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A wave is propagated in a waveguide at frequency of 9 GHz and separation is 2 cm between walls Calculate group velocity for dominant mode.
- a)1.8 x 108 m/sec
- b)5 x 108 m/sec
- c)3 x 108 m/sec
- d)1.5 x 108 m/sec
Correct answer is option 'A'. Can you explain this answer?
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A wave is propagated in a waveguide at frequency of 9 GHz and separati...
Vg . Vp = C2
where 
.Most Upvoted Answer
A wave is propagated in a waveguide at frequency of 9 GHz and separati...
Given:
- Frequency of the wave, f = 9 GHz = 9 x 10^9 Hz
- Separation between walls, d = 2 cm = 0.02 m
To find:
Group velocity for the dominant mode
Formula:
The group velocity (vg) can be calculated using the formula:
vg = c / β
Where:
c is the speed of light in a vacuum = 3 x 10^8 m/s
β is the phase constant
Solution:
1. Calculate the phase constant (β) using the formula:
β = 2πf√(εμ - (λ/2d)^2)
Where:
ε is the permittivity of the waveguide
μ is the permeability of the waveguide
λ is the wavelength of the wave in the waveguide
2. Calculate the wavelength (λ) using the formula:
λ = c / f
3. Substitute the values of ε, μ, λ, and d into the formula for β and calculate its value.
4. Substitute the value of β and the speed of light (c) into the formula for group velocity (vg) and calculate its value.
Now, let's calculate step by step:
Step 1:
- The permittivity of air (ε0) is approximately 8.854 x 10^-12 F/m.
- The permeability of free space (μ0) is approximately 4π x 10^-7 T/m.
- Therefore, the permittivity (ε) and permeability (μ) of the waveguide can be assumed to be ε0 and μ0, respectively.
Step 2:
- λ = c / f
= (3 x 10^8 m/s) / (9 x 10^9 Hz)
= 1/3 m = 0.333 m
Step 3:
- β = 2πf√(εμ - (λ/2d)^2)
= 2π(9 x 10^9 Hz)√(8.854 x 10^-12 F/m)(4π x 10^-7 T/m) - ((0.333 m)/(2 x 0.02 m))^2
Simplifying the equation:
= 2π(9 x 10^9 Hz)√(8.854 x 10^-12 F/m)(4π x 10^-7 T/m) - (0.333/0.04)^2
= 2π(9 x 10^9 Hz)√(35.416 x 10^-19) - (8.325)^2
= 2π(9 x 10^9 Hz)(5.95 x 10^-10) - 69.281
= 2π(5.355 x 10^9) - 69.281
= 33.718 x 10^9π - 69.281
= 106.128 x 10^9 - 69.281
= 106.128 x 10^9 - 69.281
≈ 106.128 x 10^9
Step 4:
- vg = c / β
= (3 x
- Frequency of the wave, f = 9 GHz = 9 x 10^9 Hz
- Separation between walls, d = 2 cm = 0.02 m
To find:
Group velocity for the dominant mode
Formula:
The group velocity (vg) can be calculated using the formula:
vg = c / β
Where:
c is the speed of light in a vacuum = 3 x 10^8 m/s
β is the phase constant
Solution:
1. Calculate the phase constant (β) using the formula:
β = 2πf√(εμ - (λ/2d)^2)
Where:
ε is the permittivity of the waveguide
μ is the permeability of the waveguide
λ is the wavelength of the wave in the waveguide
2. Calculate the wavelength (λ) using the formula:
λ = c / f
3. Substitute the values of ε, μ, λ, and d into the formula for β and calculate its value.
4. Substitute the value of β and the speed of light (c) into the formula for group velocity (vg) and calculate its value.
Now, let's calculate step by step:
Step 1:
- The permittivity of air (ε0) is approximately 8.854 x 10^-12 F/m.
- The permeability of free space (μ0) is approximately 4π x 10^-7 T/m.
- Therefore, the permittivity (ε) and permeability (μ) of the waveguide can be assumed to be ε0 and μ0, respectively.
Step 2:
- λ = c / f
= (3 x 10^8 m/s) / (9 x 10^9 Hz)
= 1/3 m = 0.333 m
Step 3:
- β = 2πf√(εμ - (λ/2d)^2)
= 2π(9 x 10^9 Hz)√(8.854 x 10^-12 F/m)(4π x 10^-7 T/m) - ((0.333 m)/(2 x 0.02 m))^2
Simplifying the equation:
= 2π(9 x 10^9 Hz)√(8.854 x 10^-12 F/m)(4π x 10^-7 T/m) - (0.333/0.04)^2
= 2π(9 x 10^9 Hz)√(35.416 x 10^-19) - (8.325)^2
= 2π(9 x 10^9 Hz)(5.95 x 10^-10) - 69.281
= 2π(5.355 x 10^9) - 69.281
= 33.718 x 10^9π - 69.281
= 106.128 x 10^9 - 69.281
= 106.128 x 10^9 - 69.281
≈ 106.128 x 10^9
Step 4:
- vg = c / β
= (3 x
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A wave is propagated in a waveguide at frequency of 9 GHz and separation is 2 cm between walls Calculate group velocity for dominant mode.a)1.8 x 108 m/secb)5 x 108 m/secc)3 x 108 m/secd)1.5 x 108 m/secCorrect answer is option 'A'. Can you explain this answer?
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