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

## 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