Math Problem Statement
- Interference and Power Problem 6B: Two copies of a plane wave are made to be present in the same region of free space and are expressed as
E ⃗_1=E_x a ̂_x e^(-jβz) E ⃗_2=E_x a ̂_x e^(-jβ(z+z_0))
with 𝐸𝑥 = 5 [V/m] and 𝑧0 = 𝜆/2 where 𝜆 is the wavelength.
- What is the net Poynting vector in this region of space?
- Plot |E ⃗_tot|2 as a function of 𝑧0 where E ⃗_tot=E ⃗_1+E ⃗_2.
If you do this problem numerically, you may choose a reasonable value of 𝛽.
Solution
The plot shows the total electric field intensity as a function of . You can observe the oscillating behavior caused by the interference of the two waves, with peaks at constructive interference points and zero intensity at destructive interference points, such as .
Summary:
- The net Poynting vector at is zero, indicating complete destructive interference.
- The plot of as a function of shows constructive and destructive interference patterns, with maximum values where the waves reinforce each other and minimum values where they cancel out.
Would you like further details or have any questions? Here are some related questions to consider:
- What happens to the Poynting vector at points of constructive interference?
- How does the wavelength affect the interference pattern?
- Can interference occur in non-planar wave geometries?
- What role does the phase difference play in this type of interference?
- How would this analysis change if the amplitude of the two waves were different?
Tip: Interference patterns are sensitive to both the phase and amplitude of the waves involved, leading to a rich variety of behaviors depending on their relative values.
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Math Problem Analysis
Mathematical Concepts
Electromagnetic Waves
Wave Interference
Poynting Vector
Plane Waves
Formulas
E⃗_tot = E⃗_1 + E⃗_2
|E⃗_tot|^2 = |E_x e^(-jβz) + E_x e^(-jβ(z+z_0))|^2
Poynting vector S⃗ = E⃗ × H⃗
Theorems
Superposition Principle
Interference Theory
Suitable Grade Level
Undergraduate (Physics/Electromagnetism)
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