On the surface of a long solenoid of radius

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Manifolds, charts and tangent spaces

Problem

On the surface of a long solenoid of radius RR, use the chart (z,ϕ)(z,\phi), zz the height and ϕ∈(0,2π)\phi\in(0,2\pi) the azimuthal angle. Outside the solenoid, the electromagnetism course wrote the vector potential as the one-form A=Φ2π dϕA=\dfrac{\Phi}{2\pi}\,\mathrm{d}\phi, with Φ=2.0×10−3 Wb\Phi=2.0\times10^{-3}\,\mathrm{Wb} the constant enclosed flux. Consider also the tilted chart (z′,ϕ′)=(z, ϕ+kz)(z',\phi')=(z,\ \phi+kz) with k=0.5 m−1k=0.5\,\mathrm{m^{-1}}, tracing a helical marking on the surface. A particle on the surface has, in the chart (z,ϕ)(z,\phi), velocity v=4.0 ∂z+3.0 ∂ϕv=4.0\,\partial_z+3.0\,\partial_\phi (units m s−1\mathrm{m\,s^{-1}} and rad s−1\mathrm{rad\,s^{-1}} respectively). ((a)) Use the chain rule to find the components vz′,vϕ′v^{z'},v^{\phi'} of vv in the tilted chart. ((b)) Evaluate A(v)A(v) directly in the chart (z,ϕ)(z,\phi) using the cotangent-space pairing dϕ(∂ϕ)=1\mathrm{d}\phi(\partial_\phi)=1, dϕ(∂z)=0\mathrm{d}\phi(\partial_z)=0.

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