In an inertial frame with coordinates

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Relativistic electrodynamics

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

Problem

In an inertial frame SS with coordinates (x0,x1,x2,x3)=(ct,x,y,z)(x^0,x^1,x^2,x^3)=(ct,x,y,z), a uniform electric field of magnitude E0E_0 along +x+x has F01=−E0/cF_{01}=-E_0/c and no other independent component, so its field-strength two-form is F=−E0c dx0∧dx1,F = -\frac{E_0}{c}\,\mathrm dx^0\wedge\mathrm dx^1, with E0E_0 a constant in V m−1\mathrm{V\,m^{-1}}, so that E0/cE_0/c is in tesla. ((a)) Verify dF=0\mathrm dF=0 in SS. ((b)) Frame S′S' moves with speed vv along the yy (that is, x2x^2) direction relative to SS, related by the boost x0=γ(x′0+βx′2),x2=γ(x′2+βx′0),x1=x′1,x3=x′3,\begin{aligned}x^0&=\gamma(x'^0+\beta x'^2),& x^2&=\gamma(x'^2+\beta x'^0),\\ x^1&=x'^1,& x^3&=x'^3,\end{aligned} with β=v/c\beta=v/c, γ=(1−β2)−1/2\gamma=(1-\beta^2)^{-1/2}, the Lorentz transformation held from the relativity course. Express FF in the primed coordinates and verify dF=0\mathrm dF=0 there too. ((c)) A potential for FF in SS is A=(E0/c) x1 dx0A=(E_0/c)\,x^1\,\mathrm dx^0, that is V=−E0xV=-E_0x with Ax=Ay=Az=0A_x=A_y=A_z=0. Check dA=F\mathrm dA=F, express AA in the primed coordinates, and verify dA′=F′\mathrm dA'=F', showing that F=dAF=\mathrm dA survives the change of frame.

Related theory

Browse the problem library