Electromagnetic Theory (v2) #28 | Problem Solving #05: Divergence of a Vector Field

Instructor: Dr. Süleyman Burak ÇELİK

Fifth problem-solving episode of the EMT (v2) series. We meet the second differential operator: the divergence. Where gradient takes a scalar and returns a vector, divergence takes a vector field and returns a scalar — a number at every point that measures the net outflow of the field per unit volume. Positive divergence means a source, negative means a sink, zero means the field is solenoidal (everything that flows in flows out). In electrostatics, ∇·E = ρ/ε₀ — divergence of the electric field tells you where the charge density lives. That is one of Maxwell's equations. Cartesian example: A = x²y x̂ + y²z ŷ + z²x ẑ at P(1, 2, 3). Three partials. ∂Aₓ/∂x = 2xy → 4 at P. ∂Aᵧ/∂y = 2yz → 12. ∂A_z/∂z = 2zx → 6. Sum: ∇·A|_P = 4 + 12 + 6 = 22. Positive divergence means P acts as a source for this field. Spherical example — the one every electromagnetics student should keep in their pocket. A = (1/r²) r̂, the inverse-square radial field. The E-field outside a point charge looks like this. Use the radial term of spherical divergence: ∇·A = (1/r²) ∂(r²A_r)/∂r. Substitute A_r = 1/r². Inside the derivative: r²·(1/r²) = 1. ∂(1)/∂r = 0. So ∇·A = 0 — everywhere with r › 0. No sources off the origin. All the source sits at r = 0 as a delta function. That is the mathematical heart of Gauss's law: a point charge produces a 1/r² field whose divergence vanishes everywhere except where the charge sits. Topics: 0:00 Cover 0:05 Concept: source, sink, solenoidal — and Maxwell's connection 1:25 Formulas: divergence in Cartesian, cylindrical, spherical (volume scale factors) 3:07 Cartesian worked example: A = x²y x̂ + y²z ŷ + z²x ẑ at P(1,2,3) → 22 4:50 Spherical worked example: A = (1/r²) r̂ → 0 (Gauss's law in disguise) 6:50 Summary Key equations: • ∇·A = ∂Aₓ/∂x + ∂Aᵧ/∂y + ∂A_z/∂z (Cartesian) • ∇·A = (1/ρ) ∂(ρA_ρ)/∂ρ + (1/ρ) ∂A_φ/∂φ + ∂A_z/∂z (cylindrical) • ∇·A = (1/r²) ∂(r²A_r)/∂r + (1/(r sinθ)) ∂(sinθ A_θ)/∂θ + (1/(r sinθ)) ∂A_φ/∂φ (spherical) • Volume scale factors live INSIDE the radial derivatives — easy to miss! Answers: Cart ∇·A = 22 at P(1, 2, 3). Sph ∇·A = 0 anywhere off the origin. Reference: Sadiku, "Elements of Electromagnetics" 7th Ed, Chapter 3 (divergence)