--- title: "Operation of infinitesimal dipole" subject: "Propagation and Antenna" chapter: 1 key: "semester_6/propagation_and_antenna:infinitesimal-dipole-operation" source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/infinitesimal-dipole-operation site: BCT Engineering --- # Operation of infinitesimal dipole Propagation and Antenna · Chapter 1: Radiation and Antenna Fundamentals - Asked in: 2080 Chaitra, 2+6 marks; 2070 Bhadra, 2+8 marks (asked 2 times) - 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-operation - Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-1 ## Question Define an antenna. Explain the operation of infinitesimal dipole with the help of mathematical relations and the field pattern. ## Short answer An infinitesimal dipole is a very short wire l≪λ carrying uniform current. Its far fields are Eθ≈jηkI0l sinθ e^(−jkr)/(4πr) and Hφ=Eθ/η. Radiated power is η(π/3)(I0l/λ)^2, radiation resistance Rr=80π²(l/λ)², directivity is 1.5, and the pattern is a doughnut shape. ## Answer An **antenna** is a transducer that converts guided electromagnetic energy on a transmission line into free-space radiated waves (and vice versa). #### Infinitesimal (Hertzian) dipole An infinitesimal dipole is a very short straight wire of length l ≪ λ (l ≤ λ/50) carrying a **uniform current** I(z) = I0 along its length, placed at the origin along the z-axis. Real antennas can be built as a sum of such elements. ```text z | P(r, θ, φ) | / ┌┼┐/ r l │I0│ θ └┼┘ | +----------- y / x ``` #### Vector potential ```text A = âz μ I0 l e^(−jkr) / (4π r) Ar = Az cosθ, Aθ = −Az sinθ ``` #### Field components (H = ∇×A/μ, E from Maxwell) ```text Hφ = j k I0 l sinθ e^(−jkr)/(4πr) · [1 + 1/(jkr)] Er = η I0 l cosθ e^(−jkr)/(2πr²) · [1 + 1/(jkr)] Eθ = jηkI0 l sinθ e^(−jkr)/(4πr) · [1 + 1/(jkr) − 1/(kr)²] Hr = Hθ = Eφ = 0 ``` Field regions: - **kr ≪ 1 (near/reactive field):** 1/r² and 1/r³ terms dominate; energy is stored, not radiated. - **kr ≫ 1 (far field):** only 1/r terms remain: ```text Eθ ≈ jη k I0 l sinθ e^(−jkr) / (4πr) Hφ ≈ j k I0 l sinθ e^(−jkr) / (4πr) Er ≈ 0, Eθ / Hφ = η = 120π Ω ``` The far fields are transverse, in phase, and form a TEM wave. #### Power, radiation resistance and directivity ```text P_rad = η (π/3) |I0 l / λ|² Rr = 2 P_rad / |I0|² = 80π² (l/λ)² Ω U(θ) ∝ sin²θ → D = 1.5 (1.76 dBi), HPBW = 90° ``` Example: l = λ/50 → Rr = 80π²/2500 ≈ 0.316 Ω (very small, so efficiency is low). #### Field pattern The far-field pattern is |Eθ| ∝ sinθ: - **E-plane** (vertical plane): figure-of-eight, maximum at θ = 90°, nulls along the axis (θ = 0°, 180°). - **H-plane** (horizontal plane): circle – omnidirectional. - 3-D: doughnut shape. ```text E-plane (sinθ) H-plane z y | .---. .-. | .-. / \ ( ) | ( ) | o | x '-' | '-' \ / | '---' null on axis circle ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which describes an infinitesimal (Hertzian) dipole? A. A half-wavelength wire with sinusoidal current B. A very short wire, l much less than λ (l ≤ λ/50), with uniform current I0 C. A loop of one wavelength circumference D. A long wire terminated in a resistor Answer: B. It is a very short wire along z with uniform current, used as a building block of real antennas. (Key point: Definition of infinitesimal dipole) 2. What is true of the far fields (kr much greater than 1) of the infinitesimal dipole? A. Er dominates and falls as 1/r^3 B. Only Hφ exists C. Eθ and Hφ vary as 1/r, are in phase and Eθ/Hφ = 120π Ω D. Fields are stored and not radiated Answer: C. Only 1/r terms remain; the fields are transverse and in phase with ratio η = 120π Ω, forming a TEM wave. (Key point: Far-field behaviour) 3. What is the radiation resistance of an infinitesimal dipole? A. 73 Ω independent of length B. 80π^2 (l/λ) Ω C. 40π (l/λ) Ω D. 80π^2 (l/λ)^2 Ω Answer: D. Rr = 80π^2 (l/λ)^2 Ω, e.g. about 0.316 Ω for l = λ/50, so efficiency is low. (Key point: Radiation resistance) 4. What are the far-field pattern shape and directivity of the infinitesimal dipole? A. sinθ field pattern, D = 1.5, HPBW 90° B. cosθ field pattern, D = 1.64 C. Circular in the E-plane, D = 1 D. sinθ pattern with D = 3 Answer: A. |Eθ| ∝ sinθ gives a doughnut: figure-of-eight in the E-plane, circle in the H-plane, D = 1.5 (1.76 dBi). (Key point: Pattern and directivity) 5. Which describes the near (reactive) field region kr much less than 1? A. Only 1/r terms remain B. 1/r^2 and 1/r^3 terms dominate and energy is stored, not radiated C. The field is zero D. Er is zero Answer: B. In the near field the higher inverse-power terms dominate and represent stored reactive energy. 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