Biot-Savart law

The Biot-Savart law describes the magnetic field of a steady line current:

B(r)=μ04π∫γI×r^r2 dl′=μ04πI∫γdI×r^r2 dl′\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4\pi}\int_{\gamma} \frac{\mathbf{I}\times \hat{\boldsymbol{\mathfrak{r}}}}{\mathfrak{r}^{2}}\ dl'=\frac{\mu_{0}}{4\pi}I\int_{\gamma} \frac{d\mathbf{I}\times \hat{\boldsymbol{\mathfrak{r}}}}{\mathfrak{r}^{2}}\ dl'

where integration is done over the current path γ\gamma, in the direction of the flow. dId\mathbf{I} is an element of length of the wire and r\boldsymbol{\mathfrak{r}} is the distance between the source element and the point r\mathbf{r}. μ0\mu_{0} is the Vacuum permeability. This is the magnetic analog of Coulomb's law, though since there are no "point currents", this law only has an integral form.

For a surface current density K\mathbf{K} it is

B(r)=μ04π∫K(r′)×r^r2 da′\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4\pi}\int \frac{\mathbf{K}(\mathbf{r}')\times \hat{\boldsymbol{\mathfrak{r}}}}{\mathfrak{r}^{2}}\ da'

and for a volume current density J\mathbf{J} it is

B(r)=μ04π∫J(r′)×r^r2 dτ′\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4\pi}\int \frac{\mathbf{J}(\mathbf{r}')\times \hat{\boldsymbol{\mathfrak{r}}}}{\mathfrak{r}^{2}}\ d\tau'

As with the electric field, the superposition principle applies.