Increase in surface charge density in different frames of reference

[lys04]

I am reading Purcell's Electricity and Magnetism and am getting confused on equation 5.22. It seems to me they are using relativistic velocity addition for u' which is u'=(u-v)/(1-uv/c^2), but aren't we solving for the velocity of the electrons in the test charge's frame of reference, so should be using the reverse transformation instead? i.e u=(u'+v)/(1+u'v/c^2)?
The only way this equation has the correct sign is if v=-v though, I guess this makes sense because if in the lab's frame the test charge is moving to the right with velocity v then in the test charge's frame of reference the lab is moving to the left with same speed?

 

[Sagittarius A-Star]

I am reading Purcell's Electricity and Magnetism and am getting confused on equation 5.22. It seems to me they are using relativistic velocity addition for u' which is u'=(u-v)/(1-uv/c^2), but aren't we solving for the velocity of the electrons in the test charge's frame of reference, so should be using the reverse transformation instead? i.e u=(u'+v)/(1+u'v/c^2)?

No, they want to calculate the primed velocity of the electrons.

In the book is misleading, that they call (before Eq. 5.22) the Eq. G.7 "formula for addition of velocities". That name fits better to the inverse of it, Eq. G.8.