# Materials Science and Engineering/Equations/Electronic Materials

## Electrons

### Conductivity and Ohm's Law

 ${\displaystyle E={\frac {U}{L}}}$

${\displaystyle U}$: applied potential difference
${\displaystyle L}$: length
${\displaystyle E}$: electric field
 ${\displaystyle a={\frac {e}{m}}E}$

 ${\displaystyle v_{\mbox{average}}=a\tau \;}$

${\displaystyle \tau }$: average time between collisions (mean free time, relaxation time, collision time)
 ${\displaystyle v_{D}=\left({\frac {e}{m}}\tau \right)E}$

${\displaystyle v_{D}}$: drift velocity
 ${\displaystyle J=N_{e}ev_{D}\;}$
${\displaystyle J={\frac {N_{e}e^{2}\tau }{m}}E}$
${\displaystyle J=\sigma E\;}$

${\displaystyle J}$: electric current density
${\displaystyle N_{e}}$: multiply the drift velocity by the density of electrons
${\displaystyle \sigma }$: electrical conductivity
 ${\displaystyle \sigma =\left({\frac {e}{m}}\tau \right)(N_{e}e)}$
${\displaystyle \sigma =\mu _{e}(N_{e}e)\;}$