You have this all wrong. The ideal inductor has zero resistance, so in series with a current-limiting resistor, R, after five time-constants equal to L/R seconds, the current in the inductor will have increased exponentially and asymptotically to approximately I = V/R, where V is the supply voltage and R is the value of the current-limiting resistor. At this point there is zero voltage across the inductor. A pretty good
introductory explanation with just a little math can be found here.
What happens
when the switch opens depends on how
fast the switch opens. If it could open instantaneously, and stay open, the current in the inductor, I = V/R, would
instantly go to zero. The voltage across the inductor would
immediately rise to Infinity because the time rate-of-change of current from some finite value to zero value occurred in zero time. In reality
this cannot happen, but the voltage can rise to a high enough value to either (1) cause an arc across the switch contacts or (2) destroy a semiconductor switch, such as a bipolar junction transistor or a MOSFET.
In reality, stray capacitance in the circuit
always limits how fast the voltage across the inductor can rise when the switch opens. In high-power pulsed-power applications every effort is made to minimize the stray capacitance to deliberately allow thousands of volts to appear across an inductor when it's current is interrupted by a fast-opening switch. For the rest of us, some protective circuits are necessary. For mechanical switches, you can place a capacitor, in series with a small-valued resistor, across either the switch contacts or the inductor. Either way, the capacitor slows down the rate at which the inductor current decreases when the switch opens, and the resistor dissipates the electromagnetic energy stored in the coil. You can also use a diode, with or without an energy-dissipating resistor, placed in parallel with the inductor to do the same thing. The diode is wired so it does not conduct when power is applied to the inductor. It only conducts (briefly) when power is removed from the inductor.