LC Circuit


$$ \large L \frac{d^2 Q}{d t^2} + \frac{1}{C}Q = 0 $$

Time Constant for an L-C Circuit

$$ \large \omega_0 \ = \ \frac{1}{\sqrt{LC}} $$

Charge of an LC Circuit

$$ \large Q(t) \ = \ Q_0 \cos (\omega t + \phi) $$

where

  • $\omega = \frac{1}{\sqrt{LC}}$
  • $\phi$: phase (depends on the initial state of the circuit

Current of an LC Circuit

$$ \large I(t) \ = \ \frac{d Q}{d t} \ = - \omega \ I_0 \sin (\omega t + \phi) $$

Energy Stored in an LC Circuit

$$ \large U = U_C + U_L = \frac{Q^2_0}{2C} \cos^2 (\omega t + \phi) + \frac{L I^2_0}{2} \sin^2 (\omega t + \phi) $$

Vs. Simple Harmonic Motion

$$ \large m\frac{d^2 x}{d t^2} + kx = 0 $$$$ \large \begin{split} L & \quad \Leftrightarrow \quad & m \\ \\ Q & \quad \Leftrightarrow \quad & Q \\ \\ \frac{1}{C} & \quad \Leftrightarrow \quad & k \end{split} $$
  • GRE Physics GR0177: Problem 059

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