Key Equations

Mutual inductance by flux	$M=\frac{N_2{\text{Φ}}_{21}}{I_1}=\frac{N_1{\text{Φ}}_{12}}{I_2}$
Mutual inductance in circuits	${\epsilon }_1=\text{−}M\frac{d{I}_2}{dt}$
Self-inductance in terms of magnetic flux	$N{\text{Φ}}_{\text{m}}=LI$
Self-inductance in terms of emf	$\epsilon =\text{−}L\frac{dI}{dt}$
Self-inductance of a solenoid	$L_{\text{solenoid}}=\frac{{\mu }_0{N}^{2}A}{l}$
Self-inductance of a toroid	$L_{\text{toroid}}=\frac{{\mu }_0{N}^{2}h}{2\pi }\text{ln}\frac{R_2}{R_1}.$
Energy stored in an inductor	$U=\frac{1}{2}L{I}^2$
Current as a function of time for a RL circuit	$I(t)=\frac{\text{ε}}{R}(1-e^{\text{−}t\text{/}{\tau }_L})$
Time constant for a RL circuit	${\tau }_L=L\text{/}R$
Charge oscillation in LC circuits	$q\left(t\right)=q_0\text{cos}(\omega t+\varphi )$
Angular frequency in LC circuits	$\omega =\sqrt{\frac{1}{LC}}$
Current oscillations in LC circuits	$i(t)=\text{−}\omega q_0\text{sin}(\omega t+\varphi )$
Charge as a function of time in RLC circuit	$q(t)=q_0{e}^{\text{−}Rt\text{/}2L}\text{cos}(\omega \text{′}t+\varphi )$
Angular frequency in RLC circuit	$\omega \text{′}=\sqrt{\frac{1}{LC}-{\left(\frac{R}{2L}\right)}^2}$