Formula Sheet Electromagnetics, Study Guides, Projects, Research of Electromagnetic Engineering

Download Formula Sheet Electromagnetics and more Study Guides, Projects, Research Electromagnetic Engineering in PDF only on Docsity! Formula Sheet Electromagnetics I (5EPA0) 1 Analysis 1.1 Vector calculus ~A · ~B = ~B · ~A, ~A · (~B× ~C) = ~B · (~C × ~A) = ~C · (~A× ~B), ~A× ~B =−(~B× ~A), ~A× (~B× ~C) = (~A · ~C)~B− (~A · ~B)~C . 1.2 Vector operators ∇Φ = ~ax ∂ ∂x Φ+ ~ay ∂ ∂ y Φ+ ~az ∂ ∂z Φ, ∇ · ~A= ∂ ∂x Ax + ∂ ∂ y Ay + ∂ ∂z Az, ∇× ~A= ~ax ~ay ~az ∂ ∂x ∂ ∂ y ∂ ∂z Ax Ay Az = ~ax( ∂ ∂ y Az − ∂ ∂z Ay) + ~ay( ∂ ∂z Ax − ∂ ∂x Az) +~az( ∂ ∂x Ay − ∂ ∂ y Ax), ∆Φ=∇2Φ =  ∂ 2 ∂x2 + ∂ 2 ∂ y2 + ∂ 2 ∂z2  Φ. 1.3 Vector identities ∇(Φ+Ψ) = ∇Φ+∇Ψ, ∇ · (~A+ ~B) = ∇ · ~A+∇ · ~B, ∇× (~A+ ~B) = ∇× ~A+∇× ~B. ∇(ΦΨ) = Φ∇Ψ+Ψ∇Φ, ∇ · (Φ~A) = Φ∇ · ~A+ ~A · ∇Φ, ∇× (Φ~A) = Φ∇× ~A− ~A×∇Φ. ∇(~A · ~B) = (~B · ∇)~A+ (~A · ∇)~B+ ~B× (∇× ~A) + ~A× (∇× ~B), ∇ · (~A× ~B) = ~B · (∇× ~A)− ~A · (∇× ~B), ∇× (~A× ~B) = (~B · ∇)~A− (~A · ∇)~B− ~B (∇ · ~A) + ~A(∇ · ~B). 1 ∇× (∇× ~A) = ∇(∇ · ~A)−∇2~A, ∇ · (∇× ~A) = 0, ∇× (∇Φ) = ~0. 2 Integral theorems ∫∫ V ∫ ∇ · ~AdV = ‡ S ~A · ~dS, (divergence/Gauss’s theorem), ∫ S ∫ ∇× ~A · ~dS = ∮ C ~A · ~dl, (Stokes’ theorem), where ~A is a vector field. 3 Circular cylindrical coordinates x z y O P ϕ ρ Coordinates: x = ρ cosϕ, y = ρ sinϕ, z = z. Ranges: ρ ≥ 0, 0≤ ϕ ≤ 2π. Line elements: dρ, ρdϕ, dz. Unit vectors expressed in cartesian components: ~aρ = cosϕ~ax + sinϕ~ay , ~aϕ = − sinϕ~ax + cosϕ~ay . 2