--- title: "Retarded potential importance and infinitesimal dipole" subject: "Propagation and Antenna" chapter: 1 key: "semester_6/propagation_and_antenna:retarded-potential-importance-dipole" source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/retarded-potential-importance-dipole site: BCT Engineering --- # Retarded potential importance and infinitesimal dipole Propagation and Antenna · Chapter 1: Radiation and Antenna Fundamentals - Asked in: 2074 Bhadra, 2+6 marks - Video: [Retarded Potential (Definition, Basics, Unit, Formula & Calculation) Explained in Antenna Parameters](https://www.youtube.com/watch?v=8D7xtQylQaQ) — Engineering Funda - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/retarded-potential-importance-dipole - Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-1 ## Question Explain retarded potential and their importance. Describe infinitesimal dipole with the help of suitable diagram, mathematical relations and the field pattern. ## Answer **Retarded potential** is the scalar or vector potential at a point P computed using the source value at the earlier (retarded) time t − R/v, because changes in a source take R/v seconds to reach P. ```text V = (1/4πε) ∫ ρ(t − R/v)/R dv A = (μ/4π) ∫ J(t − R/v)/R dv Phasor form: ρ, J times e^(−jkR) ``` #### Importance - Gives a simple way to find the fields of any current distribution: find A (and V), then H = ∇×A/μ, E = −jωA − ∇V. - Explains **finite propagation speed** and the phase delay e^(−jkr) of radiated fields. - Produces the **1/r radiation field** term; static potentials give only 1/r² fields, which carry no net power. - Basis for analysing dipoles, loops, arrays and all wire antennas. #### Infinitesimal dipole A very short wire, l ≪ λ (≤ λ/50), on the z-axis with uniform current I0. ```text z | P(r,θ,φ) ┌┼┐ / l │I0│/ θ └┼┘----- y | ``` Vector potential: Az = μ I0 l e^(−jkr)/(4πr). Fields: ```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 = jη k I0 l sinθ e^(−jkr)/(4πr) [1 + 1/(jkr) − 1/(kr)²] ``` Far field (kr ≫ 1): ```text Eθ ≈ jη k I0 l sinθ e^(−jkr)/(4πr), Hφ = Eθ/η ``` Radiation parameters: ```text P_rad = η(π/3)|I0 l/λ|² Rr = 80π²(l/λ)² Ω D = 1.5 (1.76 dBi), HPBW = 90° ``` #### Field pattern |Eθ| ∝ sinθ. ```text E-plane (vertical) H-plane (horizontal) z .---. .-. | .-. / \ ( ) | ( ) | o | '-' | '-' \ / | '---' ``` - Maximum at θ = 90°, nulls along the dipole axis. - Omnidirectional in the H-plane; 3-D pattern is a doughnut. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why must source values be taken at t - R/v? A. Because the source is delayed by the receiver B. Because changes in a source take R/v seconds to reach the point C. Because potentials are always negative D. Because R is large Answer: B. Propagation is at finite speed, so the point sees the source as it was R/v seconds earlier. (Key point: Retarded potential meaning) 2. Why are retarded potentials important for radiation? A. They give only static 1/r^2 fields B. They remove phase delay C. They produce the 1/r radiation field that carries net power D. They make fields independent of distance Answer: C. Static potentials give only 1/r^2 fields with no net power; retardation gives the 1/r radiation term. (Key point: Importance: radiation field) 3. What is the procedure to find fields of a current distribution? A. Find E first, then A B. Guess the pattern C. Find A (and V), then H = ∇×A/μ and E = -jωA - ∇V D. Use only Ohm's law Answer: C. The potentials are found first, then both fields follow by differentiation. (Key point: Method of finding fields) 4. Which set of values is correct for the infinitesimal dipole? A. Rr = 80π²(l/λ) Ω, D = 1.64 B. Rr = 73 Ω, D = 2.15 C. Rr = 80π²(l/λ)² Ω, D = 1.5, HPBW = 90° D. Rr = 120π Ω, D = 1 Answer: C. The short dipole has Rr = 80π²(l/λ)², D = 1.5 (1.76 dBi) and HPBW 90°. (Key point: Infinitesimal dipole parameters) 5. What is the H-plane pattern of the infinitesimal dipole? A. Figure-of-eight B. Circle (omnidirectional) C. Two narrow lobes D. Single lobe Answer: B. |Eθ| ∝ sinθ is independent of φ, so the H-plane is a circle; the E-plane is a figure-of-eight. 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