Introduction / Context:
Radiation resistance is the equivalent resistance of an antenna that represents the power radiated as electromagnetic waves. Short dipoles (length ≪ λ) have radiation resistance that depends on their electrical length. This problem requires applying the standard formula for a dipole with length λ/10.
Given Data / Assumptions:
- Antenna length L = λ/10.
- Type: short dipole (length much less than half a wavelength).
- Free-space radiation resistance formula is applicable.
Concept / Approach:
For a short dipole antenna: Rr ≈ 80π^2 (L/λ)^2. Substituting L/λ = 1/10 gives a numerical value.
Step-by-Step Solution:
Step 1: Formula: Rr = 80π^2 (L/λ)^2.Step 2: Substitute L/λ = 1/10 = 0.1.Step 3: Compute (0.1)^2 = 0.01.Step 4: Multiply: 80 * π^2 * 0.01 ≈ 80 * 9.8696 * 0.01 ≈ 7.9 Ω.Step 5: Closest option is 7.8 Ω.
Verification / Alternative check:
Cross-check with known rule: Radiation resistance for λ/2 dipole ≈ 73 Ω; for λ/10 dipole it should be much smaller (~8 Ω), consistent with result.
Why Other Options Are Wrong:
8 Ω: Approximation but less precise than 7.8 Ω.780 Ω and 78 Ω: Orders of magnitude too high.1 Ω: Too low compared with formula prediction.
Common Pitfalls:
Forgetting the squared term (L/λ)^2; using λ/2 dipole values; confusing input impedance with radiation resistance.
Final Answer:
7.8 Ω
Discussion & Comments