6. 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. 


The plot shows the total electric field intensity \( |E_{\text{tot}}|^2 \) as a function of \( z_0 \). 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 \( z_0 = \frac{\lambda}{2} \).

### Summary:
1. The net Poynting vector at \( z_0 = \frac{\lambda}{2} \) is zero, indicating complete destructive interference.
2. The plot of \( |E_{\text{tot}}|^2 \) as a function of \( z_0 \) shows constructive and destructive interference patterns, with maximum values where the waves reinforce each other and minimum values where they cancel out.

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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.

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) and 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.

Explore the interference of two plane waves with detailed calculations of the Poynting vector and electric field intensity. Learn how the superposition principle and interference theory apply to electromagnetic waves, and view a plot of |E_tot|^2 as a function of z_0. Suitable for undergraduate physics students.

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