Circuits with Capacitors and Inductors - AP Physics C: Electricity and Magnetism

When capacitor is fully discharged. All energy stored in inductor's magnetic field.

Electrical to Magnetic and vice versa. Energy oscillates between electric field and magnetic field.

$f_0 = \frac{1}{2\pi\sqrt{LC}}$. Derived from setting $X_L = X_C$ at resonance.

Energy. Energy alternates between electric and magnetic forms.

$E = \frac{1}{2}CV^2 = 100 \text{ J}$. Use $E = \frac{1}{2}CV^2$ with $C = 2$ F, $V = 10$ V.

Current leads by $90^\circ$. Current flows before voltage builds up across capacitor.

Current lags by $90^\circ$. Current cannot change instantly through an inductor.

Capacitor. Electric field between capacitor plates stores electric energy.

$Q_{max} = \sqrt{2E/C}$. Maximum charge when all energy is electric.

Current is maximum. At resonance, impedance is minimum so current reaches maximum value.

$E = \frac{1}{2}LI^2 = 12.5 \text{ J}$. Use $E = \frac{1}{2}LI^2$ with $L = 1$ H, $I = 5$ A.

$I = \frac{dq}{dt}$. Current is the time derivative of charge.

$\omega L = \frac{1}{\omega C}$. Inductive and capacitive reactances are equal at resonance.

$\tau = \sqrt{LC}$. Characteristic time scale for LC oscillations.

$\omega = \sqrt{\frac{1}{LC}} = 1 \text{ rad/s}$. Use $\omega = \frac{1}{\sqrt{LC}}$ with given values.

$V_C = \frac{q}{C}$. Voltage is proportional to stored charge.

Decreases resonant frequency. Higher L makes $f_0 = \frac{1}{2\pi\sqrt{LC}}$ smaller.

$X_C = \frac{1}{\omega C}$. Capacitive reactance decreases with increasing frequency.

Zero. Voltage and current oscillate in phase at resonance.

It remains constant. No energy loss in ideal LC circuit, only conversion between forms.

Capacitor. Electric field between capacitor plates stores electric energy.

When current is zero. All energy stored in capacitor's electric field.

$E = \frac{1}{2}CV^2 + \frac{1}{2}LI^2$. Sum of electric energy in capacitor and magnetic energy in inductor.

$X_L = \omega L$. Inductive reactance increases with increasing frequency.

Zero. Equal and opposite reactances cancel at resonance.

Zero. Voltage and current are in phase when $X_L = X_C$.

Inductor. Magnetic field in inductor coil stores magnetic energy.

$q(t) = Q_0 \cos(\omega t + \phi)$. Cosine function represents oscillating charge with amplitude $Q_0$.

$V_L = L \frac{di}{dt}$. Voltage across inductor depends on rate of current change.