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Jont B. Allen; UIUC Urbana IL, USA ... Lecture 39: Review of vector field calculus. 39.14.2 ... Closure: Fundamental Theorems of Integral Calculus.
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Definition: R^1 7 → ∇
E ( x , y , z ) = [ ∂x , ∂y , ∂z ] T^ φ ( x , y , z ) =
[ (^) ∂φ
∂x
∂φ ∂y
∂φ ∂z
] T
x ,y ,z
E ⊥ plane tangent at φ ( x , y , z ) = φ ( x 0 , y 0 , z 0 ) Unit vector in direction of E is ˆn = (^) || EE || , along isocline Basic definition
∇ φ ( x , y , z ) ≡ lim |S|→ 0
{ ∫∫∫^ φ ( x , y , z ) ˆn dA |S|
}
ˆn is a unit vector in the direction of the gradient S is the surface area centered at ( x , y , z )
Definition: R^3 7 → ∇·
∇· D ≡ [ ∂x , ∂y , ∂z ] · D =
[ ∂Dx ∂x
∂Dy ∂y
∂Dz ∂z
] = ρ ( x , y , z )
Examples: Voltage about a point charge Q [SI Units of Coulombs]
φ ( x , y , z ) =
Q ǫ 0
√ x^2 + y^2 + z^2
=
Q ǫ 0 R
φ [Volts]; Q = [C]; Free space ǫ 0 permittivity ( μ 0 permeability ) Electric Displacement (flux density) around a point charge ( D = ǫ 0 E )
D ≡ −∇ φ ( R ) = − Q ∇
{ 1 R
} = − Q δ ( R )
General case of a Compressible vector field Volume integral over charge density ρ ( x , y , z ) is total charge enclosed Qenc ∫∫∫
V
∇· D dV =
∫∫
S
D · ˆ n dA = Qenc
Examples When the vector field is incompressible ρ ( x , y , z ) = 0 [C/m^3 ] over enclosed volume Surface integral is zero ( Qenc = 0) Unit point charge: D = δ ( R ) [C/m^2 ]
Definition: R^3 7 → ∇×
∣∣ ∣∣ ∣∣ ∣
x ˆ y ˆ ˆ z ∂x ∂y ∂z Hx Hy Hz
∣∣ ∣∣ ∣∣ ∣
Examples: Maxwell’s equations: ∇× E = − B ˙, ∇× H = σ E + D ˙, H = − y ˆ x + x ˆ y then ∇× H = 2ˆ z constant irrotational H = 0ˆ x + 0ˆ y + z^2 ˆ z then ∇× H = 0 is irrotational
The variables have the following names and defining equations:
Symbol Equation Name Units E ∇ × E = − B ˙ Electric Field strength [Volts/m] D ∇ · D = ρ Electric Displacement (flux density) [Col/m^2 ] H ∇ × H = ˙ D Magnetic Field strength [Amps/m] B ∇ · B = 0 Magnetic Induction (flux density) [Weber/m^2 ]
In vacuo B = μ 0 H , D = ǫ 0 E , c = √ μ^10 ǫ 0 [m/s], r 0 =
√ (^) μ 0 ǫ 0 = 377 [Ω].
Notation: v ( x , y , z ) = −∇ φ ( x , y , z ) + ∇× w ( x , y , z )
Vector identities: ∇×∇ φ = 0; ∇ · ∇× w = 0
Field type Generator: Test (on v ): Irrotational v = ∇ φ ∇ × v = 0 Rotational v = ∇× w ∇ × v = J Incompressible v = ∇ × w ∇ · v = 0 Compressible v = ∇ φ ∇ · v = ρ
Source density terms: Current: J ( x , y , z ), Charge: ρ ( x , y , z ) Examples: ∇× H = D ˙( x , y , z ), ∇· D = ρ ( x , y , z )
Definition: ka ≪ 1 where a is the size of object, λ = c/f wavelength This is equivalent to a ≪ λ or ω ≪ c/a which is a low-frequency approximation The QS approximation is widely used, but infrequently identified. All lumpted parameter models (inductors, capacitors) are based on QS approximation as the lead term in a Taylor series approximation.
