# OpenStax University Physics/E&M/Magnetic Forces and Fields

## Chapter 11

#### Magnetic Forces and Fields

▭ ${\vec {F}}=q{\vec {v}}\times {\vec {B}}$  is the force due to a magnetic field on a moving charge.
▭ For a current element oriented along ${\overrightarrow {d\ell }},\;d{\vec {F}}=I{\overrightarrow {d\ell }}\times {\vec {B}}$ .

▭ The SI unit for magnetic field is the Tesla: 1T=104 Gauss.
▭ Gyroradius $r={\tfrac {mB}{qB}}.\;$  Period $T={\tfrac {2\pi m}{qB}}.\;$
▭ Torque on current loop ${\vec {\tau }}={\vec {\mu }}\times {\vec {B}}$  where ${\vec {\mu }}=NIA{\hat {n}}$  is the dipole moment. Stored energy $U={\vec {\mu }}\cdot {\vec {B}}.$
▭ Drift velocity in crossed electric and magnetic fields $v_{d}={\tfrac {E}{B}}$
▭ Hall voltage = $V$  where the electric field is $E=V/\ell =Bv_{d}={\tfrac {IB}{neA}}$
▭ Charge-to-mass ratio $q/m={\tfrac {E}{BB_{0}r}}$  where the $E$  and $B$  fields are crossed and $E=0$  when the magnetic field is $B_{0}$

#### For quiz at QB/d_cp2.11

$|{\vec {a}}\times {\vec {b}}|$ $=ab\sin \theta \Leftrightarrow$  $({\vec {a}}\times {\vec {b}})_{x}=(a_{y}b_{z}-a_{z}b_{y})$ , $({\vec {a}}\times {\vec {b}})_{y}=(a_{z}b_{x}-a_{x}b_{z})$ , $({\vec {a}}\times {\vec {b}})_{z}=(a_{x}b_{y}-a_{y}b_{x})$
Magnetic force: ${\vec {F}}=q{\vec {v}}\times {\vec {B}},\;$ $d{\vec {F}}=I{\overrightarrow {d\ell }}\times {\vec {B}}$ .
${\vec {v}}_{d}={\vec {E}}\times {\vec {B}}/B^{2}$ =EXB drift velocity
Circular motion (uniform B field): $r={\tfrac {mv}{qB}}.\;$  Period=$T={\tfrac {2\pi m}{qB}}.\;$
Dipole moment=${\vec {\mu }}=NIA{\hat {n}}$ . Torque=${\vec {\tau }}={\vec {\mu }}\times {\vec {B}}$ . Stored energy=$U={\vec {\mu }}\cdot {\vec {B}}$ .
Hall field =$E=V/\ell =Bv_{d}={\tfrac {IB}{neA}}$
Lorentz force =$q\left({\vec {E}}+{\vec {v}}\times {\vec {B}}\right)$