#
Formula: **Biot-Savart Law for a Thin Wire**

$$\class{violet}{\boldsymbol{B}}(\boldsymbol{r}) ~=~ \frac{\mu_0 \, \class{red}{I}}{4\pi} \int_{S} \frac{\boldsymbol{r}-\boldsymbol{R}}{|\boldsymbol{r}-\boldsymbol{R}|^3} \times \text{d}\boldsymbol{s}$$
$$\class{violet}{\boldsymbol{B}}(\boldsymbol{r}) ~=~ \frac{\mu_0 \, \class{red}{I}}{4\pi} \int_{S} \frac{\boldsymbol{r}-\boldsymbol{R}}{|\boldsymbol{r}-\boldsymbol{R}|^3} \times \text{d}\boldsymbol{s}$$

## Magnetic field

`$$ \class{violet}{\boldsymbol{B}}(\boldsymbol{r}) $$`Unit

`$$ \mathrm{T} $$`

Magnetic flux density tells how strong the magnetic field is at the location \( \boldsymbol{r} \) generated by a steady-state current \(I\) through the conductor.

## Position vector to field point

`$$ \boldsymbol{r} $$`Unit

`$$ \mathrm{m} $$`

Position vector from the coordinate origin to any point in space at which the magnetic field is to be calculated.

## Position vector

`$$ \boldsymbol{R} $$`Unit

`$$ \mathrm{m} $$`

Location vector points from the coordinate origin to the infinitesimal conductor element \(\text{d}\boldsymbol{s}\).

Here \(\boldsymbol{r} - \boldsymbol{R}\) is the connection vector pointing from the infinitesimal conductor element \(\text{d}\boldsymbol{s}\) to the field point. \(|\boldsymbol{r} - \boldsymbol{R}|\) is the distance of the infinitesimal conductor element \(\text{d}\boldsymbol{s}\) to the field point.

## Electric current

`$$ \class{red}{\boldsymbol I} $$`Unit

`$$ \mathrm{A} $$`

Constant electric current inside the conductor.

## Conductor line

`$$ S $$`

The conductor through which the current flows.

Here \(\text{d}\boldsymbol{s}\) is an infinitesimal length element. This length element runs along the conductor.

## Vacuum permeability

`$$ \mu_0 $$`Unit

`$$ \frac{\mathrm{Vs}}{\mathrm{Am}} = \frac{ \mathrm{kg} \, \mathrm{m} }{ \mathrm{A}^2 \, \mathrm{s}^2 } $$`

The vacuum permeability is a physical constant and has the following experimentally determined value:

`$$ \mu_0 ~=~ 1.256 \, 637 \, 062 \, 12 ~\cdot~ 10^{-6} \, \frac{\mathrm{Vs}}{\mathrm{Am}} $$`