A long solenoid of radius carries a steady current giving inside and (ideally) zero…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Vector calculus and field geometry

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

Problem

A long solenoid of radius

a=1.0\,\mathrm{cm}

carries a steady current giving

B=0.1005\,\mathrm{T}

inside and (ideally) zero outside. Someone claims that outside the solenoid,

s>a

A_\phi(s)=\frac{Ba^2}{2s^2},

reasoning that

A_\phi

must fall off faster than

1/s

because

\mathbf{B}=0

there. ((a)) Use the curl in cylindrical coordinates,

B_z=\dfrac1s\dfrac{d}{ds}(sA_\phi)

, to test the claim. ((b)) Find the correct

s

-dependence of

A_\phi

outside, and evaluate it at

s=5.0\,\mathrm{cm}

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