Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 4 / 12C / b / iii

Table of contents
- Solution iii

Solution

u = c (cos θ, sin θ, 0) .

Part (ii) gives the first-order changes

δ u_{x} = - V sin^{2} θ, δ u_{y} = V sin θ cos θ .

For a vector of fixed leading-order magnitude

c

, its small change of angle is

δ θ = \frac{u _{x} δ u _{y} - u _{y} δ u _{x}}{c ^{2}} .

Substitution yields the relativistic aberration formula

θ^{'} - θ = \frac{V}{c} sin θ

to leading order.

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