ECE 390 Final Notesheet Electrostatic → − → − Force on an Electric Charge: F Q1 = Q1 E Q2 Electrostatic in Materials (Maxwell’s Equations): ‹ − → − → → − D · ds = Qf ree ∇ · D = ρf ree ⇐⇒ ˛S → − ∇× E =0 ⇐⇒ − → − → E · dl = 0 (Gauss’ Law) (KVL) C → − → − → − D = o E + P Electrostatic in Linear Isotropic Materials (Dielectric): → − → − → − → − r = 1 + xe D = o r E P = o xe E → − − Dipole Moment: → p =Qd Q Capacitance: C = ∆Φ Point Charge Line Charge Plane Charge Q R̂ 4πR2 ρl R̂ 2πR ρs ẑ 2 Displacement Field → − Coul D m2 Arbitrary Condition ˆ ρl R̂ dl 4πR2 C ¨ ρs R̂ ds 4πR2 S ˚ ρ R̂ dv 4πR2 V ˆ Electric Potential J Φ∞ [V ] or C Q 4πo r R ˆ C ρl dl 4πo r R ¨ S C ¨ ρs ds 4πo r R S ˚ V Electric Potential: ˆP2 − → − → ∆Φ = − E · dl = Φ2 − Φ1 P1 ˆr Φ∞ = − ∞ → − E = −∇Φ − → − → E · dl = − ˆr Er dr ∞ ρl dl 4πo r R ρs ds 4πo r R ρ dv 4πo r R Parallel Plates: → − ρs ρs ∆Φ = d E = o o o A d C= Coaxial Cable: For a ≤ r ≤ b, → − Q E = r̂ 2πo rl Q ∆Φ = ln 2πo l C= Assume l >> r b a Q 2πo l = ∆Φ ln ab Parallel Wires: For 0 ≤ r ≤ d, Assume d >> a → − ρl ρl E = − + x̂ 2πr x 2πo (x − d) ρl d ∆Φ = ln πo a C ρl πo = = l ∆Φ ln ad l l ∆Φ = ρsh = A W I 1 Sheet resistivity: ρsh = σt Resistance: R = ρ Conductivity: → − → − → − dρ J = σE ∇· J =− dt ‹ I= − → − → J · ds τ = RC = S Steady State Conduction: → − ∇· J =0 ⇐⇒ ∇× → −! J =0 σ ‹ S ˛ ⇐⇒ C − → − → J · ds = 0 → −! → − J · dl = 0 σ (KCL) (KVL) o r σ Magnetostatic → − → − I dl × R̂ Biot-Savart Law: d H = → − 4π| R |2 Maxwell’s Equation: → − ∇· B =0 ⇐⇒ → − → − ∇×H = J ‹ ˛S ⇐⇒ → → − − − → With: B = µo H + M − → − → B · ds = 0 − → − → H · dl = I (Ampere’s Law) C In Free Space: − → → − p =0 M =0 → − → − → − → − B = µo H D = o E → − → − − Magnetic Dipole Moment: → m = I A = qm d Infinitely Long Line Current Sheet I φ̂ 2πR JS φ̂ 2 Magnetic Field → − A H m Solenoid JS φ̂ ' NI L Arbitrary Current − ˆ → I dl × R̂ → − dl 4π| R |2 C ¨ JS × R̂ → − ds 4π| R |2 S ˚ → − J × R̂ → − dv 4π| R |2 V Susceptibility and Permeability: − → → − M = xH → − → − B = µo µr H µr = 1 + x Inductance: Impedance: Φ N I r L Zo = C L= (where I = JS W for current through parallel plates) Symbols and Units for Basic Quantities Symbol Quantity Unit/Value Wb [T esla] or m2 C m2 V olts m − → B Magnetic flux density − → D Electric flux intensity, Displacement field − → E Electric field intensity − → F Force − → H Magnetic field intensity I Current − → J Current density − → JS Surface current density L Inductance − → M Magnetization vector − → m Magnetic dipole moment − → P Electric polarization vector − → p Electric dipole moment [C · m] Q Electric charge [C] qe Electron charge 1.602 × 10−19 C R Resistance [Ω] RS Surface resistance [Ω] Zo Characteristic impedance [Ω] a Bohr radius 5.2918 × 10−11 m o Permittivity of free space 8.854 × 10−12 r Relative permittivity [unitless] µo Permeability of free space 4π × 10−7 µr Relative permeability ρ Volume charge density ρs Surface charge density ρl Line charge density σ Conductivity [unitless] C m3 C m2 C m 1 Ω Φ Electrostatic potential [N ] Amps m [Amps] Amps m2 Amps m [H] Amps m Amps · m2 C m2 H m [V olts] or F m J C