Sketching equipotentials for a moving charge

[CAF123]

Homework Statement

A point charge remains stationary at the origin for a long time but then, at time t = 0, is displaced, at uniform speed c/2, to a new position R, where it stops.

Sketch with care the equipotentials of $\phi$ at a time t = R/c, and then again at time t = 4R/c. Your sketches should including distances both large and small compared to R.

Homework Equations

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retarded time $t_R = t - |r-r'|/c$

The Attempt at a Solution

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What is t? the position of the charge or the retarded position as measured by some observer? Assuming the former, when $t=R/c,$ the charge is at position R/2. So ct=R and so we have circles centered around the point R/2. Outside R, the equipotentials are those from when the charge was at origin. So should draw circles centered at R/2 and with radii between 0 and R?

Similar analysis for t=4R/c, the charge is at 2R and ct=4R so have circles centered at 2R until max radius 4R. Beyond that, have circles centered at origin. Is it right?

 

[mfb]

Such a fast motion will introduce magnetic fields, and make the concept of a potential alone problematic, but I guess we have to ignore this problem here.

t_R determines your fields, t is the current time. That will give different circles.

Outside R, the equipotentials are those from when the charge was at origin.

Right.

Similar analysis for t=4R/c, the charge is at 2R and ct=4R so have circles centered at 2R until max radius 4R. Beyond that, have circles centered at origin. Is it right?

The transition region is more complicated.