{m}\int {{\frac {\mathbf {j} \times \mathbf {\hat {r}} }{r^{2}}}dv}$

Với

$\mathbf {\hat {r}} ={\mathbf {r} \over r}$ is the unit vector in the direction of r.

$dv$ = is the differential unit of volume.

Dòng điện ổn

$\mathbf {B} =K_{m}I\int {\frac {d\mathbf {l} \times \mathbf {\hat {r}} }{r^{2}}}$

Điện tích điểm với vận tốc không đổi

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

$\mathbf {B} =K_{m}{\frac {q\mathbf {v} \times \mathbf {\hat {r}} }{r^{2}}}$

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

$\mathbf {H} =\epsilon \mathbf {v} \times \mathbf {E}$

and hence,

$\mathbf {B} =\mathbf {v} \times {\frac {1}{c^{2}}}\mathbf {E}$