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

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## Chapter 12

#### Sources of Magnetic Fields

▭ Permeability of free space $\mu _{0}=4\pi \times 10^{-7}$ T·m/A
▭ Force between parallel wires ${\tfrac {F}{\ell }}={\tfrac {\mu _{0}I_{1}I_{2}}{2\pi r}}$ ▭ Biot–Savart law ${\vec {B}}={\tfrac {\mu _{0}}{4\pi }}\int \limits _{wire}{\frac {Id{\vec {\ell }}\times {\hat {r}}}{r^{2}}}$ ▭ Ampère's Law:$\oint {\vec {B}}\cdot d{\vec {\ell }}=\mu _{0}I_{enc}$ ▭ Magnetic field due to long straight wire $B={\tfrac {\mu _{0}I}{2\pi R}}$ ▭ At center of loop $B={\tfrac {\mu _{0}I}{2R}}$ ▭ Inside a long thin solenoid $B=\mu _{0}nI$ where $n=N/\ell$ is the ratio of the number of turns to the solenoid's length.
▭ Inside a toroid $B={\tfrac {\mu _{0}N}{2\pi r}}$ ▭ The magnetic field inside a solenoid filled with paramagnetic material is $B=\mu nI$ where $\mu =(1+\chi )\mu _{0}$ is the permeability

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

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