EMI & AC — Advanced Multi-Step Practice (Hard)

Quick Recap — Advanced EMI & AC (Multi-step)

• Rod on rails: $\varepsilon=BLv$, current $\dfrac{BLv}{R}$, force to keep it moving $\dfrac{B^2L^2v}{R}$, power dissipated $\dfrac{(BLv)^2}{R}$.
• LC oscillations: $\omega_0=\dfrac{1}{\sqrt{LC}}$; energy sloshes between $\tfrac12 LI^2$ and $\tfrac{q^2}{2C}$, total conserved.
• Series LCR numbers: $Z=\sqrt{R^2+(X_L-X_C)^2}$, $I_{rms}=\dfrac{V_{rms}}{Z}$, $P=I_{rms}^2R$, quality factor $Q=\dfrac{1}{R}\sqrt{\dfrac{L}{C}}$.
• Inductor combinations: series $L_1+L_2$; parallel $\dfrac{L_1L_2}{L_1+L_2}$ (no coupling).
• Induced charge: $q=\dfrac{N\,\Delta\Phi}{R}$; rotating-coil peak emf $NBA\omega$.

Worked mini-example. An LC circuit with $L=2$ H and $C=8\ \mu$F oscillates at $\omega_0=\dfrac{1}{\sqrt{LC}}=\dfrac{1}{\sqrt{16\times10^{-6}}}=250$ rad/s, i.e. $f_0=\dfrac{250}{2\pi}\approx40$ Hz.