# OpenStax University Physics/E&M/Inductance

## Chapter 14

#### Inductance

The SI unit for inductance is the Henry: 1H=1V·s/A ▭ Mutual inductance: ${\displaystyle M{\tfrac {dI_{2}}{dt}}=N_{1}{\tfrac {d\Phi _{12}}{dt}}=-\varepsilon _{1}}$  where ${\displaystyle \Phi _{12}}$  is the flux through 1 due to the current in 2 and ${\displaystyle \varepsilon _{1}}$  is the emf in 1. Likewise, it can be shownSEE TALK that, ${\displaystyle M{\tfrac {dI_{1}}{dt}}=-\varepsilon _{2}}$ .

▭ Self-inductance ${\displaystyle N\Phi _{m}=LI\rightarrow \varepsilon =-L{\tfrac {dI}{dt}}}$  ▭  ${\displaystyle L_{\text{solenoid}}\approx \mu _{0}N^{2}A\ell ,\,}$ ${\displaystyle L_{\text{toroid}}\approx {\tfrac {\mu _{0}N^{2}h}{2\pi }}\ln {\tfrac {R_{2}}{R_{1}}}.}$  Stored energy ${\displaystyle U={\tfrac {1}{2}}LI^{2}.}$  ▭ ${\displaystyle I(t)={\tfrac {\varepsilon }{R}}\left(1-e^{-t/\tau }\right)}$ is the current in an LR circuit where ${\displaystyle \tau =L/R}$  is the LR decay time.
▭ The capacitor's charge on an LC circuit ${\displaystyle q=q_{0}\cos(\omega t+\phi )}$  where ${\displaystyle \omega ={\sqrt {\tfrac {1}{LC}}}}$  is angular frequency
▭ LRC circuit ${\displaystyle q(t)=q_{0}e^{-Rt/2L}\cos(\omega 't+\phi )}$  where ${\displaystyle \omega '={\sqrt {{\tfrac {1}{LC}}+\left({\tfrac {R}{2L}}\right)^{2}}}}$

#### For quiz at QB/d_cp2.14

Unit of inductance = Henry (H)=1V·s/A

Mutual inductance: ${\displaystyle M{\tfrac {dI_{2}}{dt}}=N_{1}{\tfrac {d\Phi _{12}}{dt}}=-\varepsilon _{1}}$  where ${\displaystyle \Phi _{12}}$ =flux through 1 due to current in 2. Reciprocity${\displaystyle M{\tfrac {dI_{1}}{dt}}=-\varepsilon _{2}}$

Self-inductance: ${\displaystyle N\Phi _{m}=LI\rightarrow \varepsilon =-L{\tfrac {dI}{dt}}}$

${\displaystyle L_{\text{solenoid}}\approx \mu _{0}N^{2}A\ell }$ , ${\displaystyle L_{\text{toroid}}\approx {\tfrac {\mu _{0}N^{2}h}{2\pi }}\ln {\tfrac {R_{2}}{R_{1}}}}$ , Stored energy=${\displaystyle {\tfrac {1}{2}}LI^{2}}$

${\displaystyle I(t)={\tfrac {\varepsilon }{R}}\left(1-e^{-t/\tau }\right)}$  in LR circuit where ${\displaystyle \tau =L/R}$ .

${\displaystyle q(t)=q_{0}\cos(\omega t+\phi )}$  in LC circuit where ${\displaystyle \omega ={\sqrt {\tfrac {1}{LC}}}}$