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Question:

A particle of mass 10⁻³ kg and charge 1.0 C, is initially at rest. At time t=0, the particle comes under the influence of an electric field E(t) = E₀sin(ωt) î where E₀ = 1.0 N/C and ω = 10³ rad/s. Consider the effect of only the electrical force on the particle. Then the maximum speed, in m/s, attained by the particle at subsequent times is __________.

Solution:

m = 10⁻³ kg, q = 1 C, t = 0
E = E₀sinωt
Force on particle will be
F = qE = qE₀sinωt
At vmax, a = 0
qE₀sinωt = 0
F = qE₀sinωt
dv/dt = qE₀m sinωt
∫₀v dv = ∫₀π/ω qE₀m sinωt dt
v = qE₀/mω [-cosωt]₀π/ω
v = qE₀/mω [(-cosπ) - (cos0)]
v = 1 × 1 / (10⁻³ × 10³) × 2
v = 2 m/s