ECE 106 - Conductors / Capacitors

E field in conductors is =

ALWAYS 0

ΔV in conductors is =

ALWAYS 0

Equipotential surface inside conductors

Where are induced charges in conductors =

Induced charge resides ONLY on the surface; there is NO induced charge in the volume (Gauss's Law).

What is E field at surface of conductor =

ONLY perpendicular

E_tang = 0

Can positive and/or negatives charges move in conductors =

Positive charges are fixed

Negative charges can freely move, will settle on surface at equilibrium

A point charge +q is suspended in a hole inside a neutral conductor, what charges appear on the two surfaces?

−q on the inner (cavity) surface (induced from outside surface)

+q on the outer surface

E-field now exists outside conductor

Conductor boundary condition purpose and formula =

shows how E-field and induced charges change at the boundary of two mediums

E_n = ρ_s / ε₀

E_n = E normal on surface

ρ_s = surface charge density

Charge density vs. curvature on a conductor =

Smaller radius of curvature = larger surface charge density (ρ₁/ρ₂ = R₂/R₁ for connected spheres).

Why do lightning rods end in sharp points? =

Sharp = small radius of curvature → highest surface charge density there

When is surface charge density positive/negative

E_n into = ρ_s negative

out of conductor = ρ_s positive

A point charge sits above a grounded infinite conducting plane, how to find e field

Method of images: a fictitious −Q mirrored below the plane gives the same E-field ABOVE the plane

Potential on a conducting sphere due to an EXTERNAL point charge Q at distance d =

V(r = R) = V(r = 0) = kQ / d

(equidistant trick: integral of dQ over the surface = 0)

if sphere has initial charge q_net

V = [kQ] / [d] + [kq_net] / [R]

Capacitance =

C = Q / V;

Farad [F]

assumed to be isolated (only object)

amount of charge needed for a conducting body to increase potential by 1V

Capacitance of a two-conductor capacitor =

C = Q / ΔV_(2→1)

(depends ONLY on geometry and dielectric permittivity, not on Q or V)

Parallel plate capacitor =

C = εA / d = ε₀ε_r A / d

(area A, separation d, dielectric ε)

Energy stored in a capacitor =

U_e = [Q²] / [2C] = (1/2)CV²

Capacitance per unit length of a coaxial cable (inner a, outer b) =

C / L = [2πε] / [ln(b/a)]

Dielectrics in parallel (side by side) =

C = [ε₁A₁] / [d] + [ε₂A₂] / [d] = C₁ + C₂

(capacitors in parallel add)

Dielectrics in series (stacked layers d₁, d₂) =

1/C = [d₁] / [ε₁A] + [d₂] / [ε₂A] = 1/C₁ + 1/C₂

(capacitors reciprocal add in series)

Energy density of the electric field =

u_e = (1/2)D→ · E→

= (1/2)εE→ · E→

= (1/2)εE²

Moving plates with battery CONNECTED vs. DISCONNECTED =

battery connected → V fixed, use U = (1/2)CV²

battery removed → Q fixed, use U = Q²/(2C)

Capacitance of a spherical capacitor (concentric shells, inner a, outer b) =

C = 4πε₀ [ab] / [b − a]