What is the difference between Instantaneous current (i) and current (I) in a purely inductive circuit?
Is the current I the rms of the instantaneous current?



Instantaneous current (i) the value of current at a certain instant in time. 
RMS current (I) is the DC-equivalent current that would cause the same power dissipation in a resistor.
So if you have a sine wave of peak amplitude 10 volts (peak to peak = 20 volts), the RMS (I) is 7.071 volts. Confused still?
The power dissipation of a resistor due to the 20 volt p-p sine wave is the same as if 7.071 DC volts is applied.