Derivation of the radiative electromagnetic field

Here we provide a geometric derivation of the radiative 1/r piece of the electromagnetic field.

First, we assume that the charge velocity is always much less than c, so that we can ignore relativistic effects such as "squeezing" of the field lines. This means that the inner moving-field part of a field line will be parallel to its outer static-field part. The diagram below uses v≈c/2 to make the geometry more obvious, but relativistic squeezing has been ignored, and our derivation will take the limit as v becomes much less than c.

Second, we will consider the radiative field at an angle θ from the axis of the charge's motion, and at a distance ct much greater than cΔt, where t is the time that the shell reaches that distance and Δt is the duration that the charge was accelerating. This means that the shell is very thin compared to its radius, the field lines in the shell are very close to transverse, and the distance from the charge's initial to its current position is very close to vt.

Now consider the green strip indicated on the diagram. All of the field lines within the shaded blue region will pass orthogonally through that strip. The angular span α of field lines that will intersect this strip can be computed from the triangle with sides ct and vt:

Taking v much less than c gives α=(v/c)sinθ.

Now at a radius r=ct, these field lines would normally be spread over a linear distance rα, but instead they have been concentrated into a strip of width cΔt. This represents an enhancement of the radiative transverse field over the static field by a factor rα/cΔt, giving:

(More formally, we can rotate the whole diagram about the axis of motion, show that the solid angle subtended by the blue region is 2π(v/c)sin²θ, and the area of the circular green strip is 2πrcΔtsinθ, and obtain the same result.)

But (v/Δt)sinθ is just the transverse component of the charge's average acceleration, a⊥, so we have:

Now of course the field lines in the shell are not purely transverse: they move radially outward a distance cΔt while sweeping transversely a distance rα. So the radial component is a factor cΔt/rα less than the transverse component. But this means that the radial component is just given by the standard formula for a static field, while the formula above gives the transverse component of the overall field.

It turns out that this formula is correct even if the charge is doing something other than a simple impulsive acceleration, as long as its velocity is always much less than c. Specifically, the transverse field at a distance r depends on the instantaneous acceleration of the charge at a time r/c in the past (called the retarded time):

If the source of the field is a collection of charges close together, we can simply add their fields. By "close together" we mean that the shortest timescales of their motion is much greater than the light travel time between them, but much less than the light travel time to the observer. This allows us to define a common origin for the source, a common (retarded) time coordinate t'=t-r/c at that origin that is valid over the entire source, and a common transverse direction. This gives:

where P=ΣQisi is called the dipole moment of the collection of charges Qi with positions si relative to the common origin, P⊥ is the transverse projection of that dipole, and overdots represent time derivatives in the local time coordinate of the source. Letting n be the unit vector in the radial direction, we can write:

where the first formula is suitable for a continuous charge distribution ρ(s), and the second for a collection of discrete charges Qi.

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Analogy: Electromagnetic fields
Electromagnetic field of an accelerated charge
  Derivation of the radiative electromagnetic field
Electromagnetic waves
Gravitational tidal field
  Equivalence between dipole and tidal field
Gravitaional waves
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Differences between gravitational and electromagnetic radiation
Gravitational wave spectrum