A long ideal solenoid of radius is wound with and carries a steady current

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Magnetostatics · Electromagnetic induction and alternating-current circuits

The Boundary-Value Problem, Ch The vector potential and the magnetostatic boundary-value problem

Problem

A long ideal solenoid of radius a=1.0 cma = 1.0\,\mathrm{cm} is wound with n=8.0×103 turns m−1n = 8.0\times10^3\,\mathrm{turns\,m^{-1}} and carries a steady current I=10 AI = 10\,\mathrm{A}, giving B=μ0nI=0.1005 TB = \mu_0nI = 0.1005\,\mathrm{T} inside and ΦB=3.2×10−5 Wb\Phi_B = 3.2\times10^{-5}\,\mathrm{Wb}. A deuteron (charge e=1.602×10−19 Ce = 1.602\times10^{-19}\,\mathrm{C}, mass md=3.344×10−27 kgm_d = 3.344\times10^{-27}\,\mathrm{kg}) sits at rest a radial distance s=10 cms = 10\,\mathrm{cm} from the axis, well outside the winding, where B=0\mathbf{B} = 0. Someone argues as follows. Claim 1: "The vector potential A\mathbf{A} exists for the same reason the electrostatic potential VV exists: B\mathbf{B} is conservative (curl-free), so it can be written as the gradient, or here the curl, of something." Claim 2: "Since B=0\mathbf{B} = 0 everywhere outside the solenoid, a gauge function λ\lambda can always be found with A−∇λ=0\mathbf{A} - \nabla\lambda = 0 throughout the exterior region. So the deuteron, sitting where B=0\mathbf{B} = 0, feels nothing at all when the current is switched off." ((a)) Identify precisely what is wrong with Claim 1. ((b)) Identify precisely what is wrong with Claim 2, and then settle the matter numerically: if the current is switched off, find the speed the deuteron acquires along the direction the winding current flowed, using AϕA_\phi at s=10 cms = 10\,\mathrm{cm} before switch-off.

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