Equations booklet for EMT, Cheat Sheet of Electromagnetism and Electromagnetic Fields Theory

Download Equations booklet for EMT and more Cheat Sheet Electromagnetism and Electromagnetic Fields Theory in PDF only on Docsity! PHAS0038 BACKGROUND INFORMATION FOR EXAMINATION PHYSICAL CONSTANTS The following data may be used if required: Speed of light in vacuum, c = 2.998× 108 m · s−1 Permittivity of free space, ε0 = 8.85× 10−12 F ·m−1 Permeability of free space, µ0 = 4π× 10−7 H ·m−1 Fundamental magnitude of electronic charge, e = 1.602× 10−19 C Mass of an electron, me = 9.11× 10−31 kg VECTOR IDENTITIES In the formulae below, F , G and C denote vector fields, and φ and ψ are scalar functions. C × (F ×G) = (C ·G) F − (C · F ) G (1) ∇ ·∇φ = ∇2φ (2) ∇ ·∇× F = 0 (3) ∇×∇φ = 0 (4) ∇× (∇× F ) = ∇(∇ · F )−∇2F (5) ∇(φψ) = (∇φ)ψ + φ(∇ψ) (6) ∇(φF ) = (∇φ) · F + φ∇ · F (7) ∇ · (F ×G) = (∇× F ) ·G − (∇×G) · F (8) ∇× (φF ) = (∇φ)× F + φ∇× F (9) ∇× (F ×G) = (∇ ·G)F − (∇ · F )G + (G ·∇)F − (F ·∇)G (10) VECTOR OPERATORS IN DIFFERENT COORDINATE SYSTEMS • Cartesian: Coordinates (x , y , z), Volume element dV = dx dy dz ∇ψ = ix ∂ψ ∂x + iy ∂ψ ∂y + iz ∂ψ ∂z (11a) ∇ · F = ∂Fx ∂x + ∂Fy ∂y + ∂Fz ∂z (11b) ∇× F = ∣∣∣∣∣∣∣∣ ix iy iz ∂ ∂x ∂ ∂y ∂ ∂z Fx Fy Fz ∣∣∣∣∣∣∣∣ (11c) ∇2ψ = ∂2Fx ∂x2 + ∂2Fy ∂y2 + ∂2Fz ∂z2 (11d) 1 • Cylindrical: Coordinates (R,ϕ, z), Volume element dV = R dR dϕdz ∇ψ = iR ∂ψ ∂R + iϕ 1 R ∂ψ ∂ϕ + iz ∂ψ ∂z (12a) ∇ · F = 1 R ∂(RFR) ∂R + 1 R ∂Fϕ ∂ϕ + ∂Fz ∂z (12b) ∇× F = 1 R ∣∣∣∣∣∣∣∣ iR Riϕ iz ∂ ∂R ∂ ∂ϕ ∂ ∂z FR RFϕ Fz ∣∣∣∣∣∣∣∣ (12c) ∇2ψ = 1 R ∂ ∂R ( R ∂ψ ∂R ) + 1 R2 ∂2ψ ∂ϕ2 + ∂2ψ ∂z2 (12d) • Spherical polar: Coordinates (r , θ,ϕ), Volume element dV = r 2 sin θdr dθdϕ ∇ψ = i r ∂ψ ∂r + iθ 1 r ∂ψ ∂θ + iϕ 1 r sin θ ∂ψ ∂ϕ (13a) ∇ · F = 1 r 2 ∂(r 2Fr ) ∂r + 1 r sin θ ∂(sin θFθ) ∂θ + 1 r sin θ ∂Fϕ ∂ϕ (13b) ∇× F = 1 r 2 sin θ ∣∣∣∣∣∣∣∣ i r r iθ r sin θiϕ ∂ ∂r ∂ ∂θ ∂ ∂ϕ Fr rFθ r sin θFϕ ∣∣∣∣∣∣∣∣ (13c) ∇2ψ = 1 r 2 ∂ ∂r ( r 2∂ψ ∂r ) + 1 r 2 sin θ ∂ ∂θ ( sin θ ∂ψ ∂θ ) + 1 r 2 sin2 θ ∂2ψ ∂ϕ2 (13d) IMPORTANT THEOREMS • Stokes’ Theorem equates the integral of the curl of a vector function F , over an open surface S, to the line integral of that same function around the boundary (closed curve Γ ) of that surface: ∫∫ S ∇× F · dS = ∮ Γ F · d`. (14) • Divergence Theorem equates the integral of a vector function G over a closed surface S, to the volume integral of the divergence of that same function over the volume V enclosed by that surface: ∫∫∫ V ∇ ·G dV = ∫∫ S G · n dS, (15) where n denotes a unit vector locally normal to the surface. PHAS0038/2022-23 CONTINUED 2