--- title: "Retarded potential and EM wave generation" subject: "Propagation and Antenna" chapter: 1 key: "semester_6/propagation_and_antenna:retarded-potential-em-generation" source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/retarded-potential-em-generation site: BCT Engineering --- # Retarded potential and EM wave generation Propagation and Antenna · Chapter 1: Radiation and Antenna Fundamentals - Asked in: 2076 Bhadra, 2+4 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-em-generation - Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-1 ## Question What is the retarded potential? Explain how electromagnetic waves are generated by a conductor. ## Short answer Retarded potential uses the source values at the earlier time t − R/v, because fields travel at finite speed c. An alternating current accelerates charges; field lines detach each half cycle, close into loops, and the changing E and H sustain each other, producing an EM wave traveling outward at speed c. ## Answer **Retarded potential:** The scalar and vector potentials at a point at distance R from a time-varying source, calculated using the value of the source at an **earlier time t − R/v**, because the effect of the source travels at a finite speed v (= c in free space). The delay R/v is called the retardation. ```text V(t) = (1/4πε) ∫ ρ(t − R/v) / R dv A(t) = (μ/4π) ∫ J(t − R/v) / R dv ``` E and H are then found from B = ∇×A and E = −∇V − ∂A/∂t. #### Generation of EM waves by a conductor 1. An **alternating source** drives current along a conductor (e.g. a dipole). Charges move back and forth, so they are continuously **accelerated** – the basic condition for radiation: l·dI/dt = l·q·dv/dt. 2. In the first quarter cycle, charges collect at the ends and **electric field lines** form between the two arms; the moving charges (current) produce **magnetic field** circles around the wire. 3. In the next quarter cycle the current reverses and charges neutralize. The field lines already sent out lose their charges, so they **close on themselves** into loops and detach. 4. New lines of opposite direction form and push the loops outward. The changing E creates H and the changing H creates E (Maxwell's equations), so the disturbance travels at speed c as an **electromagnetic wave**. ```text source near field far field (~)--||-- --> ( ( ) ) --> ( ) ( ) --> dipole lines form loops detach ``` Close to the conductor the reactive (induction) field stores energy; far away the 1/r radiation field carries power away (E ⊥ H ⊥ direction of travel, E/H = 120π Ω). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the retarded potential? A. Potential computed using the source value at the earlier time t - R/v B. Potential computed at a later time t + R/v C. Potential that decreases with time D. Static potential with no time dependence Answer: A. Effects travel at finite speed v, so the potential at distance R depends on the source at t - R/v. (Key point: Meaning of retarded potential) 2. Which is the retarded vector potential? A. A = (μ/4π) ∫ J R dv B. A = (μ/4π) ∫ J(t - R/v)/R dv C. A = (1/4πε) ∫ ρ/R dv D. A = μ ∫ J dv Answer: B. A(t) = (μ/4π) ∫ J(t - R/v)/R dv, with J taken at the retarded time. (Key point: Vector potential expression) 3. In wave generation by a dipole, why do the earlier field lines form closed loops? A. The conductor's resistance cuts them B. A static field holds them C. The frequency drops D. When current reverses and charges neutralize, the lines lose their charges and close on themselves Answer: D. Without charges to end on, lines close into loops; new opposite lines push them outward. (Key point: Why field lines detach) 4. How does the disturbance keep travelling as an EM wave? A. Changing E creates H and changing H creates E, travelling at c B. Charges follow it through space C. The conductor guides it indefinitely D. Only the electric field is needed Answer: A. Maxwell's equations couple changing E and H, so the wave propagates at speed c. (Key point: Propagation of the wave) 5. Which is true of the field far from the conductor? A. Reactive field stores energy and dominates B. The 1/r radiation field carries power away, with E/H = 120π Ω C. E and H are parallel to the direction of travel D. The field falls as 1/r^3 Answer: B. Far away the 1/r radiation field dominates, transverse with E/H = 120π Ω. 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