Dielectrics and Capacitors

Dielectrics and Capacitors

1. Dielectric — What Changes

Dielectric = insulating material that polarizes in $E$ (bound charges shift, dipoles align) reducing net field:

where dielectric constant $κ$ (relative permittivity $ε_r$) >1 (e.g., glass 4, water 80, Si 11.7). Polarization $P = ε_0(κ-1)E$ creates bound surface charge that opposes $E$.

Displacement field

obeys $∯ D·n dS = Q_{free}$ (Gauss for dielectrics).

2. Capacitor with Dielectric

For parallel plates area $A$, separation $d$, vacuum capacitance

For dielectric fully filling:

Energy stored

So energy density

If battery connected ($V$ fixed), inserting dielectric $κ$ increases $Q=κ Q_0$, $C$ up, $U$ up. If isolated ($Q$ fixed), $V$ drops $V=V_0/κ$, $U$ reduces by $/κ$ (work done polarizing).

3. Types (exam familiarity)

Ceramic, electrolytic, film — but physics: polar vs non-polar dielectrics (permanent vs induced dipoles).

4. Worked Examples

(a) Parallel plate A=1 cm², d=1 mm, air C0=8.85e-12·1e-4/1e-3=0.885 pF; with glass κ=4 ⇒ C=3.54 pF ✓

(b) Battery 10 V, C0=1 pF ⇒ Q0=10 pC; insert κ=5 with battery → Q=50 pC, U0=50 pJ → U=250 pJ (battery does work)

(c) Isolated Q=10 pC, V0=10 V ⇒ V=2 V after κ=5, U drops 50→10 pJ

(d) Energy density at E=1e6 V/m in Si κ=11.7: u=0.5·11.7·8.85e-12·1e12≈51.8 J/m³ ✓

(e) Need to find polarization: P=ε0(κ-1)E =8.85e-12·4·1e6≈35.4 µC/m²

5. Exam Points

• Define $κ=ε/ε_0$, $D=εE$, $P$ — 3-5 marks, frequent.
• $C=κC_0$, $C_0=ε_0A/d$ — 5 marks, very frequent.
• Distinguish $V$ fixed vs $Q$ fixed when inserting dielectric — 5 marks, very frequent (many get wrong).
• Energy $U=Q²/2C$, density $u=1/2 εE²$ — 5 marks, frequent.