---
title: "Infinitesimal dipole fields and basic parameters"
subject: "Propagation and Antenna"
chapter: 1
key: "semester_6/propagation_and_antenna:infinitesimal-dipole-fields-parameters"
source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/infinitesimal-dipole-fields-parameters
site: BCT Engineering
---

# Infinitesimal dipole fields and basic parameters

Propagation and Antenna · Chapter 1: Radiation and Antenna Fundamentals

- Asked in: 2071 Magh, 4+4 marks
- Video: [Ideal wire antennas: infinitesimal dipoles. | 17/62 | UPV](https://www.youtube.com/watch?v=MeDLPZswYVI) — Universitat Politècnica de València - UPV
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/infinitesimal-dipole-fields-parameters
- Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-1

## Question

Describe the operation of infinitesimal dipole with necessary mathematical relation for electric field and magnetic field. What are basic antenna parameters? Explain briefly on any four parameters.

## Answer

#### Infinitesimal dipole
An infinitesimal (Hertzian) dipole is a very short conductor, l ≪ λ (≤ λ/50), placed along the z-axis with a **uniform current** I0 along its length.

Retarded vector potential:

```text
Az = μ I0 l e^(−jkr) / (4π r)
```

Magnetic field (H = ∇×A/μ):

```text
Hφ = j k I0 l sinθ e^(−jkr)/(4πr) · [1 + 1/(jkr)]
Hr = Hθ = 0
```

Electric field (from ∇×H = jωεE):

```text
Er = η I0 l cosθ e^(−jkr)/(2πr²) · [1 + 1/(jkr)]
Eθ = jηkI0 l sinθ e^(−jkr)/(4πr)
     · [1 + 1/(jkr) − 1/(kr)²]
Eφ = 0
```

Far field (kr ≫ 1):

```text
Eθ ≈ jη k I0 l sinθ e^(−jkr)/(4πr)
Hφ ≈ Eθ / η,  η = 120π Ω,  Er ≈ 0
```

The pattern is ∝ sinθ (doughnut), D = 1.5, Rr = 80π²(l/λ)² Ω.

#### Basic antenna parameters
Radiation pattern, directivity, gain, efficiency, beamwidth, bandwidth, input impedance, radiation resistance, polarization, effective aperture.

1. **Radiation pattern:** graph of the radiated field or power versus direction (θ, φ) in the far field. It shows main lobe, side lobes, back lobe and nulls.
2. **Directivity (D):** ratio of the maximum radiation intensity to the average radiation intensity:

```text
D = U_max / U_avg = 4π U_max / P_rad
```

   Isotropic: D = 1; λ/2 dipole: D = 1.64.
3. **Gain (G):** directivity including losses: G = η_e · D, where η_e = radiation efficiency. Expressed in dBi.
4. **Radiation efficiency:**

```text
η_e = P_rad / P_in = Rr / (Rr + RL)
```

   RL = ohmic loss resistance.

Other parameters: **HPBW** – angle between half-power points of the main lobe; **bandwidth** – frequency range where parameters stay within limits; **input impedance** Z_in = R_in + jX_in at the terminals; **polarization** – orientation of the E-field vector with time.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the size condition of an infinitesimal dipole?
   A. l = λ/2
   B. l = λ
   C. l much less than λ (≤ λ/50) with uniform current
   D. l much greater than λ
   Answer: C. It is a very short conductor with uniform current I0. (Key point: Dipole size)

2. Which components are nonzero for the infinitesimal dipole along z?
   A. Eφ, Hr, Hθ
   B. Er, Eθ and Hφ only
   C. Eφ and Hθ only
   D. Hr and Er only
   Answer: B. Hr = Hθ = Eφ = 0; the nonzero components are Hφ, Er and Eθ. (Key point: Field components)

3. What is the radiation efficiency of an antenna?
   A. RL/(Rr + RL)
   B. Rr/RL
   C. P_in/P_rad
   D. Rr/(Rr + RL)
   Answer: D. η = P_rad/P_in = Rr/(Rr + RL), where RL is the loss resistance. (Key point: Radiation efficiency)

4. Which is the definition of directivity?
   A. Ratio of maximum to average radiation intensity, D = 4πU_max/P_rad
   B. Ratio of input power to radiated power
   C. Ratio of P_rad to P_in
   D. Angle between half-power points
   Answer: A. D = U_max/U_avg = 4πU_max/P_rad; isotropic D = 1, λ/2 dipole 1.64. (Key point: Directivity definition)

5. How are gain and directivity related?
   A. G = D/η
   B. G = η·D, with η the radiation efficiency
   C. G = D + η
   D. G = D^2
   Answer: B. Gain is directivity including losses: G = η_e D. (Key point: Gain and directivity)

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