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

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## 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 =$q\left({\vec {E}}+{\vec {v}}\times {\vec {B}}\right)$ 