A small circular loop antenna has radiation resistance Rr = 0.01 Ω for one turn. What is the radiation resistance for a 5-turn loop of the same size and frequency (assume tightly wound so area effectively scales with turns)?

Introduction:For electrically small loop antennas, radiation resistance is proportional to the square of the magnetic dipole moment, which itself scales with the number of turns N. Consequently, Rr scales as N^2 when the loop dimensions and frequency remain constant.

Given Data / Assumptions:

• One-turn radiation resistance Rr1 = 0.01 Ω.
• Five-turn loop, same geometry and frequency.
• Tight winding so total effective magnetic moment scales linearly with N.

Concept / Approach:

For small loops, Rr ∝ (N * A)^2, where A is the loop area. Holding A fixed and multiplying turns by N gives RrN = N^2 * Rr1.

Step-by-Step Solution:

Verification / Alternative check:

General small-loop expression Rr ≈ k * (N * A / λ^2)^2 (k is a constant) also shows quadratic scaling with N; the ratio method is therefore exact for fixed geometry.

Why Other Options Are Wrong:

• 0.002/0.01/0.05 Ω: underestimate due to ignoring the N^2 dependence.
• 0.5 Ω: would correspond to N ≈ 7.07 turns, not 5.

Common Pitfalls:

Assuming linear scaling with N; for small loops it is quadratic because radiated power depends on the square of current moment.

Final Answer:

0.25 Ω