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

## Chapter 11

#### Magnetic Forces and Fields

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

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

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

${\displaystyle |{\vec {a}}\times {\vec {b}}|}$ ${\displaystyle =ab\sin \theta \Leftrightarrow }$  ${\displaystyle ({\vec {a}}\times {\vec {b}})_{x}=(a_{y}b_{z}-a_{z}b_{y})}$ , ${\displaystyle ({\vec {a}}\times {\vec {b}})_{y}=(a_{z}b_{x}-a_{x}b_{z})}$ , ${\displaystyle ({\vec {a}}\times {\vec {b}})_{z}=(a_{x}b_{y}-a_{y}b_{x})}$
Magnetic force: ${\displaystyle {\vec {F}}=q{\vec {v}}\times {\vec {B}},\;}$ ${\displaystyle d{\vec {F}}=I{\overrightarrow {d\ell }}\times {\vec {B}}}$ .
${\displaystyle {\vec {v}}_{d}={\vec {E}}\times {\vec {B}}/B^{2}}$ =EXB drift velocity
Circular motion (uniform B field): ${\displaystyle r={\tfrac {mv}{qB}}.\;}$  Period=${\displaystyle T={\tfrac {2\pi m}{qB}}.\;}$

Dipole moment=${\displaystyle {\vec {\mu }}=NIA{\hat {n}}}$ . Torque=${\displaystyle {\vec {\tau }}={\vec {\mu }}\times {\vec {B}}}$ . Stored energy=${\displaystyle U={\vec {\mu }}\cdot {\vec {B}}}$ .
Hall field =${\displaystyle E=V/\ell =Bv_{d}={\tfrac {IB}{neA}}}$
Lorentz force =${\displaystyle q\left({\vec {E}}+{\vec {v}}\times {\vec {B}}\right)}$