OpenStax University Physics/E&M/Inductance

${\mathcal {E}}\&{\mathcal {M}}$ (Vol. 2): 5:Electric Charges and Fields 6:Gauss's Law 7:Electric Potential 8:Capacitance 9:Current and Resistance 10:Direct-Current Circuits 11:Magnetic Forces and Fields 12:Sources of Magnetic Fields 13:Electromagnetic Induction 14:Inductance 15:Alternating-Current Circuits 16:Electromagnetic Waves

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Chapter 14

Inductance

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

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

For quiz at QB/d_cp2.14

Unit of inductance = Henry (H)=1V·s/A

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

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

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

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

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

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