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What is the approximate radiation resistance of a λ/16 wire dipole antenna in free space?

Difficulty: Medium

Correct Answer: 3 Ω

Explanation:


Introduction / Context:
Radiation resistance is the effective resistance representing power radiated as electromagnetic energy. For dipoles shorter than half a wavelength, the radiation resistance decreases sharply with decreasing length. This question asks for the resistance of a dipole λ/16 long.


Given Data / Assumptions:

  • Dipole length = λ/16.
  • Operating in free space.
  • Standard approximation formula applies.


Concept / Approach:
The approximate formula for short dipoles: Rr ≈ 80π^2 (L/λ)^2. For a dipole, when length = λ/16, we substitute to calculate the value.


Step-by-Step Solution:

Step 1: Length ratio: L/λ = 1/16 = 0.0625.Step 2: Square it: (0.0625)^2 = 0.00390625.Step 3: Multiply constant: 80π^2 ≈ 789.57.Step 4: Compute: 789.57 * 0.00390625 ≈ 3.08 Ω.Step 5: Therefore Rr ≈ 3 Ω.


Verification / Alternative check:

Known reference: λ/2 dipole ≈ 73 Ω, λ/16 dipole ≈ few ohms; result is consistent.


Why Other Options Are Wrong:

1 Ω: Underestimates resistance; requires even shorter length.13 Ω and 30 Ω: Overestimates by factors of 4–10.7.3 Ω: Closer to λ/10 dipole, not λ/16.


Common Pitfalls:

Applying half-wave dipole resistance values to short dipoles; ignoring quadratic dependence on (L/λ).


Final Answer:

3 Ω
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