Electromagnetic Theory (v2) #29 | Problem Solving #06: Curl of a Vector Field

Instructor: Dr. Süleyman Burak ÇELİK

Sixth problem-solving episode of the EMT (v2) series. We meet the third differential operator: the curl. Where divergence took a vector field and asked "where are the sources?", curl takes a vector field and asks "where does it rotate?". Drop a tiny paddle wheel into the field at any point. If it spins, ∇×A is non-zero — the curl vector points along the rotation axis by the right-hand rule. If it stays still, the field is irrotational at that point. Curl shows up in two of Maxwell's equations: ∇×E = -∂B/∂t and ∇×H = J + ∂D/∂t. It also tells you whether a field is conservative — if curl is zero everywhere, a scalar potential exists. Cartesian example — same field as PS05. We deliberately reuse A = x²y x̂ + y²z ŷ + z²x ẑ at P(1, 2, 3) to drive home a key message. Last time, divergence of this field at this point came out to 22. Now we compute its curl. Three slots, six partials. ∂Az/∂y - ∂Ay/∂z = 0 - y² = -y². ∂Ax/∂z - ∂Az/∂x = 0 - z² = -z². ∂Ay/∂x - ∂Ax/∂y = 0 - x² = -x². So ∇×A = -y² x̂ - z² ŷ - x² ẑ. At P: -4 x̂ - 9 ŷ - ẑ. Lesson: divergence and curl are independent windows into the same vector field — same A, the field has both a source-like outflow AND local rotation, and one number can't tell you both. Spherical example — also the same field as PS05: A = (1/r²) r̂, the inverse-square radial field. The electric field outside a point charge. Compute the curl. The radial-only A with Ar depending only on r kills every term: r̂ slot needs Aphi and Atheta (both zero), theta slot needs ∂Ar/∂phi (zero), phi slot needs ∂Ar/∂theta (zero). All three slots give zero. So ∇×A = 0 everywhere for positive r — the inverse-square radial field is irrotational. Now feel the consequence. PS05 showed div A = 0 off the origin. PS06 shows curl A = 0 off the origin. The second condition is exactly what guarantees a scalar potential V exists. And it does: V = 1/r, since -∇(1/r) = (1/r²) r̂ = A. The Coulomb potential exists because curl is zero. Coulomb integrals are path-independent because curl is zero. Pretty much every clean result in electrostatics rests on this one fact. Topics: 0:00 Cover 0:03 Concept: paddle wheel rotation + Maxwell's curl equations 1:36 Formulas: curl in Cartesian, cylindrical, spherical (determinant template) 3:11 Cartesian worked example: A = x²y x̂ + y²z ŷ + z²x ẑ at P(1,2,3) 5:46 Spherical worked example: A = (1/r²) r̂ → 0 (Coulomb's potential) 8:01 Summary Key equations: • ∇×A = (∂Az/∂y − ∂Ay/∂z) x̂ + (∂Ax/∂z − ∂Az/∂x) ŷ + (∂Ay/∂x − ∂Ax/∂y) ẑ (Cartesian) • Cylindrical and spherical: same determinant template, scale factors h_ρ=1, h_φ=ρ, h_z=1 / h_r=1, h_θ=r, h_φ=r sinθ • Conservative field theorem: ∇×A = 0 everywhere ⟺ A = -∇V for some scalar V Answers: Cart ∇×A = -y² x̂ - z² ŷ - x² ẑ; at P(1,2,3): -4 x̂ - 9 ŷ - ẑ. Sph ∇×A = 0 everywhere off the origin (irrotational, conservative). Reference: Sadiku, "Elements of Electromagnetics" 7th Ed, Chapter 3 (curl)