---
title: "Retarded time and retarded potentials"
subject: "Propagation and Antenna"
chapter: 1
key: "semester_6/propagation_and_antenna:retarded-time-vector-scalar-potential"
source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/retarded-time-vector-scalar-potential
site: BCT Engineering
---

# Retarded time and retarded potentials

Propagation and Antenna · Chapter 1: Radiation and Antenna Fundamentals

- Asked in: 2080 Asoj, 2+5 marks
- Video: [Retarded Potential in time-varying fields | Antenna Parameters | Engineering Funda](https://www.youtube.com/watch?v=iT7MYpkETUs) — 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-time-vector-scalar-potential
- Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-1

## Question

What is the retarded time? Explain the retarded vector and scalar potential with mathematical expression.

## Short answer

Retarded time is t' = t − R/v, the earlier time at which the source state must be taken. The retarded potentials are V = (1/4πε)∫ρ(t−R/v)/R dv and A = (μ/4π)∫J(t−R/v)/R dv; they satisfy the wave equation and give E and B via E = −∇V − ∂A/∂t, B = ∇×A.

## Answer

**Retarded time** is the earlier time at which a signal must leave the source to reach the observation point at the present time t:

```text
t' = t − R/v
```

where R = distance from source to point and v = speed of propagation (c in free space). The field at P now depends on what the source was doing at t'.

#### Retarded potentials
For static sources:

```text
V = (1/4πε) ∫ ρ / R dv,     A = (μ/4π) ∫ J / R dv
```

For time-varying sources, ρ and J are taken at the retarded time:

```text
V(r, t) = (1/4πε) ∫ [ρ(r', t − R/v)] / R  dv'
A(r, t) = (μ/4π)  ∫ [J(r', t − R/v)] / R  dv'
```

These are the **retarded scalar potential** and **retarded vector potential**. They satisfy the wave equations:

```text
∇²V − με ∂²V/∂t² = −ρ/ε
∇²A − με ∂²A/∂t² = −μJ
```

with the Lorenz gauge ∇·A = −με ∂V/∂t.

**Phasor (time-harmonic) form**, with e^(jωt) and k = ω/v = 2π/λ:

```text
V = (1/4πε) ∫ ρ e^(−jkR) / R  dv'
A = (μ/4π)  ∫ J e^(−jkR) / R  dv'
```

The factor e^(−jkR) is the phase delay of retardation.

For a line current I (wire antenna): A = (μ/4π) ∫ I e^(−jkR)/R dl.

Fields are then found from

```text
B = ∇ × A
E = −∇V − ∂A/∂t   (= −∇V − jωA)
```

**Importance:** retarded potentials explain why time-varying currents produce fields that travel outward with finite speed and a 1/r part – the radiation field. All antenna fields (dipole, loop, arrays) are derived from them.

## Check yourself

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

1. What is the retarded time?
   A. t' = t + R/v
   B. t' = R/v
   C. t' = t - R/v
   D. t' = t·v/R
   Answer: C. It is the earlier time at which the signal must leave the source to reach the point at time t. (Key point: Retarded time)

2. In phasor form, how does retardation appear in the potentials?
   A. As a factor e^(-jkR), the phase delay
   B. As a factor 1/R^2
   C. As a factor jωR
   D. It disappears
   Answer: A. A = (μ/4π) ∫ J e^(-jkR)/R dv', where e^(-jkR) is the phase delay, k = 2π/λ. (Key point: Phasor form)

3. Which wave equation does the retarded vector potential satisfy?
   A. ∇²A + με ∂²A/∂t² = 0
   B. ∇²A = -μJ only
   C. ∇²A - με ∂²A/∂t² = -μJ
   D. ∇×A = 0
   Answer: C. With the Lorenz gauge, A satisfies ∇²A - με ∂²A/∂t² = -μJ. (Key point: Wave equation for A)

4. How are B and E obtained from the potentials?
   A. B = ∇V, E = -∇×A
   B. B = ∇×A, E = -∇V - ∂A/∂t
   C. B = A, E = V
   D. B = ∂A/∂t, E = ∇×V
   Answer: B. B = ∇×A and E = -∇V - ∂A/∂t (= -∇V - jωA). (Key point: Finding the fields)

5. What do retarded potentials explain?
   A. Only static fields
   B. Why antennas must be grounded
   C. Why DC radiates
   D. Why time-varying currents give fields travelling with finite speed and a 1/r radiation part
   Answer: D. They give finite propagation speed and the 1/r radiation field used to derive all antenna fields. (Key point: Importance)

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