In a laboratory

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Maxwell's equations and conservation laws

Invariance, Ch Differential forms and the exterior derivative

Problem

In a laboratory, a uniform electric field E=(E0,0,0)\mathbf{E}=(E_0,0,0) with E0=200 V m−1E_0=200\,\mathrm{V\,m^{-1}} and a uniform magnetic flux density B=(0,0,B0)\mathbf{B}=(0,0,B_0) with B0=0.5 TB_0=0.5\,\mathrm{T} fill a region of space. ((a)) Write E\mathbf{E} as the work one-form α=Ei dxi\alpha=E_i\,\mathrm{d}x^i appropriate to it, and evaluate ∫γα\int_\gamma\alpha along the straight path γ\gamma from the origin to the point (2,3,0) m(2,3,0)\,\mathrm{m}. ((b)) Write B\mathbf{B} as the flux two-form β\beta appropriate to it, and evaluate ∫Sβ\int_S\beta over the unit square 0≤x≤10\le x\le1, 0≤y≤10\le y\le1, z=0z=0, oriented so that dx∧dy\mathrm{d}x\wedge\mathrm{d}y is positive. ((c)) In one sentence, say why E\mathbf{E} belongs with a one-form and B\mathbf{B} with a two-form in this dictionary.

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