What is retarded potential in EM theory

In electromagnetic (EM) theory, the term retarded potential refers to a specific formulation used in the context of the propagation of electromagnetic fields, particularly in the study of radiation from moving charges.

Retarded Potentials in Electromagnetism

In electrodynamics, the retarded potential is used to describe the electromagnetic potentials that account for the fact that changes in the electromagnetic field (such as a moving charge) propagate at a finite speed—specifically, at the speed of light.

When a charge moves, it generates electromagnetic fields that propagate outward from the source. However, these changes in the field do not propagate instantaneously. Instead, they are delayed by the time it takes for the information (or the field disturbance) to travel from the source to the observation point.

To account for this, the concept of retarded time is introduced. The retarded time refers to the time at which an event (such as a charge moving) must have occurred in the past for its effect (the electromagnetic field) to be observed at a specific point in space at the present time. In other words, the potential at an observation point at time $t$ depends on the values of the charge and current at an earlier time $t’$, which is related to $t$ by the retarded time equation:

where:

• $t$ is the time of observation,

• $mathbf{r}$ is the location of the observer,

• $mathbf{r}’$ is the location of the source (charge or current),

• $c$ is the speed of light,

• $|mathbf{r} – mathbf{r}’|$ is the distance between the source and the observer.

Retarded Potentials in the Electromagnetic Field

The retarded potentials specifically are the scalar potential $phi$ and vector potential $mathbf{A}$, which can be expressed as:

1. Scalar Potential:

where $rho(mathbf{r}’, t’)$ is the charge density, and the integration is over the entire volume of charge distribution.

2. Vector Potential:

where $mathbf{J}(mathbf{r}’, t’)$ is the current density at the retarded time.

These potentials take into account the time delay it takes for the electromagnetic influence of the charge or current to propagate to the observation point. The fields $mathbf{E}$ and $mathbf{B}$ can be derived from these potentials, and they describe how the electromagnetic field behaves at the observation point, considering the finite speed of light.

Why the Retarded Potential is Important

1. Causality: The use of retarded potentials ensures that the theory obeys the principle of causality. That is, the electromagnetic effects are caused by events that happened in the past and propagate forward in time at the speed of light. The effects are not instantaneous, which would violate causality.

2. Radiation: When dealing with radiation from accelerating charges (such as in the case of antennas or moving particles), the retarded potentials allow us to understand how the radiation propagates from the source to the observer, accounting for the time delays.

3. Radiation Reaction: The retarded potentials also play a key role in understanding the reaction force on a particle due to its own electromagnetic radiation. This is a part of the broader concept of the radiation reaction force, which describes how the emitted radiation affects the motion of the particle.

In summary, the retarded potential in EM theory is a method to describe the potentials (and subsequently the fields) generated by moving charges while accounting for the finite speed of propagation of electromagnetic interactions.