Electromagnetic Theory (v2) #31 | Problem Solving #08: Stokes' Theorem & Ampère's Law

Instructor: Dr. Süleyman Burak ÇELİK

Eighth problem-solving episode of the EMT (v2) series, and the twin of last time. Where the divergence theorem turned local sources into Gauss's law, Stokes' theorem turns local swirls into Ampère's law. The statement: the circulation of a vector field around a closed curve C equals the flux of its curl through any surface S that hangs on that curve. ∮ A · dl = ∫ (∇×A)·dS. The intuition is the same trick one dimension down: tile the surface into millions of tiny cells, each with its own little circulation. Every interior edge is walked twice by neighboring cells in opposite directions and cancels — only the outer edges survive, and together they are exactly the boundary curve C. Square example — clean verification, sign and all. Take the running polynomial field we have carried since PS05: A = x²y x̂ + y²z ŷ + z²x ẑ. The region is the unit square in the z=0 plane, walked counter-clockwise, so the right-hand rule gives the normal n̂ = ẑ. Surface side: the curl was already computed in PS06, ∇×A = −y² x̂ − z² ŷ − x² ẑ. Its z-component on z=0 is −x², and ∫∫(−x²) dx dy over the square is −1/3. Line side: A has y-component y²z, which is zero on z=0, so both vertical edges vanish; the bottom edge gives zero too. Everything hangs on the top edge (y=1), where A_x = x² and we walk from x=1 back to x=0: ∫ x² dx from 1 to 0 = −1/3. Both sides give −1/3 — even the minus sign matches, because the curl points along −z while the walk circles the +z way. Vortex example — the lesson lives in the singularity. Take A = (1/ρ) φ̂, a field that swirls around the z axis and grows as you approach it, like water around a drain. Let C be a circle of radius R and S the disk it bounds. Surface side, naively: for an azimuthal field the curl is (1/ρ) ∂(ρ·A_φ)/∂ρ ẑ, but ρ·A_φ = 1 is constant, so the curl is zero off the axis and the naive surface integral is zero. Line side: on the circle A·dl = (1/R)(R dφ) = dφ, and ∮ dφ from 0 to 2π = 2π — independent of R. Mismatch! 0 ≠ 2π. All the swirl hides on the axis as a delta function: ∇×(φ̂/ρ) = 2π δ²(ρ) ẑ. Integrate that over the disk and the surface side gives 2π too. The theorem is restored once we are honest about the singularity. And here is the payoff. Multiply the field by I/(2π). The vortex becomes H = I/(2πρ) φ̂ — exactly the magnetic field of a straight wire carrying current I. The circulation of H around any loop enclosing the wire becomes the enclosed current: ∮ H · dl = I_enc. That is Ampère's law. Every Amperian-loop calculation in magnetostatics rides on Stokes' theorem plus the delta-function resolution of the singularity at the wire. With Gauss (PS07) and Ampère (PS08), the vector-calculus toolbox is complete — these two theorems are the bridges between Maxwell's differential and integral forms. Topics: 0:00 Cover 0:03 Theorem statement + interior-edge cancellation intuition 1:29 Square worked example: both sides equal −1/3 (sign too) 4:16 Vortex worked example: 0 vs 2π → delta on the axis → Ampère's law 7:09 Summary Key equations: • Stokes' theorem: ∮ A · dl = ∫ (∇×A)·dS • ∇×(φ̂/ρ) = 2π δ²(ρ) ẑ (singular distributional identity) • Ampère's law: ∮ H · dl = I_enc Answers: Square ∮ = ∫ = −1/3. Vortex ∮ = 2π = ∫ (with delta on the axis). → Ampère's law for any loop enclosing the current. Reference: Sadiku, "Elements of Electromagnetics" 7th Ed, Chapter 3 (Stokes' theorem); Griffiths Ch 1.