A charged bead moves in a region of the -plane where the electrostatic field (from…

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

A charged bead moves in a region of the xyxy-plane where the electrostatic field (from the electromagnetism course) is uniform, E=(Ex,Ey)=(300, 400) V m−1\mathbf E=(E_x,E_y)=(300,\,400)\,\mathrm{V\,m^{-1}}, related to the potential by E=−∇φ\mathbf E=-\nabla\varphi, so the one-form −dφ=300 dx+400 dy-\mathrm d\varphi=300\,\mathrm dx+400\,\mathrm dy. Take also the polar chart (r,θ)(r,\theta), x=rcos⁡θx=r\cos\theta, y=rsin⁡θy=r\sin\theta. At the point (x,y)=(3.0,4.0) m(x,y)=(3.0,4.0)\,\mathrm m the bead has Cartesian velocity vx=−2.0 m s−1v^x=-2.0\,\mathrm{m\,s^{-1}}, vy=1.0 m s−1v^y=1.0\,\mathrm{m\,s^{-1}}. ((a)) Use the chain rule to find the velocity components vr,vθv^r,v^\theta in the polar chart at that point. ((b)) Evaluate −dφ(v)-\mathrm d\varphi(v) directly in the Cartesian chart (the power delivered per unit charge), then express −dφ-\mathrm d\varphi in the polar chart at that point using the lower-index transformation law and confirm the pairing with vr∂r+vθ∂θv^r\partial_r+v^\theta\partial_\theta gives the same number.

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