Analyzing a rail design circuit that must have constant current?
It seems to me that if you use a CCS as your power supply you will have a diagram as follows:







simulate this circuit &ndash; Schematic created using CircuitLab

The voltage across the rod will be constant but the voltage across the terminals of the CCS will be slightly different depending on the position of the rod. 


  as the rod begins to move and R &amp; L increase, will a constant
  current source(CCS) maintain Ic throughout the process?


As long as your CCS is good at regulating current, then yes. The key is to know how the CCS is generating its current. Just like CVS, they must have a regulation loop to maintain constant current across a range of voltages. 


  how will the increased inductance affect the CCS/or the circuit( self
  inductance)?


In steady-state, the (potentially small) change in inductance will have no effect on the voltage or current through the rod. However, if the rod were to move suddenly and quickly (high dI/dt), then the voltage across the CCS will spike to accommodate the changing inductance. This effect will become more pronounced as the loop inductance increases.