SME
Engineering Electromagnetics
Wave propagation
Use this for “A 3 GHz plane wave travels in a medium with εᵣ = 4. Find λ and η.”
Diagram
H₀
=
E₀
η
=
10
V/m
÷
188.3
Ω
=
0.05309
A/m
S
=
E₀
2
2η
=
10
2
2 × 188.3 Ω
S
=
0.2655
W/
m
2
E ⟂ H ⟂ the direction of travel, in step; the Poynting vector S = E × H points along it.
This is an example. Type your own values to start a new problem.
v (m/s)
=
c
÷
εᵣ
λ (cm)
=
v (m/s)
f (GHz)
η (Ω)
=
η₀
÷
εᵣ
E₀
E amplitude
example
H₀
H amplitude
calculated
S
Power density
calculated
Formulas
v
=
3 ×
10
8
÷
√
εᵣ
1.5 ×
10
8
=
3 ×
10
8
÷
√
4
λ
=
v
f
0.05
=
1.5 ×
10
8
3,000,000,000
η
=
376.7
÷
√
εᵣ
188.35
=
376.7
÷
√
4
H₀
=
E₀
η
0.05309
=
10
188.35
S
=
E₀
2
2 ×
η
0.2655
=
10
2
2 × 188.35
Formulas are worked in Hz, m, A/m. The step-by-step shows the conversions.
Assumptions
•
A uniform plane wave in a lossless, nonmagnetic medium (μᵣ = 1): E and H are in step and at right angles.
•
In vacuum c = 3.00 × 10⁸ m/s and η₀ = 376.7 Ω; the medium divides both by √εᵣ.
•
E₀ and H₀ are peak amplitudes, so S = E₀² ÷ (2η) is the time-averaged power density.
Step-by-step
WE KNOW
f = 3 GHz, εᵣ = 4, E₀ = 10 V/m
FIND
speed (v), wavelength (λ), wave impedance (η), H amplitude (H₀), power density (S)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to Hz, m, A/m).
f = 3 GHz = 3,000,000,000 Hz (1 GHz = 10⁹ Hz)
Step 2 · Find speed (v)
Use v = 3 × 10⁸ ÷ √(εᵣ)
Light slows by √εᵣ in the medium (μᵣ = 1).
v
=
3 ×
10
8
÷
√
4
v
=
3 ×
10
8
2
v
=
1.5 ×
10
8
m/s
Step 3 · Find wavelength (λ)
Use λ = v ÷ f
One wavelength passes each cycle.
λ
=
1.5 ×
10
8
3,000,000,000
λ
=
0.05
m
Step 4 · Find wave impedance (η)
Use η = 376.7 ÷ √(εᵣ)
η
=
√(
μ
ε
):
the
vacuum’s
376.7
Ω,
smaller
by
√εᵣ.
η
=
376.7
÷
√
4
η
=
376.7
2
η
=
188.35
Ω
Step 5 · Find H amplitude (H₀)
Use H₀ = E₀ ÷ η
The wave impedance is the ratio of E to H.
H₀
=
10
188.35
H₀
=
0.05309
A/m
Step 6 · Find power density (S)
Use S = E₀² ÷ (2 × η)
Average
E
×
H
over
a
cycle:
half
the
peak
product,
and
H₀
=
E₀
η
.
S
=
10
2
2 × 188.35
S
=
10
2
376.7
S
=
100
376.7
S
=
0.2655
W/
m
2
Step 7 · Convert the answers
Change each answer back to the unit you chose.
λ = 0.05 m = 5 cm (1 m = 100 cm)
H₀ = 0.05309 A/m = 53.0926 mA/m (1 A/m = 1000 mA/m)
Check
Put the numbers back in. Both sides should match.
1.5 ×
10
8
=
3 ×
10
8
÷
√
4
✓
0.05
=
1.5 ×
10
8
3,000,000,000
✓
188.35
=
376.7
÷
√
4
✓
0.05309
=
10
188.35
✓
0.2655
=
10
2
2 × 188.35
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.