### Key Equations

Permeability of free space | ${\mu}_{0}=4\pi \phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-7}}\text{T}\cdot \text{m/A}$ |

Contribution to magnetic field from a current element |
$dB=\frac{{\mu}_{0}}{4\pi}\phantom{\rule{0.2em}{0ex}}\frac{I\phantom{\rule{0.2em}{0ex}}dl\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}\theta}{{r}^{2}}$ |

Biot–Savart law | $\overrightarrow{B}=\frac{{\mu}_{0}}{4\pi}{\displaystyle \underset{\text{wire}}{\int}\frac{Id\overrightarrow{l}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\widehat{r}}{{r}^{2}}}$ |

Magnetic field due to a long straight wire |
$B=\frac{{\mu}_{0}I}{2\pi R}$ |

Force between two parallel currents | $\frac{F}{l}=\frac{{\mu}_{0}{I}_{1}{I}_{2}}{2\pi r}$ |

Magnetic field of a current loop | $B=\frac{{\mu}_{0}I}{2R}\phantom{\rule{0.2em}{0ex}}\text{(at center of loop)}$ |

Ampère’s law | $\oint \overrightarrow{B}}\xb7d\overrightarrow{l}={\mu}_{0}I$ |

Magnetic field strength inside a solenoid |
$B={\mu}_{0}nI$ |

Magnetic field strength inside a toroid | $B=\frac{{\mu}_{o}NI}{2\pi r}$ |

Magnetic permeability | $\mu =(1+\chi ){\mu}_{0}$ |

Magnetic field of a solenoid filled with paramagnetic material |
$B=\mu nI$ |