Classical Electromagnetics
Classical Electrodynamics - Retarded position of moving point charge
Retarded position of moving point charge (2D animation)
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At time a spherical light wave is emitted from field point [position vector ] to the past, and the point charge is moving backwards in time from its present position [position vector ] ("video backwards"). The point where this wave "arrests" the charge is the retarded point. The retarded time and position are
Note that both these retarded quantities are implicit functions of and for given function for the motion of .
Now, with "video forward" : when the charge is on the retarded point at the retarded time , the electromagnetic disturbance (field) caused by it is travelling with speed and arrives on field point [position vector ] at the present time the charge being on the present position .
For charge living deep in the past (not created close to the present time) the retarded position exists and it's unique for given , and function . The proof is based on the assumption that the speed of the charge is at any time less than the speed of light :