Last year at finishing.com plating guru Ted Mooney offered the
answer to your exact question.
I think there is little possibility of measuring the thickness of
rhodium anti-tarnish plating on silver costume jewelry “manually” –
which I interpret to mean with a micrometer or like instrument. And
viewing it by metallographic cross-section sounds quite unlikely
unless you run a world-class lab.
It is probably possible to track thickness with an x-ray
fluorescence machine, but these are expensive, so suitable for
manufacturers but not for custom jewelry designers. The most
practical technique is probably as described by TK Mohan; for
Metals are electrodeposited in proportion to the applied
ampere-seconds, as discovered by Faraday with his Law of
Electrolysis. 96,485 ampere-seconds (coulombs) deposits exactly one
gram equivalent weight of metal if the bath is operating at 100%
efficiency. From the valence state and atomic weight, you can
determine exactly how much weight you have deposited; and from the
density of the metal, what volume you have deposited. Dividing by
the surface area of the work, you can determine the thickness. All of
this conversion effort can be saved by going to the “Electrochemical
Equivalents” appendix in the Metal Finishing Guidebook , which
incorporates all those conversion factors for you and offers the
answer that it takes 22.9 ampere-hours per square foot to deposit
0.001" of rhodium and 6.2 ampere-hours to deposit 0.001" of silver.
But most plating baths do not operate at 100% efficiency. In this
case Faraday’s Law of Electrolysis still holds but a portion of the
ampere-seconds are going towards liberating hydrogen from the water
of the plating solution instead of depositing silver or rhodium. TK
suggests that you can determine the efficiency that you are operating
at, or simply incorporate the efficiency into your calculations, by
measuring how much rhodium you deposit once (or periodically), and
then know how much rhodium you are depositing on your jewelry by
simply applying a scaling factor to your ampere-seconds measurement.