Past exam of the symmetry and relativity course of the University of Oxford / 2017 / 3 / d

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- Solution d

Solution

Take the stated static vector potential with zero scalar potential. In the laboratory,

E = 0, B = \nabla \times A = (A_{0} k_{u} sin (k_{u} z), 0, 0) .

Let

S^{'}

move along

z

at the electron's reference velocity

v

. The inverse transformation is

z = γ (z^{'} + v t^{'})

. The Lorentz transformation of electromagnetic fields gives

B_{x}^{'} = γ A_{0} k_{u} sin [γ k_{u} (z^{'} + v t^{'})], E_{y}^{'} = γ v A_{0} k_{u} sin [γ k_{u} (z^{'} + v t^{'})],

with all other components zero. The sign follows from

v \times B = v B_{x} e_{y}

.

Although an observer fixed in

S^{'}

sees an oscillating field, this is not an exact source-free propagating electromagnetic wave. The phase covector has

ω^{'} = - γ v k_{u}

,

k_{z}^{'} = γ k_{u}

, so its squared norm is

(\frac{ω ^{'}}{c})^{2} - (k_{z}^{'})^{2} = - k_{u}^{2} < 0.

It is the boost of a static spatial grating, not a null vacuum wave. Equivalently,

E^{'} / B^{'} = v < c

and

B^{'2} - E^{'2} / c^{2} = B^{2}

is nonzero away from nodes; a vacuum plane wave would have vanishing electromagnetic field invariants. In the ultrarelativistic limit it can be a useful nearly wave-like approximation along the electron's path, but the exact distinction remains.

A charged electron cannot actually maintain the given constant velocity through this nonzero transverse field without an external constraint: the Lorentz force deflects it. The calculation therefore uses the prescribed straight reference trajectory, a weak-deflection approximation, or an instantaneously comoving inertial frame. A globally fixed electron rest frame is not available for its exact deflected motion.

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