and capacitance as active and passive elements

and capacitance as active and passive elements

1. Capacitance ($C$, farad $F$)

Capacitance = ability to store charge/energy in electric field between two conductors separated by dielectric:

where $Q$ = charge on positive plate, $V$ = voltage across.

(current = rate of change of voltage).

Energy stored:

in electric field.

• DC: $C$ is open at steady DC (no current if $dV/dt=0$); transient charging/discharging via $τ=RC$.
• AC: reactance

, current leads voltage by $90°$ (opposite to $L$).

Factors: parallel plate

(ε = κ ε0). So large area, small gap, high κ ⇒ large $C$.

2. Active vs Passive Classification (exam focus)

Element	Active or Passive?	Why	Can it generate net energy?
Resistance ($R$)	Passive	Absorbs ($I²R$) always, $p≥0$	No
Inductance ($L$)	Passive	Stores $½LI²$ in $B$ field, returns, no net generation (ideal $L$ no loss)	No (can transiently deliver stored energy, not generate)
Capacitance ($C$)	Passive	Stores $½CV²$ in $E$ field, returns	No
Voltage/Current Source	Active	Delivers power ($p<0$ for some time), can have $V$ independent of $I$	Yes
Transistor, Op-amp	Active	Can amplify (needs external supply)	Yes (controlled)

Key test: Passive satisfies Tellegen: $∯ p(t) dt ≥0$ over all time (cannot deliver net energy). Active violates. Some textbooks call ideal independent source active, dependent source active, R/L/C passive.

Note: $R,L,C$ are passive (even though $L,C$ can briefly act as source when discharging stored field, net over cycle zero).

3. Worked Examples

(a) Parallel plate A=1 cm²=1e-4, d=1 mm=1e-3, κ=4 (glass): C=8.85e-12·4·1e-4/1e-3=3.54 pF ✓
    At V=10 V, W=0.5·3.54e-12·100=1.77e-10 J ✓

(b) C=10 µF, dV/dt=1000 V/s ⇒ i=10e-6·1000=10 mA ✓

(c) X_C at f=50 Hz, C=10 µF: X_C=1/(2π·50·10e-6)=318 Ω ✓

(d) RC charging: 10kΩ·10µF τ=0.1 s, after 5τ considered charged.

(e) Which is active? Battery + resistor + inductor + capacitor circuit: only battery active.

4. Exam Points

• Define $C=Q/V$, $i=C dV/dt$, $W=½CV²$, $X_C=1/ωC$ — 5 marks, very frequent.
• Classify $R,L,C$ as passive, source as active — 5 marks, very frequent (common active/passive question).
• Give $C=εA/d$ — 5 marks, frequent.
• Show $R$ dissipates, $L/C$ store — 3-5 marks, frequent.