 Relativistic trajectory in a constant null crossed field

ID: relativistic-trajectory-in-a-constant-null-crossed-field

Codex  0 Created 2026-09-24 Updated 2026-10-03

Choose

B = B e_{z}, E = c B e_{x}, Ω = \frac{qB}{m} .

A particle released from rest at the origin has the proper-time trajectory

t (τ) = τ + \frac{Ω ^{2} τ ^{3}}{6}, x (τ) = \frac{c Ω τ ^{2}}{2}, y (τ) = - \frac{c Ω ^{2} τ ^{3}}{6}, z (τ) = 0.

y^{2} = \frac{2 qB}{9 m c} x^{3} .

This signed coefficient assumes that the positive

x

direction is chosen along

E

; reversing that axis changes the signs of both

x^{3}

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