# Magnetic Field and Magnetic Forces

• Magnetic forces: $\vec{F}=q\vec{v}\times\vec{B}$.
• Gauss’s law for magnetism: $\oint\vec{B}\cdot d\vec{A}=0$.
• Magnetic force on a conductor: $d\vec{F}=Id\vec{l}\times\vec{B}$.
• Magnetic torque: $\vec{\tau}=\vec{\mu}\times\vec{B}$, $\vec{\mu}=I\vec{A}$.  $U=-\vec{\mu}\cdot\vec{B}$.
• Magnetic field of a moving charge: $\vec{B}=\frac{\mu_0}{4\pi}\frac{q\vec{v}\times\hat{\vec{r}}}{r^2}$.
• The law of Biot and Savart: $d\vec{B}=\frac{\mu_0}{4\pi}\frac{Id\vec{l}\times\hat{\vec{r}}}{r^2}$.
• Magnetic field of a long, straight, current-carrying conductors: $\vec{B}=\frac{\mu_0 I}{2\pi r}$.
• Magnetic force between two long, parallel, current-carrying conductors: $\frac{F}{L}=\frac{\mu_0 I I\prime}{2\pi r}$.
• Magnetic field at the center of a current loop: $B=\frac{\mu_0 I}{2r}$.
• Long solenoid that has $n$ turns per unit length: $B=\mu_0 n I$.
• Ampere’s law: $\oint\vec{B}\cdot d\vec{l}=\mu_0 I_{encl}$.