Math Problem Statement

  1. 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.

  1. What is the net Poynting vector in this region of space?
  2. 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 Etot2|E_{\text{tot}}|^2 as a function of z0z_0. 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 z0=λ2z_0 = \frac{\lambda}{2}.

Summary:

  1. The net Poynting vector at z0=λ2z_0 = \frac{\lambda}{2} is zero, indicating complete destructive interference.
  2. The plot of Etot2|E_{\text{tot}}|^2 as a function of z0z_0 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:

  1. What happens to the Poynting vector at points of constructive interference?
  2. How does the wavelength affect the interference pattern?
  3. Can interference occur in non-planar wave geometries?
  4. What role does the phase difference play in this type of interference?
  5. 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)