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Identify the expression that is not a traveling electromagnetic wave in one spatial dimension.

Difficulty: Easy

Correct Answer: e = Em sin(βx)

Explanation:


Introduction / Context:
Traveling waves vary in both space and time with a specific phase relationship. Recognizing traveling versus standing or static patterns is a key skill in electromagnetics and transmission-line theory.


Given Data / Assumptions:

  • Expressions are scalar field examples along x.
  • β is phase constant, ω is angular frequency, Em is amplitude.


Concept / Approach:
A one-dimensional traveling wave has dependence on (βx ± ωt); any expression missing time or space dependence cannot represent a traveling wave. Purely spatial (or purely temporal) dependence corresponds to static distributions (or uniform oscillations everywhere) rather than propagation.


Step-by-Step Solution:

Forms like sin(βx − ωt), cos(βx − ωt), sin(ωt − βx) clearly include both x and t with a constant phase velocity v = ω / β.The term sin(βx) has no time dependence; it is a spatial standing pattern or static profile, not a propagating wave.Therefore the non-traveling expression is e = Em sin(βx).


Verification / Alternative check:
For a traveling wave, points of constant phase satisfy βx − ωt = constant ⇒ x = (ω/β)t + constant, i.e., motion in x over time. No such relation exists for sin(βx) alone.


Why Other Options Are Wrong:

All other options contain (βx ± ωt), representing waves moving in ±x. Both sine and cosine forms are equivalent representations shifted in phase.


Common Pitfalls:

Assuming sin(ωt) alone is a traveling wave; it has no spatial dependence and thus no propagation.


Final Answer:

e = Em sin(βx)
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