Biot-Savart law

• $\vec{\mathbf{B}}=\frac{{\mu }_0}{4\pi }\frac{q\vec{v}\times \hat{r}}{r^2}$ is the magnetic field (in $\mathrm{T}$) at distance $r$ from a point charge $q$ moving with velocity $\vec{v}$ in the direction $\hat{r}$. $\vec{\mathbf{B}}$ is perpendicular to both $\vec{v}$ and $\hat{r}$.
  • (superposition) The total magnetic field caused by multiple point charges is the vector sum of the magnetic fields caused by each individual charge.
• $\vec{\mathbf{B}}=\frac{{\mu }_0}{4\pi }{\oint }_{\mathcal{C}}\frac{Id\vec{\mathbf{\ell }}\times \hat{\mathbf{r}}}{r^2}$
• $B={\mu }_{0}nI$ is the magnetic field inside a long solenoid with $n$ turns per unit length ($n=\frac{N}{L}$) carrying current $I$.
• $B=\frac{{\mu }_{0}I}{2\pi r}$ is the magnitude of magnetic field at distance $r$ from a long straight wire carrying current $I$.
• $B=\frac{{\mu }_{0}I}{2R}$ is the magnitude of magnetic field at the center of a circular loop of radius $R$ carrying current $I$.
• $B={\mu }_{0}nI$ is the magnetic field inside of a long solenoid with $n$ turns per unit length ($n=\frac{N}{L}$) carrying current $I$.

see also: