# OpenStax University Physics/E&M/Sources of Magnetic Fields

## Chapter 12

#### Sources of Magnetic Fields

▭ Permeability of free space ${\displaystyle \mu _{0}=4\pi \times 10^{-7}}$ T·m/A
▭ Force between parallel wires ${\displaystyle {\tfrac {F}{\ell }}={\tfrac {\mu _{0}I_{1}I_{2}}{2\pi r}}}$

▭ Biot–Savart law ${\displaystyle {\vec {B}}={\tfrac {\mu _{0}}{4\pi }}\int \limits _{wire}{\frac {Id{\vec {\ell }}\times {\hat {r}}}{r^{2}}}}$

▭ Ampère's Law:${\displaystyle \oint {\vec {B}}\cdot d{\vec {\ell }}=\mu _{0}I_{enc}}$
▭ Magnetic field due to long straight wire ${\displaystyle B={\tfrac {\mu _{0}I}{2\pi R}}}$ ▭ At center of loop ${\displaystyle B={\tfrac {\mu _{0}I}{2R}}}$
▭ Inside a long thin solenoid ${\displaystyle B=\mu _{0}nI}$ where ${\displaystyle n=N/\ell }$ is the ratio of the number of turns to the solenoid's length.
▭ Inside a toroid ${\displaystyle B={\tfrac {\mu _{0}N}{2\pi r}}}$

▭ The magnetic field inside a solenoid filled with paramagnetic material is ${\displaystyle B=\mu nI}$ where ${\displaystyle \mu =(1+\chi )\mu _{0}}$ is the permeability

#### For quiz at QB/d_cp2.12

Free space permeability ${\displaystyle \mu _{0}=4\pi \times 10^{-7}}$ T·m/A
Force between parallel wires ${\displaystyle {\tfrac {F}{\ell }}={\tfrac {\mu _{0}I_{1}I_{2}}{2\pi r}}}$
Biot–Savart law ${\displaystyle {\vec {B}}={\tfrac {\mu _{0}}{4\pi }}\int \limits _{wire}{\frac {Id{\vec {\ell }}\times {\hat {r}}}{r^{2}}}}$
Ampère's Law:${\displaystyle \oint {\vec {B}}\cdot d{\vec {\ell }}=4\pi \mu _{0}I_{enc}}$
Magnetic field inside solenoid with paramagnetic material =${\displaystyle B=\mu nI}$ where ${\displaystyle \mu =(1+\chi )\mu _{0}}$= permeability