# 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

cross product

${\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=${\displaystyle T={\tfrac {2\pi m}{qB}}.\;}$

Hall effect

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