Plane-Wave Tracker — which way does E₀ sin(3y + 4z + ωt) travel?

Answer: —

1. Read the wave vector. The phase is k·r + ωt with k = 3ĵ + 4k̂. So |k| = √(9 + 16) = 5 m⁻¹, and (B), which says 0.5 m⁻¹, is false. The wavelength is 2π/5 ≈ 1.257 m.
2. Direction of travel. A crest keeps its phase: 3y + 4z + ωt = constant. As t grows, 3y + 4z must fall, so the wave moves against k, along −(3ĵ + 4k̂)/5. (A) is true. (With a "+ωt" phase, the wave travels opposite to k.)
3. Frequency. In vacuum ω = c|k| = 3 × 10⁸ × 5 = 1.5 × 10⁹ rad/s, so (C) is true.
4. Magnetic field (Faraday's law). ∂B/∂t = −∇ × E, and ∇ × (E_x î) = (∂E_x/∂z) ĵ − (∂E_x/∂y) k̂ = E₀ cos(…)(4ĵ − 3k̂). Integrating in time:
   B = (E₀/ω) sin(3y + 4z + ωt)(−4ĵ + 3k̂) = (E₀/c) sin(…) · (−4ĵ + 3k̂)/5.
   Option (D) has the opposite sign and is 5 times too large: its |B| would be 5E₀/c, not E₀/c. (D) is false. Check: E × B ∝ î × (−4ĵ + 3k̂) = −(3ĵ + 4k̂), along the direction of travel.

Understand the concept

Key takeaways

• sin(k·r + ωt) travels along −k̂; sin(k·r − ωt) travels along +k̂.
• |k| = √(k_x² + k_y² + k_z²), λ = 2π/|k|, and ω = c|k|.
• B = (1/c) n̂ × E, with |B| = |E|/c. Use a unit vector for n̂, and check that E × B points along n̂.

Verification: an independent derivation, a sympy route (Maxwell's equations applied to the given field), a numerical route (a crest followed through time and the curl found by finite differences), and the page's own engine, run by tests/verify_p1phyq08.py. The official key (A, C) and the MathonGo solution (A, C) agree. Status: script-verified by the PrayogX simulation factory; no review by a person has been recorded. The wave is shown in slow motion (the real period is 4.2 ns), and arrows are drawn to relative scale, with |B| shown as c|B|.