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=2.0cm is wound with n=5.0×103turnsm−1 and carries a steady current I=20A, giving B=μ0nI≈0.1257T inside and negligible field outside. An alpha particle (charge 2e=3.204×10−19C, mass mα=6.644×10−27kg) rests at a radial distance s=15cm from the axis, outside the winding, where B=0. Two claims are made about this setup. Claim 1: "A vector potential A can be written for B because B is a conservative field, in the same sense that E=−∇V follows from E being conservative." Claim 2: "Outside the solenoid B vanishes identically, so by a suitable choice of gauge A can be transformed to zero everywhere in that region; consequently switching off the current can do nothing to a charge parked there."
((a)) State exactly which property of B, not curl-freeness, is what licenses writing B=∇×A, and say where Claim 1 goes wrong.
((b)) Show that Claim 2 is false by computing the speed the alpha particle acquires, along the direction the winding current flowed, when the current is switched off, using the flux enclosed by the solenoid.