Can someone say why it is said that in electrostatics,the test charge may be moving very slowly?Specifically,what is wrong if the test charge moves at a speed which is a significant proper fraction of speed of light?

In electrostatics,we neglect magnetic effects and time varying effects.Now,a single moving charge,even running at a constant speed,results in a non-steady current,and that in turn,will result in a time-varying magnetic field.This,again,will induce a time varying electric field.
Now,if the speed is very small,we will have a slight fluctuation and if the speed is high,the magnitude of current,and hence,the resulting E and B fields will be big.So,electrostatics will make a mess with big fluctuations in fields and magnetic effects.
There will be relativistic effects as well.

If a charge is moving very quickly then the time derivatives in Maxwell's equations typically become important, but these terms are neglected in the formulation of electrostatics (they couple electric and magnetic fields). Furthermore, accelerating charges actually produce radiation which is a qualitatively different sort of thing from electrostatic and magnetostatic fields.

Does electrostatics imply that charges aren't moving at all?

It does not imply it, that is what 'static' means.

As for thought experiments with a charge moving, I think that can be a valid and useful approach to solving or understanding certain problems or situations.

Griffiths says we need not at all worry about the test charge.electrostatics means the source distribution produces static field and that's all.

neeklash, i think if another charge is moving with uniform speed with respect to a certain body of charge then you have to use special relativity to predict what forces occur in what frames.

even if the velocity of the approaching charge is non-relativistic.

Oh,consider a '?' sign in the last line.

This sounds like Quantum Mechanics, but here if the observer is moving or is at rest with the charge makes the difference.

>> If a charge is moving very quickly then the time derivatives in Maxwell's equations typically become important, but these terms are neglected in the formulation of electrostatics (they couple electric and magnetic fields). >>>

can you tell me about the concepts involved, please ?

>>> In electrostatics,we neglect magnetic effects and time varying effects. >>

Why do you say 'neglect' ?
and what effects are produced if timing is varied ?

Thanks

This one seems to be a final blow to Relativists.

Why ?

Why ?

Because the relativists are always talking about some kinda imaginary frames, now they cant do that with this one.