Past exam of the optics and electrodynamics course of the University of Cambridge / 2021 / Section B / B6 / c

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- Solution c

Solution

A uniform Lorentz boost preserves the plane wave form, while changing both its frequency and its vector wave amplitude. Decompose vectors parallel and perpendicular to

v

, so

k = k_{∥} + k_{⊥}

. In the observer's primed coordinates,

E^{'} (r^{'}, t^{'}) = E_{0}^{'} exp {i [k_{⊥} \cdot r_{⊥}^{'} + γ (k_{∥} - \frac{ω}{c ^{2}} v) \cdot r_{∥}^{'} - γ (ω - v \cdot k) t^{'}]} .

The transformed wave amplitude is specified in part (d). The unprimed “rest frame” in this question is the original coordinate frame, not a rest frame of a vacuum light wave, which has no inertial rest frame.

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