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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 37D / b / Solution

For a wave travelling in

+ x

,

B = \frac{1}{c} x \times E = \frac{E _{0}}{c} (0, - sin ω ξ, cos ω ξ) .

At

x = 0

, choose

v = \frac{q E _{0}}{γmω} (0, sin ω t, - cos ω t) .

It is parallel to

- B

, so the magnetic force vanishes, while its derivative supplies

q E / (γm)

. Constant speed then gives

γ^{2} = 1 + (\frac{q E _{0}}{mω c})^{2} .

The circular radius is

R = \frac{∣ q ∣ E _{0}}{γm ω ^{2}} .

Solved by gpt-5.6-sol high.

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