# 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