Magnetic Dipole vs Electric Dipole Explained

Magnetic Dipole Antenna:

Final Standard Formula

R_r = 31200 × (A / λ²)²

Where:

• A = Area of loop antenna
• λ = Wavelength

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Step-by-Step Derivation Idea

The derivation starts from the fundamental expression of radiated power of a small loop (magnetic dipole model).

1. Radiated Power Expression

P_rad = (η k⁴ I² A²) / (6π)

Where:

• η = intrinsic impedance of free space = 120π
• k = wave number = 2π / λ

2. Radiation Resistance Definition

R_r = (2 P_rad) / I²

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Substitution of Constants

Substitute η and k into the power expression:

P_rad = [120π × (2π/λ)⁴ × I² A²] / (6π)

Simplification Steps

(2π)⁴ = 16π⁴

P_rad = (120π × 16π⁴) / (6π) × (I² A² / λ⁴)

Simplify constants:

• 120 / 6 = 20
• π cancels partially → π⁴ remains

P_rad = 320 π⁴ × (I² A² / λ⁴)

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Convert Power to Radiation Resistance

R_r = (2 P_rad) / I²

R_r = 640 π⁴ × (A² / λ⁴)

Numerical Evaluation

Since:

π⁴ ≈ 97.41

640 × 97.41 ≈ 31200

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Final Result

R_r = 31200 × (A / λ²)²

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Key Insight

The constant 31200 is not arbitrary. It is derived from:

• Intrinsic impedance of free space (120π)
• Wave number (2π/λ)
• Radiation integral of a small loop (magnetic dipole theory)

In simple terms, it represents the combined effect of electromagnetic wave propagation in free space as predicted by Maxwell’s equations.

Can the Formula Rr = 31200 × (A / λ²)² Be Used for a Half-Wave Dipole?

The short answer is: No, this formula cannot be used for a half-wave dipole antenna.

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Why This Formula Works Only for Small Loop Antennas

The formula:

R_r = 31200 × (A / λ²)²

is derived specifically for a small loop antenna (magnetic dipole), where:

• The loop circumference is much smaller than wavelength (≪ λ)
• The antenna behaves like a magnetic dipole
• The field is dominated by loop current and enclosed area (A)

So the key parameter is area of loop (A), not length of conductor.

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Why It Does NOT Apply to a Half-Wave Dipole

A half-wave dipole (λ/2 antenna) is completely different:

• It is a linear electric dipole, not a loop
• Radiation comes from charge separation along a wire
• Current distribution is sinusoidal along length
• Not defined by enclosed area (A = not applicable)

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Correct Radiation Resistance Formula for Half-Wave Dipole

For a half-wave dipole, the radiation resistance is derived from electromagnetic field integration and is approximately:

R_r ≈ 73 Ω

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Mathematical Comparison (Why Loop Formula Fails)

Loop Antenna (Magnetic Dipole)

R_r = 31200 × (A / λ²)²

Depends on:

• Enclosed area A
• Very small physical structure (A ≪ λ²)

Half-Wave Dipole (Electric Dipole)

R_r ≈ 73 Ω

Depends on:

• Length distribution (≈ λ/2)
• Sinusoidal current distribution
• No enclosed area concept

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Physical Reason (Key Insight)

The fundamental reason the formula cannot be reused is:

• Small loop → radiation from magnetic dipole moment
• Half-wave dipole → radiation from electric dipole moment

These are two different electromagnetic source models with different field equations, so their radiation resistance formulas are not interchangeable.

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Conclusion

The formula Rr = 31200 × (A / λ²)² is strictly valid for electrically small loop antennas only.

For a half-wave dipole antenna, the correct radiation resistance is approximately 73 ohms, derived from a completely different electromagnetic model.

Radiation Resistance Formulas for Dipole and Other Antennas

Different antennas have different radiation resistance formulas because their radiation mechanisms are different (electric dipole, magnetic loop, etc.). Below is a clear summary of the most important cases.

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1. Small Loop Antenna (Magnetic Dipole)

Used when loop size is much smaller than wavelength (≪ λ).

R_r = 31200 × (A / λ²)²

• A = area of loop (m²)
• λ = wavelength
• Valid only for electrically small loops

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2. Short Dipole (Very Small Length ≪ λ)

A short dipole is a linear antenna much shorter than wavelength.

R_r = 80π² (l / λ)²

Approximation:

R_r ≈ 197 (l / λ)² Ω

• l = length of dipole
• Valid for very small dipoles (l ≪ λ)

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3. Half-Wave Dipole (λ/2 Antenna)

This is the most commonly used antenna in practice.

R_r ≈ 73 Ω

• Length ≈ λ/2
• Sinusoidal current distribution
• Maximum practical efficiency for simple dipoles

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4. Quarter-Wave Monopole

A monopole above a ground plane behaves like half a dipole.

R_r ≈ 36.5 Ω

• Equivalent to half-wave dipole image theory
• Used in antennas mounted on ground planes

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5. General Full-Wave Dipole (Length ≈ λ)

When dipole length increases beyond λ/2, radiation resistance increases and becomes more complex.

R_r varies (typically 100 Ω to 200+ Ω depending on length)

• Depends on current distribution
• Requires numerical methods for exact values

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Quick Comparison Table

Antenna Type	Radiation Resistance	Condition
Small Loop	31200 (A/λ²)²	Loop ≪ λ
Short Dipole	≈ 197 (l/λ)² Ω	l ≪ λ
Half-Wave Dipole	≈ 73 Ω	l ≈ λ/2
Quarter-Wave Monopole	≈ 36.5 Ω	Ground plane present

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Key Insight

Radiation resistance depends strongly on antenna geometry:

• Loops → magnetic dipole radiation
• Dipoles → electric dipole radiation
• Size relative to wavelength controls efficiency

That is why different formulas are required for different antenna types.

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