ECE 390 Final Notesheet Electrostatic − → − → Force on an Electric Charge: F Q1 = Q1 E Q2 Electrostatic in Materials (Maxwell’s Equations): ‹ − − → ρ − → → Q ∇· E = ⇐⇒ E · ds = enc Eo Eo S ‹ − → − → → − ∇ · D = ρf ree ⇐⇒ D · ds = Qf ree → − ∇× E =0 ˛S ⇐⇒ → − − → → − D = Eo E + P → → − − E · dl = 0 (Gauss’ Law) (Gauss’ Law) (KVL) C Electrostatic in Linear Isotropic Materials (Dielectric): − → − → → − − → P = Eo x e E D = Eo Er E Er = 1 + x e − → → Dipole Moment: − p =Qd Q Capacitance: C = ∆Φ Electric Field − V → E m Point Charge Line Charge Plane Charge Q R̂ 4πEo Er R2 ρl R̂ 2πEo Er R ρs ẑ 2Eo Er Arbitrary Condition ˆ ρl R̂ dl 4πEo Er R2 C ¨ ρs R̂ ds 4πEo Er R2 S ˚ ρ R̂ dv 4πEo Er R2 V Displacement Field − C oul → D m2 ρs ẑ 2 ρl R̂ 2πR Q R̂ 4πR2 ˆ C ¨ S ˚ ρl R̂ dl 4πR2 ρs R̂ ds 4πR2 ρ R̂ dv 4πR2 V ˆ Electric Potential J Φ∞ [V ] or C Q 4πEo Er R ˆ C ρl dl 4πEo Er R ¨ S ρs ds 4πEo Er R C ¨ S ˚ V ρl dl 4πEo Er R ρs ds 4πEo Er R ρ dv 4πEo Er R Electric Potential: ˆP2 − → → − ∆Φ = − E · dl = Φ2 − Φ1 P1 ˆr Φ∞ = − → − − → E · dl = − ∞ ˆr Er dr ∞ → − E = −∇Φ Parallel Plates: − → ρ ρs E = s ∆Φ = d Eo Eo Eo A d C= Coaxial Cable: For a ≤ r ≤ b, Assume l >> r → − E = Q r̂ 2πEo rl b Q ∆Φ = ln a 2πEo l 2πEo l Q C= = ∆Φ ln ab Parallel Wires: For 0 ≤ r ≤ d, Assume d >> a → − ρl ρl E = − + x̂ 2πEr x 2πEo (x − d) ∆Φ = ρl πEo ln d a πEo C = ρl = l ∆Φ ln d a 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 σ ‹ S =0 ⇐⇒ ˛ C → − − → J · ds = 0 → − J σ − → · dl = 0 (KCL) (KVL) Eo Er σ Magnetostatic − → ˆ − → I dl × R Biot-Savart Law: d H = − 2 → 4π| R | Maxwell’s Equation: − → ∇· B =0 ⇐⇒ → → − − ∇×H = J ‹ ˛S ⇐⇒ → − − → B · ds = 0 → − − → H · dl = I (Ampere’s Law) C (→ → − − − →\ B = µo H + M → − − → → − With: D = Eo E + P In Free Space: − → − → p =0 M =0 − → − → − → − → D = Eo E B = µo H − → − → → 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 A − 2 ds m → 4π| R | S ˚ → − J × R̂ A − 2 dv m2 → 4π| R | V Susceptibility and Permeability: − → − → M = xH − → − → B = µo µr H µr = 1 + x Inductance: L = Impedance: Zo = Φ N I (where I = JS W for current through parallel plates) L C Magnetic Circuits: (Assume flux is uniform and contained inside permeable material) ˛ − → → − H · dl = N I φ B = µo µr H = A N I = φ� l �= µ oµ r A φ: flux �: reluctance A: cross sectional area Symbols and Units for Basic Quantities Symbol Quantity Unit/Value − → B Magnetic flux density [T esla] or − → D Electric flux intensity, Displacement field C m2 − → E Electric field intensity − → F Force − → H Magnetic field intensity I Current − → J Current density − → JS Surface current density L Inductance − → M Magnetization vector Amps m − → m Magnetic dipole moment Amps m2 − → P Electric polarization vector Wb m2 V olts m [N ] Amps m [Amps] Amps m2 Amps m [H ] C m2 − → p Electric dipole moment Q Electric charge [C ] qe Electron charge 1.602 × 10 R Resistance [Ω] RS Surface resistance [Ω] Zo Characteristic impedance [Ω] a Bohr radius 5.2918 × 10 Eo Permittivity of free space 8.854 × 10 Er Relative permittivity [unitless] µo Permeability of free space 4π × 10 µr Relative permeability [unitless] ρ Volume charge density ρs Surface charge density C m2 ρl Line charge density C m σ Conductivity [S] Φ Electrostatic potential [V olts] or C m −19 C −11 −11 m F m −7 H m C m3 J C