# Superposition of Electric Field(s) Generated by a moving particle?

1. Apr 21, 2012

### genxium

I've just watch MIT open course 8.03, vibrations and waves, lecture 14, and think of a weird problem,

if a particle(with charge) in vacuum, say a proton or an electron, regardless of the size because of far-field consideration, like what's shown in the attachment, is moving along a straight line with a constant velocity $v,$

assume the current time is $t,$ current position of the particle is $P(t),$ and a view point which is very far away from the particle's current position, so that the EM wave emitted at $<t,P(t)>$ has to travel for a non-ignorable time $\triangle t$ to reach my view point, and say the time when this EM wave reaches my view point will be $T.$

Could anyone tell me, if there exist 1 or more points on the straight line, say $P(t'),$ that the EM wave emitted at $<t',P(t')>$ will also reach my view point at $T.$

P.S. From my calculation, a point $P(t')$ satisfies that $|P(t')-P(t)|=\frac{2 \cdot vR \cdot (c-v \cdot cos a)}{c^2-v^2},$ by cosine law, the reason I don't trust this result is obvious:

From my calculation , there's only 1 such point that matches the requirement, but think of it recursively, if 1 such point is found, we can always find a new one based on the previous one with arguments $<t,P(t), R, a>,$ then the series have to be infinite.

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