Capacitance and Dielectrics — Notes

• In electrostatic equilibrium: E = 0 inside a conductor; excess charge resides on the outer surface; the conductor is an equipotential; just outside, E is normal to the surface. Free charges move through conductors; bound charges in insulators only shift within molecules.
• Capacitance $C = Q/V$; unit farad (C V⁻¹); μF, nF, pF in practice. Depends on geometry and medium, not on Q.
• Isolated sphere: $C = 4\pi\varepsilon_0R$ (R = 9 cm gives 10 pF; the Earth about 711 μF).

Plates of area A, separation d, charges ±Q: uniform field $E = \sigma/\varepsilon_0 = Q/(\varepsilon_0A)$; $V = Ed = Qd/(\varepsilon_0A)$; so $C = \dfrac{\varepsilon_0A}{d}$ (edge effects neglected).

• Series: same Q on each; $V = \sum V_i$; $\dfrac{1}{C_s} = \sum \dfrac{1}{C_i}$; smaller than the smallest; the smallest C takes the largest V.
• Parallel: same V; $Q = \sum Q_i$; $C_p = \sum C_i$; larger than the largest.
• Mixed networks: reduce step by step; find Q on series branches and V across parallel ones.

• $U = \dfrac12CV^2 = \dfrac12QV = \dfrac{Q^2}{2C}$ (area under the V–Q line; formulae used, not derived, in the ISC scope).
• Energy density of the field: $u = \dfrac12\varepsilon_0E^2$ (J m⁻³).
• Connecting a charged $C_1$ (at $V_1$) to an uncharged $C_2$: charge conserved, common $V = C_1V_1/(C_1 + C_2)$; energy is lost as heat.

• In a field a dielectric polarises (polar molecules align; non-polar molecules become induced dipoles); bound surface charges appear (− facing the + plate) and reduce the field inside to $E_0/K$.
• Dielectric constant $K = C'/C_0 = \varepsilon/\varepsilon_0 = \varepsilon_r$; filled capacitor $C' = K\varepsilon_0A/d$.
• Battery disconnected (Q fixed): C × K; V, E, U ÷ K. Battery connected (V fixed): C, Q, U × K; E unchanged.

Slab of thickness t (t < d), constant K: $V = E_0(d - t) + \dfrac{E_0}{K}t$, so $C = \dfrac{\varepsilon_0A}{d - t + t/K}$. t = 0 gives $C_0$; t = d gives $KC_0$; a metal slab (K → ∞) gives $\varepsilon_0A/(d - t)$.