To tackle your question about the plane electromagnetic wave in a dielectric medium, we can break it down into several parts. Given that the relative permittivity (ε_r) is 3 and the peak electric field intensity (E₀) is 6 V/m, we can find the speed of the wave, intrinsic impedance, magnetic intensity, and the peak Poynting vector step by step.
Speed of the Wave
The speed of an electromagnetic wave in a dielectric medium can be calculated using the formula:
v = c / √ε_r
Where:
- c is the speed of light in vacuum (approximately 3 x 10^8 m/s)
- ε_r is the relative permittivity of the medium
Substituting the values:
v = (3 x 10^8 m/s) / √3 ≈ (3 x 10^8 m/s) / 1.732 ≈ 1.732 x 10^8 m/s
Intrinsic Impedance
The intrinsic impedance (η) of a medium is given by the formula:
η = √(μ/ε)
In a lossless dielectric, we can express this in terms of relative permittivity and permeability:
η = √(μ₀/ε₀) * √ε_r
Where:
- μ₀ is the permeability of free space (approximately 4π x 10^-7 H/m)
- ε₀ is the permittivity of free space (approximately 8.854 x 10^-12 F/m)
Calculating ε:
ε = ε_r * ε₀ = 3 * (8.854 x 10^-12 F/m) ≈ 2.6562 x 10^-11 F/m
Now, substituting into the intrinsic impedance formula:
η = √((4π x 10^-7 H/m) / (2.6562 x 10^-11 F/m)) ≈ 377 / √3 ≈ 217.5 Ω
Magnetic Field Intensity
The magnetic field intensity (H) can be determined from the electric field intensity (E) using the relationship:
H = E / η
Substituting the known values:
H = 6 V/m / 217.5 Ω ≈ 0.0276 A/m
Peak Poynting Vector
The Poynting vector (S) represents the power per unit area carried by the wave and is calculated as:
S = E × H
Using the peak values:
S = 6 V/m × 0.0276 A/m ≈ 0.1656 W/m²
Summary of Results
To summarize the calculations:
- Speed of the wave: approximately 1.732 x 10^8 m/s
- Intrinsic impedance: approximately 217.5 Ω
- Magnetic field intensity: approximately 0.0276 A/m
- Peak Poynting vector: approximately 0.1656 W/m²
This breakdown should give you a clear understanding of how to approach problems involving electromagnetic waves in dielectric media. Each step builds on the previous one, allowing you to see the relationships between electric fields, magnetic fields, and wave propagation. If you have any further questions or need clarification on any of these points, feel free to ask!
