Unit 2 Notes: Capacitors, Dielectric Materials, and Stored Electric Energy

Capacitor

Any two-conductor system that can store separated electric charge; placing +Q on one conductor and −Q on the other creates an electric field and a potential difference between them.

Capacitance (C)

The amount of charge separation a system can hold per volt of potential difference: C = Q/ΔV; depends on geometry and the material between conductors (ideal capacitor: independent of Q).

Farad (F)

SI unit of capacitance; 1 F = 1 C/V.

Potential difference (ΔV)

Voltage between the capacitor conductors (often taken as potential of the positive plate minus the negative plate).

Permittivity (ε)

Material property that controls electric field response in a medium; relates to dielectric behavior via ε = κε0.

Vacuum permittivity (ε0)

Permittivity of free space; numerical value ε0 = 8.85×10⁻¹² F/m (used in many capacitor formulas).

Equivalent capacitance (Ceq)

Single capacitance that replaces a network of capacitors while preserving the same external Q–ΔV behavior.

Capacitors in parallel

Connection where each capacitor has the same ΔV; total capacitance adds: Ceq = C1 + C2 + ··· (charges add).

Capacitors in series

Connection where each capacitor carries the same charge magnitude Q; reciprocals add: 1/Ceq = 1/C1 + 1/C2 + ··· (voltages add).

Parallel plate capacitor

Idealized capacitor with two large conducting plates of area A separated by distance d (with d small compared with plate size), giving approximately uniform field between plates.

Surface charge density (σ)

Charge per unit area on a plate: σ = Q/A (magnitude for a capacitor plate).

Electric field between capacitor plates (vacuum, ideal)

For two large oppositely charged plates: E = σ/ε0 = Q/(ε0A), approximately uniform between plates.

Single infinite sheet field (common pitfall)

Field magnitude from one infinite sheet: E = σ/(2ε0); not the correct between-plates field for an ideal parallel plate capacitor.

Uniform-field potential relation

For uniform field between plates separated by d: ΔV = Ed.

Parallel plate capacitance (vacuum)

Capacitance of an ideal vacuum-filled parallel plate capacitor: C = ε0A/d.

Geometry dependence of capacitance

For parallel plates: increasing A increases C; increasing separation d decreases C (since C ∝ A and C ∝ 1/d).

Edge (fringing) effects

Non-uniform electric field near plate edges; small when plate dimensions are large compared with separation d, enabling the uniform-field approximation.

Dielectric

Insulating material placed between conductors; polarizes in an electric field and increases capacitance while reducing the internal field for a given free charge.

Polarization

Slight separation/shift of positive and negative charge within dielectric molecules in response to an external electric field.

Bound charge

Charge that appears on dielectric surfaces due to polarization (e.g., bound negative near the positive plate and bound positive near the negative plate), producing a field that partially cancels the original field inside the dielectric.

Dielectric constant (κ) / relative permittivity

Dimensionless factor describing a dielectric: ε = κε0; if fully filling a capacitor, capacitance scales as C = κC0.

Battery-connected capacitor (fixed ΔV)

Scenario where the capacitor remains connected to a battery, so ΔV stays constant; inserting a dielectric increases C and therefore increases free charge Q = CΔV by factor κ.

Isolated capacitor (fixed Q)

Scenario where the capacitor is disconnected, so free charge Q stays constant; inserting a dielectric increases C and therefore decreases ΔV = Q/C by factor κ.

Energy stored in a capacitor (U)

Electric potential energy associated with separated charge/field; equivalent formulas: U = Q²/(2C) = (1/2)C(ΔV)² = (1/2)QΔV.

Energy density of an electric field (u)

Energy per unit volume stored in a field: u = (1/2)εE² (in vacuum, ε = ε0); for parallel plates, U = u(Ad).