Question about Retarded Time and Moving Charges

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SUMMARY

The discussion centers on the concept of retarded time in the context of moving charges, defined by the equation t' = t - r/c, where r represents the distance between the observation point and the charge's position. The key question posed is whether r should be expressed as a function of normal time t or retarded time t'. The consensus is that if the observation point is stationary and the charge is moving, r should be a function of t', leading to the equation t' = t - r(t')/c. In scenarios where both the charge and observation point are in motion, the distance must account for both positions, resulting in the equation t' = t - ||r_c(t') - r_o(t)||/c.

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I have a quick question about the retarded time when dealing with moving charges.

The retarded time is:

[itex]t' = t - \frac{r}{c}[/itex]

where [itex]r[/itex] is the distance between the point of observation and the position of the charge.

My question is very simple, is [itex]r[/itex] a function of the normal time [itex]t[/itex], or the retarded time [itex]t'[/itex]?

That is, which equation is correct?

1. [itex]t' = t - \frac{r(t)}{c}[/itex]

2. [itex]t' = t - \frac{r(t')}{c}[/itex]

Thanks.
 
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Depends on which one is moving. If the observation point stays still, and the charge is moving, then you are interested in r(t'), the distance at time the signal was emitted.

More generally, suppose both the charge and observation point are moving, with rc and ro being positions of charge and observation point respectively relative to some fixed origin. In that case, the distance traveled by the wave will be function of both.

[tex]t' = t - \frac{||\vec{r}_c(t')-\vec{r}_o(t)||}{c}[/tex]
 

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