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:

[Biot-Savart Law]

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

Where:

$K_{m}={\frac {\mu _{0}}{4\pi }}\,$, 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

This formula is only true for a steady current, which means that electric charge is not building up anywhere. This is analogous to Coulombs law, which is only true for static charge distributions. When the current is not steady and charge distribution is changing, we have to add correction terms to the Biot-Savart Law. However, the Biot-Savart law is unreasonably robust, more so than Coulomb's Law. This is because most of the errors (the first-order error) will cancel out. Thus, Biot-Savart Law can be applied to cases that are clearly not steady; for example, household currents that alternate at 60 Hz.

Forms

General

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

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

where

$\mathbf {\hat {r}} ={\mathbf {r} \over r}$ 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

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

Point charge at constant velocity

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).

Microscopic Scale

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

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

and hence,

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