⚡ AC Voltage Applied to a Capacitor

A capacitor behaves very differently in direct-current (DC) and alternating-current (AC) circuits:

  • DC: current flows only while the plates charge; once fully charged, current drops to zero and the lamp (if any) goes dark. 📴:contentReference[oaicite:0]{index=0}
  • AC: the capacitor charges and discharges every half-cycle, so current keeps flowing and the lamp shines. 💡:contentReference[oaicite:1]{index=1}

1. Core Equations 🧮

For a sinusoidal source \(v = v_m \sin\!\omega t\):

  • Charge–voltage relation: \(q = C\,v\) (7.15):contentReference[oaicite:2]{index=2}
  • Current: \(i = \dfrac{dq}{dt} = i_m \sin\!\Bigl(\omega t + \dfrac{\pi}{2}\Bigr)\) (7.16):contentReference[oaicite:3]{index=3}
  • Peak current: \(i_m = \omega C\,v_m\) or \(i_m=\dfrac{v_m}{X_c}\):contentReference[oaicite:4]{index=4}

2. Capacitive Reactance \(X_c\) 🚧

The capacitor’s “effective resistance” in AC is called capacitive reactance:

\[ X_c \;=\; \frac{1}{\omega C}\qquad\text{(ohms)} \]:contentReference[oaicite:5]{index=5}

  • Lower frequency or smaller capacitance ⇒ larger \(X_c\) (harder for current to flow).:contentReference[oaicite:6]{index=6}
  • Higher frequency or bigger capacitance ⇒ smaller \(X_c\) (easier for current).📉:contentReference[oaicite:7]{index=7}

3. Phase Relationship ⏩

Current leads voltage by \(\dfrac{\pi}{2}\) (a quarter-cycle): the blue current wave peaks before the red voltage wave.🏁:contentReference[oaicite:8]{index=8}

4. Instantaneous & Average Power 🔋

Power delivered at any moment:

\[ p_c \;=\; i\,v \;=\; \tfrac{1}{2}\,i_m v_m \sin 2\omega t \]:contentReference[oaicite:9]{index=9}

The positive and negative halves cancel over a cycle, so

\[ \langle P\rangle = 0 \]

— the capacitor does not dissipate net energy; it just stores and releases it each cycle.🔄:contentReference[oaicite:10]{index=10}

5. Everyday Insight 🌟

  • Series lamp experiment: with AC the lamp glows; if you halve the capacitance, \(X_c\) doubles and the lamp dims.💡➡️💡:contentReference[oaicite:11]{index=11}
  • Numerical check (15 µF, 220 Vrms, 50 Hz): \(X_c≈212 \Omega\); \(I_{rms}≈1.04 A\); doubling the frequency halves \(X_c\) and doubles the current.📈:contentReference[oaicite:12]{index=12}

🎯 High-Yield NEET Nuggets

  1. \(X_c = \dfrac{1}{\omega C}\) — inversely proportional to both frequency and capacitance.:contentReference[oaicite:13]{index=13}
  2. In a pure capacitive circuit, current leads voltage by \(\dfrac{\pi}{2}\).:contentReference[oaicite:14]{index=14}
  3. Peak current formula \(i_m = \omega C v_m\) (or \(V_m/X_c\)).:contentReference[oaicite:15]{index=15}
  4. Average power over a cycle is zero — energy is stored and returned, not consumed.:contentReference[oaicite:16]{index=16}
  5. Doubling frequency ⇒ \(X_c\) halves ⇒ current doubles — a favorite quick-math item.⚡:contentReference[oaicite:17]{index=17}

Keep practicing — capacitors may “oppose” changes, but with these ideas you’re always ahead of the curve! 🚀