2B29 Electromagnetic Theory Formulae (UCL)

2B29 Electromagnetic Theory. iii) Some useful Mathematical Tools VECTORS

a ⋅ (b × c) = b ⋅ (c × a) = c ⋅ (a × b) a × (b × c) = (a ⋅ c) b − (a ⋅ b) c (a × b) ⋅ (c × d) = (a ⋅ c) (b ⋅ d) − (a ⋅ d) (b ⋅ c)

VECTOR CALCULUS

* *

∇×∇ φ = 0 ∇ ⋅ (∇ × A) = 0 ∇ ⋅ (φ A) = φ ∇ ⋅ A + A ⋅ ∇φ ∇ × (φ A) = ∇φ × A + φ∇ × A ∇ ( A ⋅ B) = ( A ⋅ ∇ )B + (B ⋅ ∇ ) A + A × (∇ × B) + B × (∇ × A) ∇ ⋅ ( A × B) = (∇ × A) ⋅ B − (∇ × B) ⋅ A ∇ × ( A × B) = A(∇ ⋅ B) − B(∇ ⋅ A) + (B ⋅ ∇ ) A − ( A ⋅ ∇ )B ∇ × (∇ × A) = ∇ (∇ ⋅ A) − ∇ 2 A

where

 ∂ ∂ ∂  ∂ ∂ ∂  ∂ ∂ ∂  ˆ ˆ Bz + Ay + Az + Ay + Az + Ay + Az ( A ⋅ ∇ )B = ˆi  Ax  Bx + j  Ax  By + k  Ax ∂y ∂z ∂y ∂z ∂y ∂ z   ∂x  ∂x  ∂x

x 2 + y 2 + z 2 , e.g. as φ(r), F(r), then r d r dφ r dF r dF , i.e. ∇φ (r ) = , ∇ ⋅ F (r ) = ⋅ , ∇ × F(r ) = × , ∇≡ r dr r dr r dr r dr 1 d  2 d φ  1 d 2 (r φ ) d 2 φ 2 d φ ∇ 2φ = 2 = 2+ . r = r d r  d r  r d r2 dr r dr

For a function that depends only on the distance r = r =

EXPLICIT FORMS OF THE VECTOR OPERATORS (a)

Cartesian (x, y, z), volume element dτ = dx dy dz.

∂ψ ˆ ∂ψ ˆ ∂ψ ∂ Ax ∂ Ay ∂ Az +j +k + + and ∇ ⋅ A = ∂x ∂y ∂z ∂x ∂y ∂z  ∂ A ∂ Ay  ˆ  ∂ Ax ∂ Az  ˆ  ∂ Ay ∂ Ax  ∇ × A = ˆi  z − − +k  − +j  ∂ z   ∂ z ∂ x  ∂y   ∂y  ∂x

∇ψ = ˆi

∇×A =

ˆi

ˆj

kˆ

∂ ∂x

∂ ∂y

∂ ∂z

Ax

Ay

Az

∇ ⋅ ∇ψ = ∇ 2ψ = 2B29 Mathematical Tools 2004

∂ 2ψ ∂ 2ψ ∂ 2ψ + + ∂ x2 ∂ y2 ∂ z 2 Section iii)

1

(b)

Cylindrical polar (ρ, φ, z), volume element dτ = ρ dρ dφ dz.

∂ψ ˆ 1 ∂ψ ∂ψ +φ + zˆ ∂ρ ρ ∂φ ∂z 1 ∂ ( ρ Aρ ) 1 ∂ Aφ ∂ Az ∇⋅A = + + ρ ∂ρ ρ ∂φ ∂z ∇ψ = ρˆ

ρˆ ∇×A =

1 ∂

ρ ∂ρ Aρ

(c)

ρφˆ ∂ ∂φ ρ Aφ

zˆ

∂ ∂z Az

Spherical polar (r, θ, φ), volume element dτ = r2 sinθ dr dθ dφ.

∇ψ = rˆ ∇⋅A =

∇ 2ψ =

∂ψ ˆ 1 ∂ψ ˆ 1 ∂ψ +θ +φ ∂r r ∂θ r sin θ ∂φ

1 ∂ (r 2 Ar ) 1 ∂ (sin θ Aθ ) 1 ∂ Aφ + + 2 r r sin θ r sin θ ∂φ ∂r ∂θ rˆ r θˆ r sin θφˆ 1 ∂ ∂ ∂ ∇×A = ∂φ r sin θ ∂ r ∂θ Ar r Aθ r sin θ Aφ

1 ∂  2 ∂ψ  1 ∂  ∂ψ  1 ∂ 2ψ r sin θ + +    r 2 ∂ r  ∂ r  r 2 sin θ ∂θ  ∂θ  r 2 sin 2 θ ∂φ 2 1 ∂  2 ∂ψ  1 ∂ 2 (rψ ) ∂ 2ψ 2 ∂ψ = + r = r2 ∂ r  ∂ r  r ∂ r2 ∂ r2 r ∂ r

INTEGRAL RELATIONS If V is a volume with volume element dτ and S is a closed surface enclosing volume V and having a surface ˆ at dS then element dS and an outward normal n *

∫ ∇ ⋅ A d τ = v∫ A ⋅ nˆ d S

V

(Divergence theorem)

S

ˆ to S is defined by If S is an open surface and C is a contour bounding it with line element dA and the normal n the right-hand rule in relation to the sense of a line integral around C then *

∫ (∇ × A) ⋅ nˆ d S = v∫ A ⋅ dA S

2B29 Mathematical Tools 2004

(Stokes's theorem)

C

Section iii)

2