When conductors reach electrostatic equilibrium:

Properties:

• All excess charge resides on surface
• Charge distributes to make interior E = 0
• Charge density varies with curvature (higher at sharp points)
• Entire conductor is equipotential (V = constant everywhere)

Charge density on curved surface:

• Higher where surface curvature is higher
• Maximum at sharp points
• Minimum on flat or smoothly curved regions

Hollow conductor shields interior from external electric fields.

Properties:

• E = 0 inside hollow conductor
• External fields terminated by charges on outer surface
• Interior can have charge without affecting exterior (if grounded)

Applications:

• Sensitive equipment protection
• Lightning protection
• Electromagnetic shielding (RF)

Cavity within conductor:

• If cavity empty: E = 0 throughout cavity
• If cavity contains charge Q: inner surface has -Q, outer surface has +Q
• Induced charges on inner and outer surfaces

$\vec{E}_{inside} = 0$

Why:

• Free electrons in conductor can move
• Any E-field causes charge redistribution
• Charges move until E = 0 everywhere inside

Consequences:

• No net force on charges inside
• Cannot store electric field inside conductor
• Charge resides entirely on surface

Excess charge distributes on outer surface only.

Distribution determined by:

• Making interior E = 0
• Minimizing potential energy
• Surface geometry (curvature affects density)

At surface of charged conductor:

• E = 0 just inside
• E = $\sigma/\varepsilon_0$ just outside (perpendicular to surface)
• Discontinuous at surface due to surface charge

Entire conductor is equipotential in electrostatic equilibrium.

$V = \text{constant throughout conductor}$

Why:

• E = 0 inside -> no potential changes inside
• Potential continuous across surface
• All points accessible via paths through conductor

Conductor shields its interior from external fields.

Mechanism:

• External field induces charge redistribution
• Induced charges create canceling field inside
• Net field inside = 0

Applications:

• Faraday cages
• Coaxial cables (shield)
• Grounding (connect to Earth = V = 0 reference)

Capacitance measures ability to store charge for given potential difference.

$C = \frac{Q}{V}$

Where:

• Q = magnitude of charge on each plate
• V = potential difference between plates

Units: Farads (F) = Coulombs/Volt (C/V)

Physical meaning: charge stored per volt

For parallel plate capacitor with vacuum between plates:

$C = \frac{\varepsilon_0 A}{d}$

Where:

• $\varepsilon_0$ = 8.85 × 10-12 C2/(N - m2)
• A = plate area (m2)
• d = plate separation (m)

With dielectric of constant κ: $C = \kappa\frac{\varepsilon_0 A}{d}$

Field between plates: $E = \frac{\sigma}{\varepsilon_0} = \frac{V}{d}$

Two concentric spherical shells (radii R1 < R2):

$C = 4\pi\varepsilon_0 \frac{R_1 R_2}{R_2 - R_1}$

Where:

• R1 = inner sphere radius
• R2 = outer sphere radius

For R2 -> ∞ (isolated sphere): $C = 4\pi\varepsilon_0 R_1$

Two coaxial cylinders (radii a < b, length L):

$C = \frac{2\pi\varepsilon_0 L}{\ln(b/a)}$

Where:

• a = inner cylinder radius
• b = outer cylinder radius
• L = length of cylinders

Energy stored in electric field of capacitor:

$U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$

Derivation from charging: $U = \int_0^Q V\,dq = \int_0^Q \frac{q}{C}\,dq = \frac{Q^2}{2C}$

Energy density in electric field: $u = \frac{U}{Volume} = \frac{1}{2}\varepsilon_0 E^2$

For parallel plate capacitor: $U = \frac{1}{2}\varepsilon_0 E^2 \cdot Ad = \frac{1}{2}CV^2$

Dielectric polarization induces bound surface charge.

Bound charge density: $\sigma_b = \left(1 - \frac{1}{\kappa}\right)\sigma$

Where σ = free charge density on conductor

Effects:

• Induced charge partially cancels field of free charges
• Net field inside dielectric reduced
• Requires more free charge for given field

Polarization P: $P = \varepsilon_0(\kappa - 1)E$

Related to induced charge density.

(relative permittivity) κ measures material's effect on capacitance.

$C = \kappa C_0$

Where C0 = capacitance without dielectric

Values:

• Vacuum: κ = 1
• Air: κ ≈ 1.0006
• Paper: κ ≈ 3.7
• Glass: κ ≈ 5-10
• Water: κ ≈ 80

Physical origin: polarization of material by electric field.

Capacitance (with dielectric): $C = \kappa C_0$

Capacitance increases by factor κ.

Electric field (with dielectric): $E = \frac{E_0}{\kappa}$

Field reduced by factor κ (intrinsic field cancelled by induced polarization).

Voltage (for fixed charge Q): $V = \frac{V_0}{\kappa}$

Voltage decreases by factor κ.

Stored energy (for fixed charge): $U = \frac{U_0}{\kappa}$