At t=0 the voltage across the Inductor will immediately jump to battery voltage. Why?
Switch closes. Electric field starts to propagate in the wire, much faster than electrons.
Maxwell equation number 4 says:


  The magnetic field induced around a closed loop is proportional to the electric current plus displacement current (rate of change of electric field) it encloses.



which is defined by the fourth formula here:



Check the differential form. At the very moment of closing the switch, the current is 0, therefore the current density (J) is 0. However, as the electric field arrives to the inductor, a sudden change in electric field induces a very high  magnitude rotational magnetic field.


Due to Lenz' law, another electric field will be induced to resist this sudden magnetic field, in the opposite direction. Thus, a counter electric potential is formed, without any currents involved. You can search the phenomena "Displacement field" to see how this magic occurs.