In a laboratory frame a wide region carries a uniform magnetic field along

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

Invariance, Ch General bases, the metric and tensors in spacetime

Problem

In a laboratory frame SS a wide region carries a uniform magnetic field B=0.50 TB = 0.50\,\mathrm{T} along zz, with no electric field. Frame S′S' moves at v=0.6cv = 0.6c along xx relative to SS. For a boost along xx the transverse fields obey Ey′=γ(Ey−vBz)E'_y = \gamma(E_y - vB_z) and Bz′=γ(Bz−vEy/c2)B'_z = \gamma(B_z - vE_y/c^2), with γ=(1−v2/c2)−1/2\gamma = (1-v^2/c^2)^{-1/2}, as derived from the field-strength tensor. ((a)) Find Ey′E'_y and Bz′B'_z in S′S'. ((b)) Evaluate the invariant B2−E2/c2B^2 - E^2/c^2 in both frames and check they agree. ((c)) Could a frame exist in which this field is purely electric? Justify the answer using the invariants B2−E2/c2B^2-E^2/c^2 and E⋅B\mathbf{E}\cdot\mathbf{B}, without further calculation.

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