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Relativistic motion in a constant electric field with transverse momentum

For

E = (0, 0, E)

,

B = 0

, and initial momentum

(p_{0}, 0, 0)

,

p_{x} = p_{0}, p_{z} = qEt, E (t) = E_{0}^{2} + c^{2} q^{2} E^{2} t^{2} .

With the initial position at the origin,

z (t) = \frac{E ( t ) - E _{0}}{qE}, x (t) = \frac{c p _{0}}{qE} arsinh \frac{c qEt}{E _{0}} .

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