Radiation resistance for a small square loop (single turn of side length l of size λ / 10 is approximately:

Radiation resistance for a small square loop (single turn of side length l) of size λ / 10 is approximately:

1. 1.38 ohms
2. 3.12 ohms
3. 73 ohms
4. 273 ohms

Correct Answer

Option (2): 3.12 ohms

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How to Solve This Problem (Fundamental Idea)

This is based on the concept of radiation resistance of small loop antennas. A small loop antenna behaves like a magnetic dipole.

Fundamental Formula

For a small loop antenna (area A, N turns):

R_r = 31,200 × ( (N A) / λ² )²

For a single-turn square loop:

• N = 1
• A = l²
• l = λ / 10

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Step-by-Step Solution

Step 1: Area of loop

A = l² = (λ / 10)² = λ² / 100

Step 2: Substitute into formula

R_r = 31,200 × ( (λ² / 100) / λ² )²

Step 3: Simplify

= 31,200 × (1 / 100)²

= 31,200 × 1 / 10,000

Step 4: Final result

R_r = 3.12 ohms

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Physical Meaning (Important Insight)

Radiation resistance represents the portion of input resistance that actually radiates electromagnetic energy.

For a small loop antenna (λ/10 size):

• Radiation is weak because the loop is electrically small
• Most energy is stored (reactive field) rather than radiated
• Hence radiation resistance is very low (~3 ohms)

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Summary

Smaller loop antennas produce lower radiation resistance, meaning they are less efficient radiators. As the loop size increases relative to wavelength, radiation resistance increases.

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