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:
But (𝑣/Δ𝑡)sinθ is just the transverse component of the charge's average acceleration 𝑎⊥, so we have:
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. But this just means that the radial component is still given by the standard form for a static field, while the formula above gives the transverse component.
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 whose (retarded) time coordinate 𝑡′ = 𝑡 − 𝑟/𝑐 is approximately valid over the entire source, and a common transverse direction. This gives:
where 𝑷 = ∑𝑄𝑖 𝒔𝑖 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 𝑡′ 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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