Retarded Potentials and Radiation
Time-varying sources produce potentials that depend on the source at an earlier time, and accelerating charges radiate power given by the Larmor formula.
Definition
Retarded potentials are the solutions of the wave equations for the electromagnetic potentials in which the source contribution is evaluated at the retarded time, accounting for the finite speed of propagation; from them follow the radiation fields and the power radiated by accelerating charges.
Scope
This topic covers the solution of Maxwell's equations for time-dependent sources: the retarded scalar and vector potentials, the Liénard-Wiechert potentials and fields of a moving point charge, the separation of near and radiation fields, and the Larmor and relativistic Larmor formulas for radiated power. It establishes the general framework from which specific radiating systems are derived.
Core questions
- How are the potentials of time-varying sources computed with retardation?
- What are the fields of an arbitrarily moving point charge?
- How much power does an accelerating charge radiate?
Key concepts
- retarded time
- retarded potentials
- Liénard-Wiechert potentials
- radiation field
- near field
- Larmor formula
- radiation reaction
Key theories
- Retarded potentials
- The potentials at a point and time are determined by the charge and current at the earlier retarded time, reflecting that electromagnetic influence propagates at finite speed.
- Liénard-Wiechert potentials and Larmor formula
- The fields of a moving point charge follow from the Liénard-Wiechert potentials and split into a velocity (near) field and an acceleration (radiation) field; the Larmor formula gives the total power radiated by an accelerating charge.
Clinical relevance
The radiation framework explains synchrotron and bremsstrahlung radiation used in X-ray and light sources, energy loss in particle accelerators, and the radiated fields that all antennas and emitters produce.
History
Liénard and Wiechert derived the potentials of a moving point charge around 1898-1900, and Larmor obtained the radiated-power formula in 1897. These results gave a complete classical account of radiation from accelerating charges and later informed the theory of radiation reaction.
Key figures
- Alfred-Marie Liénard
- Emil Wiechert
- Joseph Larmor
Related topics
Seminal works
- jackson1998
- landau1975
Frequently asked questions
- Why are the potentials called retarded?
- Because electromagnetic effects travel at finite speed, the potential observed now depends on what the source was doing at an earlier time, retarded by the light-travel time from source to observer.
- What does the Larmor formula tell us?
- It gives the total power radiated by a non-relativistic accelerating charge as proportional to the square of its acceleration, showing that only accelerating charges radiate.