Someone studies the magnetic field outside a long straight wire carrying current…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Topology and global structure · Topology and global structure

Invariance, Ch Integrating forms: Stokes's theorem, holes and Maxwell's equations

Problem

Someone studies the magnetic field outside a long straight wire carrying current II along the zz-axis, written in cylindrical coordinates (ρ,ϕ,z)(\rho,\phi,z) with ρ>0\rho>0 as B=μ0I2πρϕ^,\mathbf{B} = \frac{\mu_0 I}{2\pi\rho}\hat{\boldsymbol\phi}, equivalently the one-form ω=μ0I2π dϕ\omega = \dfrac{\mu_0 I}{2\pi}\,\mathrm d\phi. They compute ∇×B=0\nabla\times\mathbf B = \mathbf 0 throughout the region ρ>0\rho>0 (no current threads any small loop that avoids the axis), and argue: 'Since ∇×B=0\nabla\times\mathbf B = \mathbf 0 everywhere in ρ>0\rho>0, the one-form ω\omega is closed, so by the Poincaré lemma there is a function ψ\psi defined on the whole of ρ>0\rho>0 with ω=dψ\omega = \mathrm d\psi. Taking ψ=μ0I2πϕ\psi = \dfrac{\mu_0I}{2\pi}\phi gives dψ=ω\mathrm d\psi=\omega, so for any closed loop CC lying in ρ>0\rho>0, ∮Cω=ψ(end)−ψ(start)=0\oint_C\omega = \psi(\text{end}) - \psi(\text{start}) = 0, because a closed loop starts and ends at the same point.' They then compute directly, for a circle of radius aa centred on the wire, ∮Cω=μ0I≠0\oint_C\omega = \mu_0 I \ne 0, contradicting their own conclusion. ((a)) Find the error in the quoted argument. ((b)) State precisely which hypothesis of the Poincaré lemma fails here, and say what is true instead: for which loops in ρ>0\rho>0 does ∮ω\oint\omega vanish, and for which is it a nonzero multiple of μ0I\mu_0I?

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