The effects of back emf on circuit current:

In proving the time constant, it was stated that, at the instant the switch is closed, the current (I) is zero.  This is because, at that instant in time the current in the coil and the flux surrounding the coil are both changing at their maximum rate.  This rate of change of flux produces maximum back emf, the value being equal and opposite to the applied voltage.  Therefore, with no potential difference across the circuit, no current can flow.  This fact is quite simple to prove using the equation for the self induced emf in a coil and elements of the 'proof of time constant' above:

          E  =  -L x dl/dt

The  current starts from zero, and would rise to its maximum value in 1 time constant if the initial rate of change could be maintained.

The rate of change of current is therefore given by the gradient of line AB.  The gradient of AB = BC/AC = I_max-0/Time constant.  

I_MAX can be calculated using V_supply/R, because when the current reaches its maximum value it is no longer producing a changing flux and therefore not producing a back-emf.  At this time, the whole supply voltage is dropped across the resistor.