Motion of Charged Particle in Alternating Electric Field

Let E = E₀ sin ωt, represent the alternating electric field. Let it act on a charged particle q and mass m.

The force acting on the particle is
        F = m d²r/dt² = mdv/dt
But        F = qE = qE = qE₀ sin ωt
. :        mdv/dt = qE₀  sin ωt                                … (1)   

On integrating with the condition v = 0 at t = 0, we get.
        v = dr/dt = - qE0/mω cos ωt + qE0/mω = qE0/mω [1 – cos ωt    ]            … (2)

On integrating with the condition r = 0 at t = 0, we get
        r = - qE0/mω2 sin ωt + qE0/mωt = qE0/mω [t – sin ωt/ω]                … (3)

This gives the displacement of the particle in time t in alternating electric field.

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