The electric field of a plane electromagnetic wave is given by E⃗ = E₀ (ı̂ + ȷ̂)/(√2) cos(kz + ω t). At t = 0, a positively charged particle is at the point (x, y, z) = (0, 0, π/k). If its instantaneous velocity at (t = 0) is v₀ k̂, the force acting on it due to the wave is

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(1) antiparallel to (ı̂ + ȷ̂)/(√2). At t = 0, z = π/k: cos(kz + ω t) = cos π = -1, so E⃗ = - E₀ (ı̂ + ȷ̂)/(√2) and qE⃗ is antiparallel to (ı̂ + ȷ̂)/√2 The phase (kz + ω t) means propagation along -k̂, so B̂ must satisfy Ê × B̂ = -k̂ ⇒ B̂ = (ı̂ - ȷ̂)/√2, and at this point B⃗ = - B₀ (ı̂ - ȷ̂)/√2 F⃗ₘ = qv⃗ × B⃗ = q v₀ k̂ × [- B₀(ı̂ - ȷ̂)/√2] = - qv₀B₀ (ı̂ + ȷ̂)/√2 — also antiparallel to (ı̂ + ȷ̂)/√2 Both parts point the same way, so the resultant is antiparallel to (ı̂ + ȷ̂)/√2

The electric field of a plane electromagnetic wave is given by E⃗ = E₀ (ı̂ + ȷ̂)/(√2) cos(kz + ω t). At t = 0, a positively charged particle is at the point (x, y, z) = (0, 0, π/k). If its instantaneous velocity at (t = 0) is v₀ k̂, the force acting on it due to the wave is

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