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Radiation resistance of a small dipole current element of length l at a frequency of 3 GHz is 3 ohms. If the length is changed by 1%, then the percentage change in the radiation resistance, rounded off to two decimal places, is _________%.
Correct answer is '2'. Can you explain this answer?
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Radiation resistance of a small dipole current element of length l at ...
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Radiation resistance of a small dipole current element of length l at ...
Given:
- Frequency of operation, f = 3 GHz = 3 × 10^9 Hz
- Radiation resistance of the small dipole current element, R = 3 Ω
To find:
- Percentage change in the radiation resistance when the length of the dipole current element is changed by 1%
Solution:
Step 1: Calculating the wavelength
- The wavelength λ of the signal can be calculated using the formula:
λ = c / f
where c is the speed of light in a vacuum, which is approximately 3 × 10^8 m/s.
λ = (3 × 10^8) / (3 × 10^9) = 0.1 meter = 10 cm
Step 2: Calculating the initial length of the dipole current element
- The length of the dipole current element is given as l.
- We can calculate l using the formula:
l = λ / 2
l = 10 cm / 2 = 5 cm = 0.05 meter
Step 3: Calculating the new length of the dipole current element
- The length of the dipole current element is changed by 1%.
- The new length l' can be calculated as:
l' = l + (1% of l)
l' = 0.05 meter + (1/100 × 0.05 meter)
l' = 0.05 meter + 0.0005 meter = 0.0505 meter
Step 4: Calculating the new radiation resistance
- The radiation resistance of a dipole antenna is given by the formula:
R' = R × (l' / l)^2
R' = 3 Ω × (0.0505 meter / 0.05 meter)^2
R' = 3 Ω × (1.01)^2
R' ≈ 3.0603 Ω
Step 5: Calculating the percentage change in the radiation resistance
- The percentage change can be calculated using the formula:
Percentage change = (|R' - R| / R) × 100%
Percentage change = (|3.0603 Ω - 3 Ω| / 3 Ω) × 100%
Percentage change = (0.0603 Ω / 3 Ω) × 100%
Percentage change ≈ 2%
Therefore, the percentage change in the radiation resistance when the length of the dipole current element is changed by 1% is approximately 2%.
- Frequency of operation, f = 3 GHz = 3 × 10^9 Hz
- Radiation resistance of the small dipole current element, R = 3 Ω
To find:
- Percentage change in the radiation resistance when the length of the dipole current element is changed by 1%
Solution:
Step 1: Calculating the wavelength
- The wavelength λ of the signal can be calculated using the formula:
λ = c / f
where c is the speed of light in a vacuum, which is approximately 3 × 10^8 m/s.
λ = (3 × 10^8) / (3 × 10^9) = 0.1 meter = 10 cm
Step 2: Calculating the initial length of the dipole current element
- The length of the dipole current element is given as l.
- We can calculate l using the formula:
l = λ / 2
l = 10 cm / 2 = 5 cm = 0.05 meter
Step 3: Calculating the new length of the dipole current element
- The length of the dipole current element is changed by 1%.
- The new length l' can be calculated as:
l' = l + (1% of l)
l' = 0.05 meter + (1/100 × 0.05 meter)
l' = 0.05 meter + 0.0005 meter = 0.0505 meter
Step 4: Calculating the new radiation resistance
- The radiation resistance of a dipole antenna is given by the formula:
R' = R × (l' / l)^2
R' = 3 Ω × (0.0505 meter / 0.05 meter)^2
R' = 3 Ω × (1.01)^2
R' ≈ 3.0603 Ω
Step 5: Calculating the percentage change in the radiation resistance
- The percentage change can be calculated using the formula:
Percentage change = (|R' - R| / R) × 100%
Percentage change = (|3.0603 Ω - 3 Ω| / 3 Ω) × 100%
Percentage change = (0.0603 Ω / 3 Ω) × 100%
Percentage change ≈ 2%
Therefore, the percentage change in the radiation resistance when the length of the dipole current element is changed by 1% is approximately 2%.
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Radiation resistance of a small dipole current element of length l at a frequency of 3 GHz is 3 ohms. If the length is changed by 1%, then the percentage change in the radiation resistance, rounded off to two decimal places, is _________%.Correct answer is '2'. Can you explain this answer?
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