It ask's to calculate the time take for the circuit to reach maximum current when the switch is in position 2. I have figured out the maximum current to be 2A but can't seem to get the time.
an inductor reaches its maximum current at its 5 time constant: T=L/R=500mH/10ohm=50ms 5T=5*50ms=25ms
While five time constants (5T) is a common rule of thumb for assuming that, from a practical standpoint, a circuit has reached steady state, there is nothing magic about it. In a first order circuit such as this, it will have reached 99.3% of its final value. But at 6T it will have reached 99.8%. Depending on the point behind the question, using the 5T rule might be enough, or the point might be to see if you recognize that, in theory, the circuit never reaches the full 2A but, mathematically, only approaches it asymptotically as time grows toward infinity.
True. One very conservative way (which is not normally used) would be to say the current has reached a steady state value when it is within the thermal noise current value for the resistance in series with the inductance (for a capacitor it would be the thermal noise voltage of the capacitor). Obviously any settling time beyond that has no practical physical meaning. But normally you just select some percentage of the final value and use that to determine the number of time-constants it takes to settle. 5T is commonly used since that value is within 1% of the final value.