An electric field on is

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

Invariance, Ch Integrating forms: Stokes's theorem, holes and Maxwell's equations

Problem

An electric field on R3\mathbb{R}^3 is E(x,y,z)=(2xy+z2, x2, 2xz)\mathbf{E}(x,y,z) = (2xy+z^2,\ x^2,\ 2xz), components in V m−1\mathrm{V\,m^{-1}} with x,y,zx,y,z in metres. ((a)) Verify that E\mathbf{E} is curl-free everywhere, so that the one-form ω=Ex dx+Ey dy+Ez dz\omega=E_x\,\mathrm{d}x+E_y\,\mathrm{d}y+E_z\,\mathrm{d}z is closed. ((b)) Since R3\mathbb{R}^3 is star-shaped about the origin, use the Poincaré lemma's radial construction f(x,y,z)=∫01ωi(tx,ty,tz) xi dtf(x,y,z)=\int_0^1\omega_i(tx,ty,tz)\,x^i\,\mathrm{d}t to build ff, and hence write the electrostatic potential φ=−f\varphi=-f satisfying E=−∇φ\mathbf{E}=-\nabla\varphi. ((c)) Check by direct differentiation that −∇φ=E-\nabla\varphi=\mathbf{E}. ((d)) Use φ\varphi to evaluate ∫ABE⋅dl\int_A^B\mathbf{E}\cdot\mathrm{d}\mathbf{l} along any path from A=(0,0,0) mA=(0,0,0)\,\mathrm{m} to B=(1,2,1) mB=(1,2,1)\,\mathrm{m}.

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